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i, {lOts^'SL
\.
HARVARD COLLEGE
SCIENCE CENTER
LIBRARY
^ri. -
o
^
Linear Associative Algebra
f
By BENJAMIN PEIRCB/ LL. D.
Late Pbrkins Professor of Astronomy and Mathematics in Harvard University
AND Superintendent of the UNitED States Coast Survey.
New Edition, with Addenda and Notes, by C. S. Peirce, Son of the Author.
[Extracted fnm The American Jdumal of MathemaHcs^l
NEW YORK : D. VAN NOSTRAND, PUBLISHER.
^ 1882. /
* ]
. f
\
/
J»A — l^ * '
PrB8 op Isaac Frikdbnwald,
Baltimore, Md.
ERRATA.
Page 10, § 31. The first formula should read
{A±:B)G=AG±BG.
Page 30. The third formula should read
k{i — h)=j.
Page 36. Foot-note, second line of second paragraph, read
Page 40. Last line of foot-note. For e , read I .
Page 52. Multiplication table of (/g) . For^i = i, read ji =j.
Page 75. Last line of foot-note, insert Z, at beginning of line.
Page 86. Foot-note. Add that on substituting k + vj for k , the algebra
{aw^) reduces to (00:5) ; and the same substitution reduces (ay^) to (az^) .
Page 91. Last line of foot-note. For i, read /.
PREFACE.
Lithographed copies of this book were distributed by Professor Peirce among his
friends in 1870. The present issue consists of separate copies extracted from Ths Ameri-
can Journal of Mathematics^ where the work has at length been published.*
The body of the text has been printed directly from the lithograph with only slight
verbal changes. Appended to it will be found a reprint of a paper by Professor Peirce,
dated 1875, and two brief contributions by the editor. The foot-notes contain transforma-
tions of several of the algebras, as well as what appeared necessary in order to complete
the analysis in the text at a few points. A relative form is also given for each algebra ; for
the rule in Addendv/m II. by which such foiTQS may be immediately written down, was
unknown until the printing was approaching completion.
The original edition was prefaced by this dedication :
To My Fbiends.
This work has been the pleasantest mathematical effort of my life. In no other have
I seemed to myself to have received so full a reward for my mental labor in the novelty
and breadth of the results. I presume that to the uninitiated the formulae will appear cold
and cheerless ; but let it be remembered that, like other mathematical formulae, they find
their origin, in the divine source of all geometry. Whether 1 shall have the satisfaction of
taking part in their exposition, or whether that will remain for some more profound
expositor, will be seen in the future.
B. P.
* To page n of this issue corresponds page n+9^ of Vol. IV. of 2^ Jimmai,
r\
LINEAR ASSOCIATIVE ALGEBRA.
1. Mathematics is the science which draws necessary conclusions.
This definition of mathematics is wider than that which is ordinarily given,
and by which its range is limited to quantitative research. The ordinary
definition, like those of other sciences, is objective ; whereas this is subjective.
Recent investigations, of which quaternions is the most noteworthy instance,
make it manifest that the old definition is too restricted. The sphere of mathe-
matics is here extended, in accordance with the derivation of its name, to all
demonstrative research, so as to include all knowledge strictly capable of dog-
matic teaching. Mathematics is not the discoverer of laws, for it is not
induction ; neither is it the framer of theories, for it is not hypothesis ; but it is
the judge over both, and it is the arbiter to which each must refer its claims ;
and neither law can rule nor theory explain without the sanction of mathematics.
It deduces from a law all its consequences, and develops them into the suitable
form for comparison with observation, and thereby measures the strength of the
argument from observation in favor of a proposed law or of a proposed form of
application of a law.
Mathematics, under this definition, belongs to every enquiry, moral as well
as physical. Even the rules of logic, by which it is rigidly boxmd, could not be
deduced without its aid. The laws of argument admit of simple statement, but
they must be curiously transposed before they can be applied to the living speech
and verified by observation. In its pure and simple form the syllogism cannot
be directly compared with all experience, or it would not have required an
2 Pbircb : Lmear Associative Algebra.
Aristotle to discover it. It must be transmuted into all the possible shapes in
which reasoning loves to clothe itself. The transmutation is the mathematical
process in the establishment of the law. Of some sciences, it is so large a
portion that they have been quite abandoned to the mathematician, — which
may not have been altogether to the advantage of philosophy. Such is the
case with geometry and analytic mechanics. But in many other sciences, as in
all those of mental philosophy and most of the branches of natural history,, the
deductions are so immediate and of such simple construction, that it is of no
practical use to separate the mathematical portion and subject it to isolated
discussion.
2. The branches of mathematics are as various as the sciences to which they
belong, and each subject of physical enquiry has its appropriate mathematics.
In every form of material manifestation, there is a corresponding form of human
thought, so that the human mind is as wide in its range of thought as the
physical universe in which it thinks. The two are wonderfully matched. But
where there is a great diversity of physical appearance, there is often a close
resemblance in the processes of deduction. It is important, therefore, to separate
the intellectual work from the external form. Symbols must be adopted which
may serve for the embodiment of forms of argument, without being trammeled
by the conditions of external representation or special interpretation. The
. words of common language are usually unfit for this purpose, so that other
• symbols must be adopted, and mathematics treated by such symbols is called
algebra. Algebra, then, is formal mathe^natics.
3. All relations are either qualitative or quantitative. Qualitative relations
can be considered by themselves without regard to quantity. The algebra of
such enquiries may be called logical algebra, of which a fine example is given
by Boole.
Quantitative relations may also be considered by themselves without regard
to quality. They belong to arithmetic, and the corresponding algebra is the
common or arithmetical algebra.
In all other algebras both relations must be combined, and the algebra must
conform to the character of the relations.
4. The symbols of an algebra, with the laws of combination, constitute its
language ; the methods of using the symbols in the drawing of inferences is its
art ; and their interpretation is its scientijic application. This three-fold analysis
of algebra is adopted from President Hill, of Harvard University, and* is made
the basis of a division into books.
Peiece: Linear AssocicUive Algebra, 5
Book I.*
The Language of Algebra.
5. The language of algebra has its alphabet, vocabulary, and grammar,
6. The symbols of algebra are of two kinds : one class represent its
fundamental conceptions and may be called its letters, and the other represent
the relations or modes of combination of the letters and are called the signs.
7. The alphabet of an algebra consists of its letters ; the vocabulary defines
its signs and the elementary combinations of its letters ; and the grammar gives
the rules of composition by which the letters and signs are united into a
complete and consistent system.
The Alphabet.
8. Algebras may be distinguished from each other by the number of their
independent fundamental conceptions, or of the letters of their alphabet. Thus
an algebra which has only one letter in its alphabet is a single algebra ; one
which has two letters is a dovhle algebra ; one^ of three letters a triple algebra ;
one of four letters a quadruple algebra, and so on.
This artificial division of the algebras is cold and uninstructive like the
artificial Linnean system of botany. But it is useful in a preliminary investiga-
tion of algebras, until a sufficient variety is obtained to afford the material for a
natural classification.
Each fundamental conception may be called a unit; and thus each imit has
its corresponding letter, and the two words, unit and letter, may often be used
indiscriminately in place of each other, when it cannot cause confusion.
9. The present investigation, not usually extending beyond the sextuple
algebra, limits the demand of the algebra for the most part to six letters ; and
the six letters, i, /, h, Z, m and ti, will be restricted to this use except in
special cases.
10. For any given letter anothe)* may be substituted, provided a new letter
represents a combination of the original letters of which the replaced letter is a
necessary component.
For example, any combination of two letters, which is entirely dependent
for its value upon both of its components, such as their sum, diflFerence, or
product, may be substituted for either of them.
*Oiily this book was ever written. [C. S. P.]
4 Peirce : Lmear Associative Algebra.
This principle of the svhstitution of letters is radically important, ftnd is a
leading element of originality in the present investigation ; and without it, such
an investigation would have been impossible. It enables the geometer to
analyse an algebra, reduce it to its simplest and characteristic forms, and
compare it with other algebras. It involves in its principle a corresponding
substitution of units of which it is in reality the formal representative.
There is, however, no danger in working with the symbols, irrespective of
the ideas attached to them, and the consideration of the change of the original
conceptions may be safely reserved for the hooTc of interpretation,
11. In making the substitution of letters, the original letter will be preserved
with the distinction of a subscript number.
Thus, for the letter i there may successively be substituted ii, *2, is, etc. In.
the final forms, the subscript numbers can be omitted, and they may be omitted
at any period of the investigation, when it will not produce confusion.
It will be practically found that these subscript numbers need scarcely ever
be written. They pass through the mind, as a sure ideal protection from erro-
neous substitution, but disappear from the writing with the same facility with
which those evanescent chemical compounds, which are essential to the theory
of transformation, escape the eye of the observer.
12. A ^^re^lgebra is one in which every letter is connected by some
indissoluble relation with every other letter.
13. When the letters of an algebra can be separated into two groups, which
are mutually independent, it is a mixed algebra. It is mixed even when there
are letters common to the two groups, provided those which are not common to
the two groups are mutually independent. Were an algebra employed for the
simultaneous discussion of distinct classes of phenomena, such as those of soimd
and light, and were the peculiar units of each class to have their appropriate
letters, but were there no recognized dependence of the phenomena upon each
other, so that the phenomena of each class might have been submitted to
independent research, the one algebra would be actually a mixture of two
algebras, one appropriate to sound, the other to light.
It may be farther observed that when, in such a case as this, the component
algebras are identical in form, they are reduced to the case of one algebra with
two diverse interpretations.
Peircb : Lmear Associative Algebra. 5
The Vocahulary.
14. Letters which are not appropriated to the alphabet of the algebra *
may be used in any convenient sense. But it is well to employ the small letters
for expressions of common algebra, and the capital letters for those of the algebra
under discussion.
There must, however, be exceptions to this notation ; thus the letter D will
denote the derivative of an expression to which it is applied, and 2 the summa-
tion of cognate expressions, and other exceptions will be mentioned as they
occur. Greek letters will generally be reserved for angular and functional
notation.
15. The three symbols J, 9, and 6 will be adopted with the signification
J=^/— 1
tl . 9 = the ratio of circumference to diameter of circle = 3.1415926536
. ; 6 = the base of Naperian logarithms = 2.7182818285, I
which gives the mysterious formula
-^^l^-rr J-»=V 6^=4.810477381.
^^-xC^ 16. All the signs of common algebra will be adopted; but any signification
(- '^ . will be permitted them which is not inconsistent with their use in common
/ ' ^ I algebra ; so that, if by any process an expression to which they refer is reduced
to one of common algebra, they must resume their ordinary signification. f
17. The sign =, which is called that of equality, is used in its ordinary sense
to denote that the two expressions which it separates are the same whole,
although they represent different combinations of parts.
18. The signs > and <] which are those of inequality, and denote **more
than " or '* less than " in quantity, will be used to denote the relations of a whole
to its part, so that the symbol which denotes the part shall be at the vertex of
the angle, and that which denotes the whole at its opening.- This involves the -
proposition that the smaller of the quantities is included in the class expressed
by the larger. * Thus
B<A or A^B
denotes that ji is a whole of which J8 is a part, so that all B is A.f
/^
*See39.
tThe formula in the text implies, also, that some A is not B, [C. S. P.]
TSV^ r. -.-i ; "^ -T» I n\d '^ ^wi ■ ■ r -~ ^ 'r
^
6 Peirce : Linear Associative Algebra.
If the usual algebra had originated in qualitative, instead of quantitative,
investigations, the use of the symbols might easily have been reversed ; for it
seems that all conceptions involved in A must also be involved in B, so that B
is more than A in the sense that it involves more ideas.
The combined expression
b:^g<a
denotes that there are quantities expressed by C which belong to th^ class A
and also to the class B. It implies, therefore, that some Bi8 A and that some A is
B* The intermediate G might be omitted if this wjere the only proposition
intended to be expressed, and we might write
BXA,
In like manner the combined expression
B< 6^>^
denotes that there is a class which includes both A and 5,f which proposition
might be written
B<>A.
19. A vertical mark drawn through either of the preceding signs reverses its
signification. Thus
A:^B
denotes that B and A are essentially different wholes ;
A:Jp>B or B<i(i A
denotes that all B is not ^ , J so that if they have only quantitative relations,
they must bear to each other the relation of
A= B or A<B.
20. The sign + is called plits in common algebra and denotes addition. It
may be retained with the same name, and the process which it indicates may be
called addition. In the simplest cases it expresses a mere mixture, in which
* This, of course, supposes that C does not vanish. [C. S. P.]
t The universe wiU be such a class unless ^ or B is the universe. [C. S. P.]
t The general interpretation is rather that either A and B are identical or that some B is not A^
[C. S. P.]
Pbircb: Linear Associative Algebra. 7
the elements preserve their mutual independence. If the elements cannot be
mixed without mutual action and a consequent change of constitution, the mere
union is still expressed by the sign of addition, although some other symbol is
required to express the character of the mixture as a peculiar compound having
properties different from its elements. It is obvious from the simplicity of the
union recognized in this sign, that the order of the admixture of the elements
cannot affect it ; so that it may be assumed that
A + B=iB + A
and
{A + B) + G=A + {B + G) = A + B + G.
21. The sign — is called minus in common algebra, and denotes svhtraciion.
Retaining the same name, the process is to be regarded as the reverse of
addition ; so that if an expression is first added and then subtracted, or the
reverse, it disappears from the result ; or, in algebraic phrase, it is canceled. This
gives the equations
A + B—B=:A — B + B = A
and
B — B=0.
' The sign minus is called the negative sign in ordinary algebra, and any term
preceded by it may be united with it, and the combination may be called a
negative term. This use will be adopted into all the algebras, with the provision
that the derivation of the word negative must not transmit its interpretation.
22. The sign x may be adopted from ordinary algebra with the name of
the sign of multiplication, but without reference to the meaning of the process.
The result of multiplication is to be called the product. The terms which are
combined by the sign of multiplication may be cslled factors ; the factor which
precedes the sign being distinguished as the multiplier, and that which follows it
being the multiplicand. The words multiplier, multiplicand, and product, may
also be conveniently replaced by the terms adopted by Hamilton, of faxdent,
fadendj shd factum. Thus the equation of the product is
multiplier X multiplicand = product ; or facient X faciend = factum.
When letters are used, the sign of multiplication can be omitted as in ordinary
algebra.
fAa-fA i7^S.^2.
]
HARVARD COLLEGE
SCIENCE CENTER
LIBRARY
12 Peirce: Linear Associative Algebra.
is cyclic because the letters are interchangeable in the order t, j, i. But neither
of these algebras is commutative.
37. When an algebra can be reduced to a form in which all the letters are
expressed as powers of some one of them, it may be called a potential algebra.
If the powers are all squares, it may be called quadratic ; if they are cubes, it
may be called cubic ; and similarly in other cases.
lAn/ear Associative Algebra.
38. All the expressions of an algebra are distribiUive^ whenever the distributive
principle eoctends to all the letters of the alphabet.
For it is obvious that in the equation
{i+j){h'\'l) = ih+jk^il+jl
each letter can be multiplied by an integer, which gives the form
{ai + bj){c1c + <^ = acik + bcjh + adil + bdjl ,
in which a^ b^ c and d are integers. The integers can have the ratios of any
four real numbers, so that by simple division they can be reduced to such real
numbers. Other similar equations can also be formed by writing for a and b , a^
and bi , or for c and d , Ci and dj , or by making both these substitutions simulta-
neously. If then the two first of these new equations are multiplied by J and
the last by — 1 ; the sum of the four equations will be the same as that which
would be obtained by substituting for a , 6 , c and d , a + Joj , b + Jby , c + J^^i
and d + ^^d^. Hence a^b^ c and d may be any numbers, real or imaginary, and in
general whatever mixtures A, By G and D may represent of the original
units imder the form of an algebraic simi of the letters i, /, k, Ac, we shall
have
{A + B){G + D)=AG + BG + AD + BD,
which is the complete expression of the distributive principle.
39. An algebra is associative whenever the associative principle extends to all the
letters of its alphabet.
For if 4 = 2 {ai) = ai + a-J + ajc + &c.
5= 2(6i) = 6i + 6J+ Ja* + &c.
G = X (ci) = ci + Cij* + Cjfc -f &c.
J
Peirce: Linear Associative Algebra. 13
it is obvious that AB = 2 {ah^ij)
BG = ^{bciij)
{AB)G= ^{ah^c^ijk) = A{BG) = ABG
which is the general expression of the associative principle.
/ 40. In every linear associative algebra^ there is at least on/e idempptmt or one
nilpotent expression. //r'/ Am^./ Al..*./ \'.^. /^ •//,
Take any combination of letters at will and denote it by A. Its square
is generally independent of A, and its cube may also be independent of A
and A^. But the number of powers of A that are independent of A and of
each other, cannot exceed the number of letters of the alphabet ; so that there
must be some least power of A which is dependent upon the inferior powers.
The mutual dependence of the powers of A may be expressed in the form of an
equation of which the first member is an algebraic sum, such as
^^{a^A^) = .
All the terms of this equation that involve the square and higher powers of A
may be combined and expressed as BA, so that B is itself an algebraic sum of
powers of A , and the equation may be written
BA + a^A = {B + a^)A = .
It is easy to deduce from this equation successively
{B + a^)A^=
{B + a^)B =
f— :?V — — :?
so that is an idempotent expression. But if ai vanishes, this expression
becomes infinite, and instead of it we have the equation
5»=0
so that 5 is a nilpotent expression.
41. When there is an idempotent expression in a linear associative algebra, it
can be assumed as one of the independent units, and be represented by one of
the letters of the alpTiabet ; and it may be called the basis.
The remaining v/nits can be so selected as to be separable into fowr distinct groups. (.
With reference to the basis, the units of the first group are idemfactors ; those of
the second group are idemfajdend and nilfadent ; those of the third group are idem-
fadent and nilfaciend; and those of the fourth group are nil factors.
14 Pbirce: Linear Associative Algebra.
First The possibility of the selection of all the remaining units as idem-
faciend or nilfaciend is easily established. For if i is the idempotent base, its
definition gives
I* = i .
The product by the basis of another expression such as A may be represented
by JB, so that
iA = B,
which gives
iB = i^A = iA = B
i{A — B)=:iA — iB = B — B=0,
whence it appears that B is idemfac iend and A — B is nilfaciend. In other
words, A is divided into two parts, of which one is idemfaciend and the other is
nilfaciend ; but either of these parts may be wanting, so as to leave A wholly
idemfaciend or wholly nilfaciend.
Secondly. The still farther subdivision of these portions into idemfacient and
nilfacient is easily shown to be possible by this same method, with the mere
reversal of the relative position of the factors. Hence are obtained the required
four groups.
The basis itself may be regarded as belonging to the first group.
42. Any algebraic sum of the letters of a group is an expression which
belongs to the same group, and may be called factorially fiomogeneous.
43. The j)rodnct of two factorially homogeneous expressions, which does not
vanish, is itself factorially homogeneous, and its fadend name is the same as thai
of its facierd, while its facierd name is the sam^ as that of its facfiend.
Thus, if A and B are, each of them, factorially homogeneous, they satisfy
the equations
i{AB) = {iA)B ,
{AB)i = A{Bi),
which shows that the nature of the product as a faciend is the same as that of
the facient A , and its nature as a facient is the same as that of the faciend B.
44. Hence, no product which does not vanish can he commutative unless both its
faotors btlong to the same group.
* t • ■
* «
J
Peibge : Linear Associative Algebra.
15
45. Every product vanishes^ of which thefacient is idemfacient while the/adend
is nU/adend ; or of which the facient is nilfojcient while the fadervd is idem/aciend.
For in either case this product involves the equation
AB = {Ai)B = A{iB) = .
46. The combination of the propositions of §§ 43 and 45 is expressed in the
following form of a multiplication table. In this table, each factor is expressed
by two letters, of which the first denotes its name as a faciend and the second as
a facient. The two letters are d and n, of which d stands for idem and n fov nil
The facient is written in the left hand column of the table and the faciend in the
upper line. The character of the product, when it does not vanish, is denoted
by the combination of letters, or when it must vanish, by the zero, which is
written upon the same line with the facient and in a column under the faciend,
dd dn nd
nn
dd
dn
nd
dd
dn
dd
dn
nd
nn
nd
nn
\
rm
47. It is apparent from the inspection of this table, that eoery expression
which belongs to the second or third group is nilpotent.
48. It is apparent that aU commutative products which do not vanish are
restricted to the first and fourth groups.
49. It is apparent that every continuous product which does not vanish, has
the same faciend name as its first facient, and the same facient name as its last
faciend.
50. Since the products of the units of a group remain in the group, they
cannot serve as the bond for uniting different groups, which are the necessary
conditions of a pure algebra. Neither can the first and fourth groups be con-
nected by direct multiplication, because the products vanish. The first and f mirth
groups^ therefore^ require for their indissolvhle union into a pu/re algebra that thefte
islkould be units in eojch of the other two grovps.
16 Peircb : Linear Associative Algebra,
61. In an algebra which has more than two independent units, it cannot
happen that all the units except the base belong to the second or to the third group.
For in this case, each unit taken with the base would- constitute a double algebra,
and there could be no bond of connection to prevent their separation into
distinct algebras.
52. The v/nits of the fomih group are svhject to independent discussion^ as if they
constituted an algebra of themselves. There must be in this group an idempotent
or a nilpotent unit. If there is an idempotent unit, it can be adopted as the
bojsis of this group, through which the group can be subdivided into subsidiary
groups.
The idempotent unit of the fourth group can even be made the basis of the
whole algebra, and the first, second and third groups will respectively become
the fourth, third and second groups for the new basis.
53. When the first group comprises any units except the basis, there is besides
the basis another idempotent expression, or there is a nilpotent expression. By a
process similar to that of § 40 and a similar argument, it may be shown that for
any expression A, which belongs to the first group, there is some least power
which can be expressed by means of the basis and the inferior powers in the
form of an algebraic sum. This condition may be expressed by the equation
If then h is determined by the ordinary algebraic equation
and if
Ai'=- A — hi
•
is substituted for A , an equation is obtained between the powers of A , from
which an idempotent expression, B, or else a nilpotent expression, can be
deduced precisely as in § 40.*
54. When there is a second idempotent unit in the first group, the ba^is can be
changed so as to free the first group from this second idempotent unit.
Thus if i is the basis, and if J is the second idempotent unit of the first
group, the basis can be changed to
* The equation in h may have no algebraic solution, in which case the new idempotent or nilpotent
would not be a direct algebraic function of t and A. [C. S. P.]
Pbiecb: Linear Associative Algebra. 17
h = *— y;
and with this new basis, j passes from the first to the fourth group. For
Firsts the new basis is idempotent, since
i\ = {% —j'Y = i* — 2ij +y* = i —j = ii ;
and secondly f the idempotent unit j passes into the fourth group, since
• • /• ava •• aA * * /\
t^j=z{%—j)j=ztj—j*z=ij—j = 0,
• • •/* *\ *■ *4 * * /\
J^l=J{^—J)=J^—r=J—J = 0.
55. With the preceding change of basis, expressions may pass from idem/adent
to nil/acient, or from idemfaciend to nilfaciend, but not the reverse.
For Jirstj if A is nilfacient with reference to the original basis, it is also, by
§ 45, nilfacient with reference to the new basis ; or if it is nilfaciend with
reference to the original basis, it is nilfaciend with reference to the new basis.
Secondly^ all expressions which are idemfacient with reference to the
original basis, can, by the process of § 41, be separated into two portions with
reference to the new basis, of which portions one is idemfacient and the
other is nilfacient ; so that the idemfacient portion remains idemfacient, and the
remainder passes from being idemfacient to being nilfacient. The same process
may be applied to the faciends with similar conclusions.
56. It is evident, then, that each group* can be reduced so as not to contain
more than one idempotent unit, which will be its basis. In the groups which
bear to the basis the relations of second and third groups, there are only
nilpotent expressions.
57. In a group or an algebra which has no idempotent expression, all the
expressions are nilpotent.
Take any expression of this group or algebra and denote it by A. If no
power of A vanished, there must be, as shown in § 40, some equation between
the powers of A of the form
in which ai must vanish, or else there would be an idempotent expression as is
shown in § 40, which is contrary to the present hypothesis. If then m^ denote
* That is, the first group as weU as each of the subsidiary groups of { 52. [C. S. P.]
18 Peiboe: Lmear Associative Algebra.
the exponent of the least power of A that entered into this equation, and m^ + A
the exponent of the highest power that occurred in it, the whole number of
terras of the equation would be, at most, h + 1. If, now, the equation were
multiplied successively by A and by each of its powers as high as that of which
the exponent is (m^ — 1)A, this highest exponent would denote the number of
new equations which would be thus obtained. If, moreover,
then the highest power of A introduced into these equations would be
The whole number of powers of A contained in the equations would be mji + 1 ,
and A + 1 of these would always be integral powers of B ; and there would
remain {m^ — \)h in number which were not integral powers of B. There
would be, therefore, equations enough to eliminate all the powers of A that
were not integral powers of B and still leave an equation between the integral
powers of B ; and this would generally include the first power of B. From
this equation, an idempotent expression could be obtained by the process of § 40,
which is contrary to the hypothesis of the proposition.
Therefore it cannot be the case that there is any equation such as that here
assumed ; and therefore there can be no expression which is not nilpotent. The
few cases of peculiar doubt can readily be solved as they occur; but they
always must involve the possibility of an equation between fewer powers of B
than those in the equation in j4.*
58. When an expression is nilpotent, all its powers which do not vanish are
mutudUy independent.
Let A be the nilpotent expression, of which the n*^ power is the highest
which does not vanish. There cannot be any equation between these powers
of the form
'm**m-
* In saying that the equation in B will generally include the first power of B , he intends to waive
the question of whether this always happens. For, he reasons, if this is not the case then the equation
in B is to be treated just as the equation in A has been treated, and such repetitions of the process must
ultimately produce an equation from which either an idempotent expression could be found, or else A
would be proved nilpotent. [C. S. P.]
J
Peibgb: Linear Associative Algebra. 19
For if ttIq were the exponent of the lowest power of A in this equation, the
multiplication of the equation by the (n — m^y^ power of A reduces it to
a^ A" = , a^ = ,
that is, the itiq^ power of A disappears from the equation, or there is no least
power of A in the equation, or, more definitely, there is no such equation.
59. In a group or an algebra which contains no idemjpotent expression^ any
expression may be selected as the basis; but one is preferable which has the
greatest nv/mber of powers which do not vanish. All the powers of the basis which
do not vanish may be adopted as independent units and represented by the
letters of the alphabet.
A nilpoterU group or algebra may be said to be of the same order with the number
of powers of its basis that do not vanish, provided the basis is selected by the
preceding principle. Thus, if the squares of all its expressions vanish, it is of
the first order ; if the cubes all vanish and not all the squares, it is of the second
order, and so on.
60. It is obvious that m a nilpotent group whose order equals the number
of letters which it contains, all the letters except the basis may be taken as the
sitccessive powers of the basis.
61. In a nilpotent group, every expression, such as A, has some least
power that is nilfacient with reference to any other expression, such as B , and
which corresponds to what may be called th^ fadent order of B relatively to
A ; and in the same way, there is some least power of A which is nilfaciend with
reference to B , and which corresponds to the faciend order of B relatively to A.
When the facient and faciend orders are treated of irrespective of any especial
reference, they must be referred to the base.
The facient order of a product which does not vanish, is not higher than that of
its facient ; and the faciend order is not higher than that of its faciend.
62. After the selection of the basis of a nilpotent group, some one from
among the expressions which are independent of the basis may be selected by the
same method by which the basis was itself selected, which, together with aU its
powers that are independerd of the basis, may be adopted as new letters ; and again,
from the independent expressions which remain, new letters may be selected by the
same process, and so on until the alphabet is completed. In making these selections,
regard should be had. to the factorial orders of the products.
20 Peircb: Linear Associative Algebra,
63. In every nilpotent group, the facient order of any letter which is indepen-
dent of the basis can be assumed to be as low as the number of letters which are
independent of the basis.
Thus, if the number of letters which are independent of the basis is denoted
by n! , and if n is the order of the group (and for the present purpose it is suffi-
cient to regard n' as being less than n), it is evident that any expression, A , with
its successive products by the powers of the basis i , as high as the n'**^, and the
powers of the basis which do not vanish, cannot all be independent of one
another ; so that there must be an equation of the form
i^ajr + ij^J^A = .
1
Accordingly, it is easy to see that there is always a value of Ai of the form
1
which will give ^
which corresponds to the condition of this section.
TTiere is a similar condition which holds in every selection of a new letter by the
method of the preceding section.
64. In a nilpotent group, the order of which is less by unity than the number of
letters, the letter which is independent of the basis and its pawers may be so selected
that its product inJho the basis shall be equal to the highest power of the basis which
does not vanish, and that its square shall either vanish or shall also be equal to the
highest power of the basis that does not vanish. Thus, if the basis is i , and if the
order of the algebra is n , and if j is the remaining letter, it is obvious, from § 63,
that j might have been assumed such that
which gives
^y=o,
y^ = ^;* = 0;
and therefore, ji = ai* + bj ,
j^ = a'i- + b'j,
=:yi^ + ^ z=: bji"" = b'^ji = b,
jH = aji"" = = b'j\ = V,
p z= a!i'' ; b '
y
Peibcb : lAnear Associative Algebra, 2 1
so that if
(^'-
kz
-)•"■
•
Jl ■ J^,
h =
■
t
J'
• •
^1*1 =
n
=jl
•
we have
and ii andyi can be substituted for i andy, which conforms to the proposition
enunciated.
It must be observed, however, that the analysis needs correction when the
group is of the second order.
65. In a nilpotent group of the first order^ the sign of a prodAict is merely
reversed hy changing the order of its factors. Thus, if
it follows by development, that
• ( j1 + jB)» = ji» + 45 + JB^ + J5» = ii JB + 54 =
BA — — AB,
which is the proposition enunciated.
66. In general^ in any nilpotent group of the w*^ order^ if {A\ JB*) dervotes the
sum of aU possible products of the form
in which
and if
it will be found that
AP& A^&' A^"B^' . . .
Xp=^s, Xq=^tj
s + tznn + 1 ,
(4', ff) = 0.
For since {A + xB)"" + ^ =
whatever be the value of x , the multiplier of each power of x must vanish, which
gives the proposed equation
(4*, 5*)=0.
67. In the first group of an algebra, having an idempotent basis , all the expres-
sions except the basis may be assumed to be nilpotent For, by the same argument
as that of §53, any equation between an expression and its successive powers
and the basis must involve an equation between another expression which is
22 Peirge : Lmear Associative Algebra.
easily defined and its successive powers without including the basis. But it
follows from the argument of §57, that such an equation indicates a corres-
ponding idempotent expression ; whereas it is here assumed that, in accordance
with § 56, each group has t>een brought to a form which does not contain any
other idempotent expression than the basis. It must be, therefore, that all the
other expressions are nilpotent.
68. No proAvjct of expressions in the first group of an algebra Tiaving an idem -
potent ba>sisj contains a term which is a mvMiple of the basis.
For, assume the equation
AB= — ad+G,
in which A , B and G are nilpotents of the orders m , n and p , respectively.
Then,
= jI'^+IjB = — xA'^ + A'^C
A'^C = xA'^
= A'^G^'^^=xA'^C^=a?A'^G^-'^=x^'^^A'^=x,
that is, the term — xi vanishes from the product AB.
69. It follows, from the preceding section, that if the idempotent basis were
taken away from the first group of which it is the basis, the remaining letters of the
first group xjoould amstitute by themselves a nilpotent algebra.
Conversely, any nilpotent algebra may be converted into an algebra with an
idempotent basis, by the simple annexation of a letter idemfa/yiend and idemfadervt
with reference to every other.*
70. However incapable of interpretation the nilfactorial and nilpotent
expressions may appear, they are obviously an essential element of the calculus
of linear algebras. Unwillingness to accept them has retarded the progress of
discovery and the investigation of quantitative algebras. But the idempotent
basis seems to be equally essential to actual interpretation. The purely nilpotent
algebra may therefore be regarded as an ideal abstraction, which requires the
introduction of an idempotent basis, to give it any position in the real universe.
In the subsequent investigations, therefore, the purely nilpotent algebras must
be regarded as the first steps towards the discovery of algebras of a higher
degree resting upon an idempotent basis.
* That every such algebra must be a pure one is plain, because the algebra (a%) is so. [C. S. P.]
Peirge! : Linear Associatwe Algebra. 28
71. Sufficient preparation is now made for the
INVESTIGATION OF SPECIAL ALGEBRAS.
The following notation will he adopted in these researches. Conformably with
§ 9, the letters of the alphabet will be denoted by i , y , Jc, I, m and n . To
these letters will also be respectively assigned the numbers 1 , 2 , 3 , 4 , 5 and
6. Moreover, their coefficients in an algebraic sum will be denoted by the
letters a , J , c , d , c and /. Thus, the product of any two letters will be
expressed by an algebraic simi, and below each coefficient will be written in
order the numbers which are appropriate to the factors. Thus,
jl = a^i + b^j + c^k + d^l + e^m +/un, ^ ^ .>
while L ^ i
?; =a4» i + *4»y + C42 fe -f ^4, Z + 64, m +/4s^n. ^
In the case of a square, only one number need be written below the coefficient,
«
thus
J(?=a^i + bsj + c^k + d^l + e^m -f/g n .
The investigation simply consists in the determination of the values of the
coefficients, corresponding to every variety of linear algebra ; and the resulting
products can be arranged in a tabular form which may be called the multipli-
cation-table of the algebra. Upon this table rests all the peculiarity of the
calculus. In each of the algebras, it admits of many transformations, and much
corresponding speculation. The basis will be denoted by i .
72. The distinguishing of the successive cases by the introduction of
numbers will explain itself, and is an indispensable protection from omission
of important steps in the discussion.
Single Algebra.
Since in a single algebra there is only one independent unit, it requires no
distinguishing letter. It is also obvious that there can be no single algebra
which is not associative and commutative. Single algebra has, however, two
cases:
[1] , when its irnit is idempotent ;
[2], when it is nilpotent.
[1]. The defining equation of this case is
24 Peircb : Linear AssocicUive Algebra.
This algebra may be called (oi) and its multiplication table is *
(«i) i
•
[2], The defining equation of this case is
This algebra may be called (bi) and its multiplication table is f
Double Algebra.
There are two cases of double algebra :
[1], when it has an idempotent expression ;
[2], when it is nilpotent.
[1]. The defining equation of this case is
By §§41 and 50, there are two cases :
[1*], when the other unit belongs to the first group ;
[12], when it is of the second group.
The hypothesis that the other unit belongs to the third group is a virtual
repetition of [12].
[1*]. The defining equations of this case are
• ■ • . •
It follows from §§ 67 and 69, that there is a double algebra derived from (bi)
which may be called (%) , of which the multiplication table is J
* This algebra may be represented hj i^ A : A ia the logic of relatives, dee Addenda. [C. S. P.]
t This algebra takes the form i^zA'.B^ in the logic of relatives. [C. S. P.]
I This algebra may be put in the form i = A : A + B : B ^ j=^ A : B . [C. S. P.]
Peirce : lAn&ir Associative Algebra.
25
(«?) * J
' I
•
•
J
•
J
' "^
[12]. The defining equations of this case are, by § 41,
whence, by § 46,
f = 0.
A double algebra is thus formed, which may be called (6g) , of which the multi-
plication table is *
t
m
•
J
[2]. The defining equation of this case is
in which n is the least power of i which vanishes. There are two cases :
[21], when n= 3;
[2«], whenn=2.
[21]. The defining equation of this case is
and by § 60,
e=o,
^=j'
This gives a double algebra which may be called (cg), its multiplication
table being f
* This algebra may be put in the form i=i A : A , jzr. A : B , [C. S. P.]
t In relative form, i=iA :B + B :C, j = A :C\ [C. S. P.]
26
Peirce: Linear Associative Algebra.
•
J
[2*]. The defining equations of this case are
«
and it follows from §§64 and 65 that
7 ^- -
so that there is no pure algebra in this ctise.*^
Triple Algebra.
There are two cases :
[1] , when there is. an idempotent basis ;
[2], when the basis is nilpotent.
[1]. The defining equation of this case is
There are, by §§ 41, 50 and 51, three cases :
[1^] , when j and k are both in the first group ;
[12], when/ is in the first, and h in the second group ;
[13], when/ is in the second, and h in the third group.
The case of j being in the first, and h in the third group, is a virtual
repetition of [12].
[1*]. The defining equations of this case .are
ij -=.ji =^ , lie •=. la^=^ k .
*Thi8 case takes the form %:=. A :B, jzzC.D. [C. S. P.]
I
Peirce : Linear Associative Algebra.
27
It follows from §§ 67 and 69, that the only algebra of this case may be derived
from (cjj) ; it may be called (03), and its multiplication table is *
(«s) *
Te
•
m
J
Tc
•
J
h
le
•
[12]. The defining equations of this case are
ji = ij =:y , ih = h , Id = ;
whence, by §§ 46 and 67,
3
;% — m —
^y = 0, j1c = c^le,
and there is no pure algebra in this case.f
[13]. The defining equations of this case are
whence, by § 46,
V —Jy ^'* = '^ » J^ — ^^^ = ;
P = T^ = Zy = , jk = «^3 i ,
and there is no pure algebra in this case. J
[2]. The defining equation of this case is
in which n is the lowest power of i that vanishes.
There are three cases :
[21], when n = 4;
[2*], whenn=3;
[23], when 71 = 2.
♦In relative form, t = ^:-4+J5:J5+C:C, jf = .4:B + B:C, k=iA:C. [C. S. P.]
t That is to say, i and j by themselveB form the algebra aj , and t and k by themselves constitute the
algebra &2 ) while the products of j and k vanish. Thus, the three letters are not indissolubly bound
together into one algebra. In relative form, this case is, iz=.A:A+B :B^ j'=.AiBy k^zAiC,
[C. 8. P.]
t In relative form, I = ^ :A + D:D,j=LA:B,kz:iC:D, [C. S. P.]
28 Peirce : Linear Associative Algebra.
[21^. The defining equation of this case is
and by § 60
This gives a triple algebra which may be called (63), the multiplication table
being *
(ftg) i j Jc
Jc
1
•
J
Je
Jc
[2*]. The defining equation of this case is
and by §§ 59 and 64, observing the exception,
i*=y, ik = 0,
Id = bsij, T^ = bsj .
There is no pure algebra when 631 vanishes,f and there are two cases :
[2*1] , when J3 does not vanish ;
[2^], when b^ vanishes.
[2*1]. The defining equation of this case can, without loss of generality,
be reduced to
This gives a triple algebra which may be called (cg), the multiplication table
being J
* In relative form, 1= A :J5 + B:C+C:i),y = ^:C+B:D, kzzAiD. [C. S. P.]
t This case takes the relative ioTm,i=A :B + B :C, j = A:C, A; = 6s^ :D + D:C. [C. S. P.]
tinrelativeform, t = -4 :B+B:C, i = ^:C, k = a. A:B + A:D + D:C. [C. S. P.]
Feirce : Linear Associatioe Algebra.
29
(Cs) i
k
k
•
J
1
«/
•
J
An interesting special example of this case is afforded by a = — 2 , when
i{k-\-i)
lk + i)i
(k + if
-J
0,
so that k-\- i might be substituted for k, and in this form, the multiplication
table of this algebra, which may be called (t/j), is *
(Ca) i
k
k
•
J
•
J
•
-J
*In relative form, i = A:B + B:C, j=A: C, & = — A :B+B:C+A :D+ D : C.
When a = + 2 , the algebra equally takes the form (c',), on subettituting k — i for k .
hand, provided a is neither 2 nor — 2 , the algebra may be put in the form
(c',') i J k
On the other
To effect the transformation, we write a = — b — v and substitute t + bk and « + r A; for i and &, and
lb — £ Jy ioTJ . Thus the algebra (Cs ) has two distinct and intransmutable species, (cg) and (c, ). [C. S. P.]
■ ■• ■ » a fa
30
Peirce : Linear Associative Algebra.
[2^. The defining equation of this case is
;5? = o,
and 6ji may be reduced to unity without loss of generality, giving a triple
algebra which may be called {d^, the multiplication table being
(<?n) »
Te
k
•
•
J
•
In this case
{i — k)k =0
Jc{i — 7e) ==♦■
(i-JcY =0,
so that i — k may be substituted for i, and in this form the multiplication
table is *
(4) i j k
k
•
J
[23]. The defining equations of this case are
and by the principles of §§ 63 and 65, it may be assumed that
V = — ^* =
Jk =
— ik = Jd = ,
— JrJ = i.
^In relative form, t = B:C, j=:A:C^ k:=A:B, This is the algebra of alio-relations in its
typical form. [C. S. P.]
Peirce : Linear Associative Algebra.
31
We thus get a triple algebra which may be called (eg), its multiplication table
being *
Te
•
•
— *
There are two cases :
Quadruple Algebra.
[1], when there is an idempotent basis ;
[2], when the base is nilpotent.
[1]. The defining equation of this case is
1? = i.
There are six cases :
[1*], wheny, h, and ?, are all in the first group;
[12], whenjT and Jc are in the first, and I in the second group ;
[13], wheny is in the first, and h and I in the second group ;
[14], wheny is in the first, h in the second, and I in the third group j
[15], wheny and k are in the second, and I in the third group ;
[16], wheny is in the second, h in the third, and I in the fourth group.
The other cases are excluded by §§ 50 and 51, or are obviously virtual repeti-
tions of those which are given.
[1*]. The defining equations of this case are
ij =ji =y , iJc = Jci = 7e, il = li = /,
and from §§ 60 and 69, the algebras (ig), (cg), (cZ^), and (e^g), give quadruple
algebras which may be named respectively (a^), (t^), (C4), and (J4), their
multiplication tables being
* In relative form, iz:z A:D , j:=^ A'.B—C'.D . k—A:C+B\D, Tliis is the algebra of alternate
numbers. [C. S. P.]
Peirce: Linear Associaiive Algebra.
1
J
*
I
(».) «
J
k
I
J
k
I
i
J
It
I
i
J
k
I
h
I
J
k
I
k
I
ak
*
{CJ «
I,) i
i
J
k
I
'■
j
k
I
i
J
*
I
J
k
J
k
k
j
I
k
t
—J
The special case (t/j) gives a corresponding special case of (64), which may
be called (Vf), of which the multiplication table is
m i j k I
i
j
k
I
j
k
k
k
(1
I
— k
Pbibce: Linear Associative Algebra.
33
The second form of {d^ gives a corresponding second form of (C4), of which
the multiplication table is
(04) i j Tc I
JL
Te
m
1
•
J
Ti
I
•
le
I
k
I
[12]. The defining equations of this case are
ij =zji =y, iJc=ihi=:ikj il= Z, Zi = ,
and it follows from §§ 67 and 69, that (c^) gives
j^ = h, jh = hj = T^ = 0,
lJ=:lk=^P = 0, jl = d^l^ Jcl:=d^l'y
jH z=ikl = d^l — d^l,
jkl = d^l = d^ = = ^24 =yZ = M,
and from § 46,
whence
and there is no pure algebra in this case.*
[13]. The defining equations of this case are
^y:=y^=y, ik=::k, il:=l, A:i = Zt = 0,
which give by §§ 46 and 67
0=p = Jc^ = Jcl=lJe = l^=:kj
and it may be assumed that
jh=il, whence y ? = .
= ?y;
This gives a quadruple algebra which may be called (64), its multiplication
table being f
*In relative form, i=:A:A + B:B+C:C+D:D, j=:A: B + B:C, & = A:C, l=zD:C.
[C. S. P.]
tin relative form, t=^:A + B:B, j=A:B,k = B:C,l=:A:C, [C. S. P.]
Vol. IV.
34
Peircb : Linear Assodatiw Algebra.
(C4) i
k
I
k
I
•
•
J
k
I
•
J
I
•
[14]. The defining equations of this case are
ij = Ji =y , ik^=ik, li = ?, H = t7 = ;
which give, by §§ 46 and 67,
o=y»
= 7/
=jJd
Jj = disJ., M
= da7j =rfiZ =
: Z/r = P,
- ««4^* + 634^',
^4% = ?y,
and &84 cannot be permitted to vanish,* so that it does not lessen the generality
to assume
M = J.
This gives a quadruple algebra which may be called (^4), its multiplication
table being f
if*) i j k I
» 1 C
k
I
•
•
J
k
•
J
•
J
?
*For then the algebra would split up into three double algebras. [C. S. P.]
1f Jn Tel&tive form, % = A :A + B:B,j=: A :B, k=iA:C, l=C:B, [C. S. P.]
Peirce: Linear Associative Algebra. 35
[15]. The defining equations of this case are
ij=j\ ik=:k, li=:^l, ji = ki =^il =^0 ,
which give, by § 46,
0=jyk = kf =J^ = lj = lk = l\
jlz^a^i^ Id •=.a^^
=y(? = a^ = a^ —jl,
= Mk = a^Jc = a84 = fc/,
and there is no pure algebra in this case."^
[16]. The defining equations of this case are
iy=y, ki=:kf ji=ik = il=zliz=0^
which give, by § 46,
o=y» = *»=*? = ?/,
jk = a^i, jl = h^J, kj = d^l, Ik = c^ , Z* = c/J,
jkj = a^ = d^h^ , jlk = a^^i = a^4^i , yP = hlj = 6,4^4^',
= 023(043 — 63J4) = 6s4(*»4 — <^) = ^(*W — ^4) = ^(^43 — <^4) =^48(^48 — ^4).
There are two cases:
[161], when d^ does not vanish;
[162], when d^ vanishes.
[161]. The defining equation of this case can be reduced to
^ = Ij
which gives
Oj3 — — J4 — C43 — — 0^4 •
There are two cases :
[161*], when d^ does not vanish ;
[1612], when d^ vanishes.
• [161*]. The defining equation of this case can be reduced to
^4=1,
which gives
jk=ii, jl'=j, lk = k, 1^ = 1]
*In relatiTe form, t= A:A,j:^A:B^ k=iA:C^ l=zD:A. There are three double algebras of
the form (&2). [C. S. P.]
36
Feibcb : Linear Associative Algebra.
and there is a quadruple algebra which may be called (g^), its multiplication
table being
is*) i j Ic I
Ic
I
1
•
%
•
•
•
J
h
I
Tc
I
This is a form of quatemionB*
[1612]. The defining equation of this case is
which gives
^4 = 0,
jk=Jl = lk = I* = Q,
*In relative form, izzAiA^jzzAiB^ k = B:A, IzzBiB. This algebra exhibits the general
system of relationship of individual relatives, as is shown in my paper in the ninth volume of the
Memoirs of the American Academy of Arts and Sciences. In a space of four dimensions, a vector may
be determined by means ol its rectangular projections on two planes such that every line in the one is
perpendicular to every line in the other. Call these planes the ui-plane and the B-plane, and let v be
any vector. Then, it; is the projection of v upon the ^-plane, and Iv is its projection upon the B-plane.
Let each direction in the ^-plane be considered as to correspond to a direction in the B-plane in such a
way that the angle between two directions in the ^-plane is equal to the angle between the correspond-
ing directions in the B-plane. Then, jv is that vector in the ^-plane which corresponds to the projection
of V upon the B-plane, and kv is that vector in the B-plane which corresponds to the projection of v upon
the ^-plane.
Professor Peirce showed that we may take t'l , ii , A^i , as three such mutually perpendicular vectors
in ordinary space, that i=j (1— jii) , J= g^ U— jfci),* = 2-(— ji — j&i), Z = ^(l-|-jt^. [See, also,
Spottiswoode, Proceedings of the London Mathematical Society, iv, 166. Cayley, in his Memoir on the
Theory of Matrices (1858), had shown how a quaternion may be represented by a dual matrix.] Thus
^y ji ^1 ' 1 have all zero tensors, and j and k are vectors. Li the general expression of the blgebra,
qzzxi + uj+zk + wl^ if x + toznl and yz=zx — x*^ we have g* = g; if » = — w'=.*/—)pt^ then
g' zz . The expression t + ^ represents scalar unity, since it is the universal idemfactor. We have, also,
S;g[= g-(aj + «?)(i+Z), Fg = -g-(a? — M?)i+«f + «fc4-g (ti? — aj)i, !Z^ = ^/ani? — y« (t + Z) .
The resemblance of the multiplication table of this algebra to the symbolical table of {46 merits
attention. [C. S. P.]
Peircb : Linear Associative Algebra.
37
and there is a quadruple algebra which may be called (A4), its multiplication
table being*
(^4) i j Jc I
k
I
•
t
•
J
k
I
[162]. The defining equation of this case is
which gives
0^3 = 0,
and there can be no pure algebra for it.f
[2]. The defining equation of this case is
There are four cases :
[21], when n
[2*], when n
[23], whenn
[24], when n
5;
4;
3;
2.
[21]. The defining equation of this case is
and by § 60 ,
i* =y , 1? ■=.k, i* :=l
»ix4<c'<-« »'*■ V
I ^^-
This gives a quadruple algebra which may be called (i^), its multiplication table
being J
i.
* In relative form, » = ^ : ^ , j=A:B, k=iC:A, lz=.C:B. [C. S. P.]
tinthiscase, t=^:.4, l=id^(B:B+ C:C) , J= A:B OT = A:D,k=C:AoT=:E:A, [C. S. P.]
tin relative form, t = ^ :B + ^:C+C:i)+D:-&, Jz=iA:C+ B iD-i-CiE, k=A:D + B:E,
l = A:E. [C. S. P.]
88
Peisce: Linear AssodcUive Algebra.
(n) »
Te
I
•
J
Tc
I
h
I
I
h
I
[2*]. The defining equation of this case is
and by §59, i*=y, i^ = k.
There are then, by § 64, two quadruple algebras, which may be called (j\) and
(kt), their multiplication tables being *
(J*) i j k I and {k^ i j k I
k
I
•
k
k
k
k
k
I
m
J
k
k
k
[23]. The defining equation of this case is
and by § 59
and it may be assumed from the principle of § 63 that
ik = 0,
which gives
jk=0.
*In either of these algebras, i= A :B + B :C+ C :D, j=.A:C+B:D, k=A:D; and
in {j\) lzzA:E + E:D + AjC,whileia(k^)l=:A:C. [C. a P.]
Peirce: Linear Associative Algebra. 39
There are two cases : [231], when il=ik]
[232], whenil=0.
[231]. The defining equation of this case is
i7 = A; ,
which gives
jI=iV = ik = 0,
Id = agii + h^j + c^ + ^1?,
^ = iTd = a^ij+d^Jc, a^=0, <^i = 0, hi = h^ -\'C^k.
So, because i^ = , A? = JajT + c^Aj ,
and because iA;Z = , A;Z = Js^J* + ^^ > ^y = ^* = ^^' = hi^zij + Ps*^ •
= A;yi = ciiJd , Cji = = Ay,
t7* = M = h^ , Zi = h^i + 64iy + C41A , Ij = Zt* = {b^ + 631^41)/ »
=: i^ =zc^ z=c^^ ilk:=J^=zb^\ ZAj = 63^ + 543^ + C43A; ,
P = a^i + 64/ + C4A; + <^4Z, =7^ = ajc + cjd + dj^ = a4Zi + bjj + ejk + (^4?.
But A^ contains no term in Z , so that ^4 = .
kl = iP = a4y , 634= ^4 > ^84 = ,
= Z^= 634A + C4&34J", J34 = a4 = = A^, P = bJ+cJc,
kil=Ji?=: b^Jl = , = M = 631^' = 63*1/ = b^i z:zki= lj\
li = b^j + c^ , ZA; = Zt7 = .
There are two cases :
[231*], when C41 does not vanish ;
[2312], when C41 vanishes.
[231*]. The defining formula of this case is
C4i4=0,
and if p is determined by the equation
C4ii>* + (C4 — &4l)i> = &4 ,
we have
i{l + pi)='k+pj,
{l + pif = (C4 + JPC41) {Tc +pj) ,
so that 1+ pi and k'\- pj may be substituted respectively for I and A; , which is
the same as to make
&4 = 0,
40
Peirce: Linear Associative Algebra.
and there are two cases :
[231'], when C4* does not vanish;
[231*2], when c^ vanishes.
[221*]. The defining equation of this case can be reduced to
This gives a quadruple algebra which .may be called (Z^), its multiplication
table being f
(li) i j k I
•
J
h
ck
h
I
[231*2], The defining equation of this case is
* J. e. the neu> C4 , or what has been written c^ +pc^i . In all cases, when new letters of the alpha-
bet of the algebra are substituted, the coefficients change with them. [C. S. P.]
t When 6 = 0, c =: 1 , we have l(i-'l)'=.{i—l)l'=.(i\ so that by the substitution of t — I for t* , the
algebra is broken up into two of the form (e,). When &=.0, 04=1) on substituting ii=i — Z,
j\ =:j — cfc, ik, = (c — 1)**, /i = (c — l)i, we have if =ji , t,Zi=0, Zjtizz/Jzzfci; so that the
algebra reduces to (r^) . When 6=1, c = , on puttingt'i = t — Z , ji zzj^-k , we have t? = ij/ = ,
/ti =ji , 2* = I; ; so that the algebra reduces to (^4) . When 6 = 1, e it: , on putting t'l = ^c"^ (^ — 1
ji=j+(c— l)fc, we have t{ =/*=*, t\/ = 0, Kj =ji ; so that the algebra reduces to (p^).
When 6(6 — l){6c + 6 — l):t=0, on putting t, = (1 — 6) 6i— (1 - 6) Z, ji = (1 — 6)»(1- 6— 6c)*,
*i =:6«(i-.6)(i — 6-.6c)i-6(l — 6)(l-6— c + c»6)*, J, =6(1 — 6)f— 6d, we get the multipU-
cation table of (04). When6(6 — 1) + 0, 6c + 6 = l ; on putting t, =6(f— Z), j, =6M1 — 6)i— 6*cfc,
ib|=6(l — 6 — e)*, 2|=6i— 2, we get the following multiplication table, which may replace that in
the text :
{h)i i h I
J
h
I
J . i i
1
1
» i 1
Inrelativeform,«= A:B+B: C+ A:D,j::^A:C, kzzAiE, i=A:B+I>:E, [a a P.]
J
Peirce : Linear Associative Algebra.
There are two cases :
[231*21], when 641 does not vanish;
[231*2*], when 641 vanishes.
[231*21]. The defining formula of this case is
There are two cases :
[231*21*], when % + 1 does not vanish ;
[231*212], when c^+l vanishes.
[231*21*]. The defining formula of this case is
41
so that
^41+1 +
80
2 f^ni + cJ _ f>id + ^41041^
C41 + I . C41+I '
bjli + C41I J _ ftllfe / b^ii + CJ \^_ ^4 1 + ^ 41 ^41 J + CgJ^
C4I + 1 C41 + 1 ' \ C41 + 1 / • C41 + i • C41 + 1
that the substitution of ^^^^^^ b^l + M ^^^ _bjc ^^^ ^^^j^^j f^^
<?« + 1 C4I + 1 C41 + 1 ' ^ -^ '
t , y , and Ic , is the same as to assume
C41 = , ha =y,
which reduces this case to [2312].
[231*212]. The defining equation of this is easily reduced to
This gives a quadruple algebra which may be called (7714), its multiplication
table being
(W4) i j Tc I
J
I
•
Tc
i-*
42
Peirce : Lmear Associative Algebra.
The substitution of i — I and y — k, respectively, for i axid J transforms this
algebra into one of which the multiplication table is *
K) *
Jc
I
k
I
Jc
•
J
[281*2*]. The defining equation of this case is
This gives a quadruple algebra which may be called (n^), its multiplication
table being f
(«4) i j k I
k
I
•
J
k
ck
[2312], The defining equation of this case is
which gives
(Z — h^xi) i = ,
so that the substitution of Z — &4ii for I passes this case virtually into [232].
*t=:A:B+C:D,i=B:D, fc = ^:C, Z = B:C. [C. S. P.]
t Inrelativefonn, t = -4:B + B \C+ D\E^ j—A\C^ k= A:E, l=:B:E+eA:D. When c =
the algebra reduces to ($4). [C. S. P.]
Peiece : Linear Associative Algebra.
43
[232]. The defining equation of this case is
t7=0,
and it may be assumed that
A^ = 0,
:=Jl=z kj == iA:* = A?i = ikl = Hi = ilk = Iki = iP
Zy = d^li , = Z/i = d^lj = eZ^Zi = d^i = ZjT .
There are two cases :
[2321], when c^ does not vanish;
[232*], when C41 vanishes.
[2321]. The defining equation of this case is easily reduced to
which gives =:Uk=zI^ z=zlil = kl
Ik = ?i = a^J-}- djc ,
= Z% = d^ % = cQZJfe = d^ , ZA; = a J = ZH' ,
1^:=ia^i •i-hj'^'cjc,
= Z^ = a4A; + C4a4y = a4 = ZA; .
There are. two cases :
[2321*], when c^ does not vanish ;
[23212], when C4 vanishes.
[2321*]. The defining equation of this case can be reduced to
which gives a quadruple algebra which may be called (04), its multiplication
table being *
iPi) i j
Te
I
h
I
J
Tt
hj+h
♦In relative form, i — A\E +E\D + B:C , j = A:D, &=A:C, l — A:B + B :C+bB:D.
When & = , this algebra reduces to (r^). When & = — 1 , the substitution oti^ltorl reduces it to (Z4).
[C. S. P.]
44
Peirce : Linear Associaiive Algebra.
[23212]. The defining equation of this case is
There are two cases :
[232121], when 64 does not vanish ;
[23212*], when 64 vanishes.
[232121]. TUe defining equation of this case can be reduced to
This gives a quadruple algebra which may be called {p^, its multiplication
table being *
(i>0 i j le I
Te
I
•
J
Te
•
[23212*]. The defining equation of this case is
Z» = 0.
This gives a quadruple algebra which may be called {q^, its multiplication
table being f
(94) * j
Te
I
Tc
I
•
h
*InrelatiTeform,t = ^:5 + B:D+C:J^,i=A:2), h^ A\E , l::zA:C-\-C:D. [C. S. P.]
t In relative form, i^A: C+ C : 2) , j = ^ : D , jfc = B : D , i = B : C. [C. S. P.]
Pbircb: Linear Associative Algebra. 45
[232*]. The defining equation of this case is
and we have
Id = h^j + 634^ + d^
Ih — b4sj + cjc + d4sl
V = hj + cJc + dj.
so that there can be no pure algebra in this case if J41 vanishes,* and it may be
assumed without loss of generality that
There are two cases :
[232*1], when d^ does not vanish ;
[232^], when rf, vanishes.
[232*1], The defining equation of this case can be reduced to
which gives
= Ar» = ;fcZ=7A; = H=?,
and there is no pure algebra in this case.f
[232^]. The defining equation of this case is
e^ = 0,
which gives = ^•' = c^ = Cg, ^ = h^jy
= It I '-^ CgiAr + d'^Jvl ^ d^ = 63^31 >
= ?^^ = C^ + ^48^:? =(^43 = 63C48 .
There are two cases :
[232^1], when 63 does not vanish;
[232*], when 63 vanishes.
[232^1], The defining equation of this case can be reduced to
which gives
= Cg4 = C43, Jd=:b^j\ lk = b^j,
Jc{i-b^:f^=o,
*ln tliis case^y, k and 2, might form any one of the algebras (&,), (c,), (d,) or (e,). [C. S. P.]
t The case is impossible because A» = and kH = J . [0. S. P.]
46
Peiecb: Linear Associative Algebra.
so that I — b^^csxi be substituted for I without loss of generality, which is the *
same as to assume
kl=Oi
and this gives
= Z^ = dj^ zzzd^zzz cjk = 04643 = ?A; = C4 ,
so that there is no pure algebra in this case.*
[232^]. The defining equation of this case is
which gives
^ = 0,
= lj = Pi = dj = d^ ,
= A;? = c^cl = C34 , hi
0=iPk=: c^lk = C43 , Ik
and there can be no pure algebra if c^ vanishes, so that it may be assumed,
without loss of generality, that
P = k,
which gives
= P = lk = kL
This gives a quadruple algebra which may be called (r^), its multiplication
table being f
(n) i j k I
k
I
•
J
•
J
k
[24]. The defining equations of this case are
* Substituting i — Z for i , this case is, i = B :Z), i=^ :D, k = A:C+C:D, Izi^AiB. [0. S. P.)
U = A:B+B:D+C:D,jz:iA:D,k=:A:E,l=zA:C+C:E. [C. S. P.]
Peirce : Linear AssocicUive Algebra. 47
and it may be assumed, from §§ 63 and 65, that
ij=zk=^ — ji , t7 = Zi = ,
which give
= ii = ki =^jk = Jcf =ikl=zJJcj
= ijl = bje = b^ =jH — — a^h + d^Jl=d^ = a,| ,
jl= — Jj = c^k,
so that there is no pure algebra in this case.*
Quintuple Algebra.
There are two cases :
[1], when there is an idempotent basis ;
[2], when the algebra is nilpotent.
[1]. The defining equation of this case is
t* = I.
There are eleven cases :
[1^], wheny, hj I and m are all in the first group ;
[12], when/, 7c and I are in the first, and m in the second group ;
[1 3] , when j and Jc are in the first, and I and m in the second group ;
[14], wheny and k are in the first, I in the second, and m in the third group ;
[15], wheny is in the first, and k, I and m in the second group ;
[16], wheny is in the first, k and I in the second, and m in the third group ;
[17], wheny is in the first, k in the second, I in the third, and m in the fourth
group;
[18], wheny, k and I are in the second, and m in the third group ;
[19], wheny and k are in the second, and I and m in the third group ;
[10^], wheny and k are in the second, I in the third, and m in the fourth group ;
[1 1^] , when y is in the second, k in the third, and I and m in the fourth group.
[1*]. The defining equations of this case are
ij^=^ji =y, ik =:ki=ik, il =U = l, im z=: mi^= m.
The algebras deduced by §69 from algebras (ij to {r^ may be named (a^) to {j\),
and their multiplication tables are respectively
*«• —
i = — A:C+B:E,j=zA:B+C:E+eD:E, k=iA:E, l = — A:D + cB:E. [C. S. P.]
48
Peircg : lAnear Associative Algebra.
(rts) i
k I m
k
I
m
•
•
J
k
I
m
•
J
k
I
m
k
I
m
I
m
m
0.
(Cb) i
t
k
I
m
{€,) i
t
k
I
k I m
•
•
J
k
I
m
•
J
k
I
k
I
I
m
I
k I m
m
•
•
J
k
I
m
•
k
I
k
I
m
k I
or
ih) i
I m
k
I
m
•
J
k
I
m
•
J
k
I
k
I
I
m
I
I
W i
k
I
m
(«b) *
k
I
m
k I m
•
•
J
k
I
m
•
J
k
I
k
I
m
ak
I
him
•
•
k
I
9\
m
•
J
I
k
I
m
k
Peibcb: Linear ABsodaHve Algebra.
49
Ud i
k I
m
J
h
I
m
m
•
J
k
I
m
•
k
I
h
I
m
oZ
ih) »
t
Jc
I
m
k I m
•
•
k
I
t
I
m
•
k
k
•
I
in
I
k
(ffs) i
k
I
m
W »
i
k
I
m
k I
m
m
%
•
k
I
m
•
k
k
a
I
n
I
l-\-ak
k I
m
m
m
J
k
I
m
•
k
k
I
m
I
U.) i
k I m
k
I
m
•
•
J
k
I
1
•
J
k
k
I
m
k
I
50 Peirce : Linear Associative Algebra.
[12]. The defining equations of this case are
ij-=zjiz=LJ^ iJc=iJci = kj ilz=ili=zl^ im = nij m = 0,
which give, by § 46 ,
= my = mk =iml-=- m^,
and if A is any expression belonging to the first group, but not involving i , we
have the form
Am = am ,
and by § 67, A is nilpotent, so that there is some power n which gives
= J.** = A^m = aA'^~^m = a'^m = a = Am ,
=y?n = Jem = Im ;
and there is no pure algebra in this case.*
[13]. The defining equations of this case are
ij =ji =y , ik = ki=^k, il = 1, im =^ m, Zi = mi = ,
which give, by § 46,
= Z)" = W; = Z' = Zm = m; = mk =ml=^ m^ j
and it may be assumed from (ofg), by § 69, that
f = k, f = 0.
It may also be assumed that
jl=zm, whence f kl =jm = . «
We thus obtain a quintuple algebra which may be called {k^)y its multiplication
table being this : J
♦In fact i and w, by themselves, form the algebra (62), while i, y, *, Z, by themselves form
one of the algebras (04), (64), (C4), (d^), the products of m with 7 , k and I vanishing. [C. S. P.]
tThisis proved as foUows : = jH=j^m=:d2jl + e2rjm = d2ie2j+ (d26 + ^2%)^- ThusdaftCza
= Oand d26 + e2\ = 0;oTd2i=0, 625 = 0, fni=:kl^O. [C. S. P.]
Xi = A:A + B:B+C:C, j=A:B+B:C, k = A:C, l = B:D, m=LA:D. [C. S. P.]
Peibce : lAnear Associative Algebra.
51
W i
•
3
Te
I
m
•
■
•
3
Te
I
m
•
J
•
3
h
m
1c
Tc
I
m
[14]. The defining equations of this case are
ij :=.ji =y, ik=:Jci=^kj il= l^ mi=^ m, li = im = ,
which give, by § 46,
It may be assumed from § 69 and (a^) that
f = k, y' = o,
whence
0=jl = M=z mj = mJc =JJm = a^J + 645^- = a^^ = 645 , Im = c^jjc ,
and there is no pure algebra in this case.* ^ . 4
[15]. The defining equations of this case are
iJ = Ji zzzj , %k = A; , il=.l^ im := m ^ Jci=^li = 7ni=^ j
which give, by §§ 46 and 67,
=y* =:kjz=z^ =1 M = km =zlj =:lk = P = lm =: mj = mk =zml =^ m*.
It may be assumed that jk = Z, Jm = ,f
whence, j>7=0,
and there is no pure algebra in this case. J
* i= A : A-i- B :B + C : C , j= A :B + B:C, k = A:C, 1 = A:D, m — cD :C. [C. S. P.]
t We cannot suppose jk=.k ^ because j'k=.0. We may, therefore, put I torjk . Then j7 =: . Then,
'='p7n =• C2 5^2 5^ + (^2 5^2 5 + C2 r, ) ^ + 62^5m . It follows that j/u = dj 5^ , and substituting m — d^sk for
m , we h&\ejm = . The algebra thus separates into (62) and (e^). [C. S. P.]
Xi = A:A + B:B,j=zA:B, k=iB:C, l=A:C, m = A:D, [C. S. P.]
62
Peibce: Linear Associative Algebra.
[16]. The defining equations of this case are
ij '=•3% =y, iAj = A;, il=:l, wii = m , A:i = ?i = tm = ,
which give, by §§ 46 and 67,
=f =jm = Jg = J^ = kl = lj = lk = P = Tnj=7nk = ml = m^,
km = a^i + b^j, Im = a^i + h^j,
and it may be assumed that
and d^ cannot vanish in the case of a pure algebra,* so that it is no loss of
generality to assume
jk=l,
which gives
There are two cases :
[161], when a^^ does not vanish ;
[162], when a^ vanishes.
[161]. The defining equation of this case can be reduced to
«85 = 1»
which gives hn=j, Jcm = i + b^j\
and i + b^' can be substituted for i , and this gives a quintuple algebra which
may be called (?^), of which the multiplication table is
ih) i
Tc
%
l6
I
m
•
A;
V
4
I
3
m
*But = m*=Awn&=(a,5t+655j)fc = a35fc+d2a&86'' Hence 0,5 = Oand either d,, or bj^zzO,
and in either case there is no pure algebra. The two algebras (Z5) and (1715) are incorrect, as may be seen
by comparing Ic . wk with hn , k. [C. S. P.]
Peirce: Linear Associative Algebra,
53
[162]. The defining equation of this case is
which gives
085 = 0*
km = 635^, Z?n = ;
and 635 cannot vanish in the case of a pure algebra, so that it is no loss of
generality to assume
km =y.
This gives a quintuple algebra which may be called (mj), of which the multipli-
cation table is
(Wj) i j k I m
k
I
m
•
•
J
k
I
m
J
I
^^
m
[17]. The defining equations of this case are
ij =ji ==y , ik = k, U^=:lj A:i = t7 = im =: mi = ,
which give, by §§ 46 and 67,
=y* ^=jk =jl=zjfn z=kf = l?:=Ij=:P^=:lm=zmj=z mk,
kl:=a^i + b^Jj km=^c^k, lk'=^e^m, ml^=d^l, m^=:e^m,
0=jkl = a^j = a^,
IE = 634^' = = 643771? = ^43(^54 , kik = huJk-= = ejcm = e^^ ,
Ikm = c^Ik = e437n* = Cs^e^^m = = e^^ ,
kml = d^kl = C35H , {d^ — C35) 634 = , km^ = ejcm = c^km , (^5 — C35) c^=0,
mV = e^ml = d^ml , (^5 — d^d^ = .
54
Peibcb : Linear Associative Algebra.
There are two cases :
[171], when eB = l;*
[172], wheiiC5 = 0.
[171]. The defining equation of this case is
which gives
m^ = m ,
U ^— ^^ —^ LrC •
There can be no pure algebra if either of. the quantities h^ , c^ or d^ vanish,
and there is no loss of generality in assuming
ld'='j^ hm'=^h, ml=zl.
This gives a quintuple algebra which may be called (wg), its multiplication table
being
(715) i j k I m
•
•
•
le
•
•
J
h
•
Te
I
I
m
I
m
[172], The defining equation of this case is
which gives
= C35 = d^ = hm = ml ;
* But on examination of the assumptions already made, it will be seen that if e^ is not zero, and
consequently 643 =0 , the algebra breaks up into two. Accordingly, the algebra (n^) is impure, for i,
J , /:and {, alone, form the algebra (/«), while m , Z, A;, J, alone, form the algebra (^4), and tm = mt
= . [C. S. P.]
Peircb : Linear Associative Algebra,
55
and there can be no pure algebra if either 634 or c^ vanishes, and it may be
assumed that
This gives a quintuple algebra which may be called (05), its multiplication table
being as follows : *
(05) i j k I m
•
•
•
t
k
•
J
m
J
k
•
J
I
I
m
m
[18]. The defining equations of this case are
ij =y , iJc = kj il=^lj mi = mj ji z=:ki = li = im=:Oj
which give, by § 46 ,
=y* =^jk=^Jl = kf =ij{? := Jd =:lf =zIk=iP =: mj = mk =zml=:m?
But if A is any expression of the second group.
which gives
Am == ai ;
= AmJ =aj =ia=:^ Am =^Jm = km = Im ,
and there is no pure algebra in this case.
[19]. The defining equations of this case are
ij=j\ ik = k, li = l, mi=-m^ il=zim=ji=iki=^0 ,
which give, by § 46.
=y* =y^ =zkj=^Ji? = Jj = Ik=:P = Im=: mj = mk = ml=z ?n*.
«•• —
% = B:B+D:D+F:F,jzzD:F, k = B:C+D:E, l = A:B + E:F, mzzA \C, [C. S. P.]
56 Peirce : Linear Associative Algebra.
But if A is an expression of the second group and B one of the third,
AB^= ai,
which gives
= ABj =:aj =:a=: AB =jl =^Jm =:kl=:lmf
and there is no pure algebra in this case.
[10']. The defining equations of this case are
iy=y, iJe = k, li=:l, ji=iJci=:il=::im=:mi=^Oy
which give, by § 46,
and it is obvious that we may assume
y?=o.
We have, then,
jm = b^J + c^, kl = a^i, km = b^j+cjc,
Ij :=: e^m , Ik = c^^m , ml = d^ , ttj? = e^m ,
There are two cases :
[lO'l], when a^ does not vanish ;
[10'2], when a^ vanishes.
[lO'l], The defining equation of this case can be reduced to
kl=:i^
which gives
c«5 = , jm=- b^\
There are two cases :
[lO'P], whenc5 = l;
[10'12], when e^ vanishes.
[l(yi*]. The defining equation of this case is
and we assume
jm=^j\ ml=^l, km:=ky
because otherwise this case would coincide with a subsequent one. We get, then,
=jlj = e^^Jm = e^=lj\ =jlk=e^m =64^ = Ik,
which virtually brings this case under [10'2].*
* This does not seem clear. But t = t' =: JdJd = , which is absurd. [C. S. P.]
Peibcb : Lirvear AssodcUive Algebra. 57
[10'12]. The defining equation of this case is
m» = 0,
which gives
=ym* = 6315^^ = &w =y^ » = mH = d^ml =a^=^ ml,
= km^ = c^km = C35 , fcm = ftggy , IJcl =zliz=:l=z c^ml = ,
which is impossible, and this case disappears.
[I(y2]. The defining equation of this case is
There are two cases :
[10'21], whene5 = l;
[10'2*], when e^ vanishes.
[10'21]. The defining equation of this case is
m^ = 7W,
and if we would not virtually proceed to a subsequent case, we must assume
jm =:y, km =z k, ml=:l,
and there is no loss of generality in assuming
SO that there is no pure algebra in this case.f
[10'2*]. The defining equation of this case is
which gives
= mH = d^ml ^=1 d'^=^ml ;
and we may assume
^86 = 0,
which gives
=jm^ = \Jm = 62B =y^ , = km^ = c^m = c^, km = h^j ,
= 643m* = Ikm = h^e^m = h^^st ; %
* In this case, the algebra at once separatee into an algebra between j, k , { and m , and three double
algebras between t and j , t and ft, and t and I , respectively. [0. S. P.]
t In fact, = Vdk = e^^yn = 64, = 2A; . So that the algebra falls into six parts of the form (&,). [C. S. P.]
tTheauthor omits to notice that = ^2^ = 64 ,Ami = 643635. Thus, either Ann = or (; = Zft=0. The
algebra (pt) inyolves an inconsistency in regard to ftZft. [C. S. P.]
58
Peibce : Linear AssocicUive Algebra.
and we have without loss of generality
Z/z=0, km=:^jy lk=:m.
This gives a quintuple algebra which may be called (p^), of which the multipli
cation table is
(jPs) i j k I m
•
m
•
J
h
m
J
k
•
J
I
I
m
m
•
[11']. The defining equations of this case are
ij =y, Jci = hf ji = ilc =:il = im = li = mi = ;
which give, by § 46,
=y* zz: A:* 1= jfc? = hm =ilj ^mj ^
jk = a^i , jl = h^j\ jm = h^J, 1^ = ^^32? + %w , Ik = c^, mk = c^.
There are two cases :
[ll'l], when I is the idempotent base of the fourth group ;
[11'2], when the fourth group is nilpotent.
[ll'l]. The defining equation of this case is
f = l.
There are two cases :
[ll'l*], when m is in the second subsidiary group of the fourth group ;
[11'12], when m is in the fourth subsidiary group of the fourth group.
[ll'l*]. The defining equations of this case are
?7n = m , w/ = ;
Peiece: Linear Associative Algebra. 59
which give = m* =:jrf? = h^^^jm = 635 =ywi ,
= mVc = 053?^^ = C53 = inA: ;
and Ogs cannot vanish in a pure algebra, so that we may assume
jh = i ,
which gives
jl=jP = h^jl, b}^ = b^=l, lh = l^h = cjk, c^ = c^^z=z\z=z d^,
jl=j\ lk = k, kfl=l =kf\
and there is no pure algebra in this case.*
[11'12]. The defining equations of this case are
Zm = mZ = ,
which give
= jIm = b^jm = 624*85^ = ^84^26 J = Imk = c^^lk = c^j^ = ^43053 ,
kjl = dy^l = 624^' = b^d^l + b^e^m , Ikf = rf^Z = c^^J = 043^32? + 043^32771 ,
kjm = e^w? = b^^Jcf = b^^l -f b^^e^m , rrilij = e^m^ = e^J{j = c^l + c^^m ,
There are two cases :
[11'121], when m is idempotent ;
[11'12*], when m is nilpotent.
[11'121]. The defining equation of this case is
m* = 7W,
which gives
and it may be assumed that
5,4 = 0.
But if the algebra is then regarded as having I for its idempotent basis, it is
evident from § 50 that the bonds required for a pure algebra are wanting, so
that there is no pure algebra in this caae.J
* In fact, t , i, k^ I form the algebra (^4), and 2, m , the algebra (^z)* [C S. P.]
t The last equation holds by i 68. [C. S. P.]
t Namely, d, 2 = , and either 63 3 = 1 , when I forms the algebra (a^) , and i, j\k^m the algebra
(94) 1 o^ ®^ ^8 2 = ) when by [18] of triple algebra a, , = , and j and k each forms the algebra (62 ) with
each of the letters i , 2 , m . [C. S. P.]
60
Peibce: Li/near Associative Algebrai
[11' 12*]. The defining equation of this case is
which gives
?n' = 0,
=ym' = h^m = hij = h^ =^jm , = nfh = c^mk = c^c = Cgg = wA; ,
l=6j^ = C43, jl=jy lk = ky = 63,,
and there is no piu^e algebra in this case.*
^ [T-1'2]. The defining equation of this case is
^=0,
in which n is 2 or 3. We must then have
= lm = ml = m*f
which give
=:jP = b^'P =: bi^jl = b^ ^=jl =-jm =:lk=: mh , = Ttgk = a^ = a^ =^jk ,
and there is no pure algebra in this case, f
[2]. The defining equation of this case is ,
There are five cases :
[21], whenn = 6
[2*], when n = 5
[23], whenn = 4
[24], when w = 3
[25] , when w = 2 .
[21]. The defining equation of this case is
i^=y, ^ = k, i^ = l, i^
and by § 60,
= tn .
This gives a quintuple algebra which may be called (55), its multiplication table
being
* Here, m forms the algebra (&i) , and the other letters form (^^4) . [C. S. P.]
t Namely, if n = 2 , j , 2 , A; , form the algebra (d,) (second form), t , j, and i^k, the algebra (bz), and
m the algebra (c^). But if n=: 8 , y. k, I and m form an algebra transformable into (J4) or (Af4), while t ,
J, and t , k form, each pair, the algebra (62 )• [^' 3* P*]
i
Peirge : lAnear Assodaiive Algebra.
61
(?5) *
k
I m
k
I
m
•
J
k
I
m
k
I
m
1
I
m
m
[2*]. The defining equation of this case is
and by § 59,
There are then by § 64 two quintuple algebras which may be called {r^) and (^5),
their multiplication tables being
(n)
t
k
I m
(«6) *
k
I m
k
I
m
J
k
i
k
I
I
0»
I
I
k
I
m
•
J
k
I
1
k
I
I
•
I
[23]. The defining equation of this case is
and by § 59,
62 Peirce: Linear Associative Algebra.
and it may be assumed, from the principle of § 63, that
which gives
=y? = Id = ill = iP = ilm
li =z c^k + ^41? + e^m , P = C4k + dj + 647^ , Im = C45AJ + ^45? + 645171 .
There are two cases :
[231], when im=^l)
[232], whenm = 0.
[231]. The defining equation of this case is
%m = Z,
whence
=-jm = km =^jmi =-jml =:jw? = 641 = 64 = 645 ,
li^ = djij = Zi* =: d^li^ = dlJf? = d^i z=d4x = lj = lk,
P =z d^ , = Z* = dj? = CZ4 , Zi = C41& , P = c^k, lm=- cjc + ^ijZ ,
imi=::U=^c^ik, ini:= c^j + c^ik + d^Jy mj ^=c^{l + d^i)k, mfc = 0,
iinZ :=P zizcJCy 7nl=: cj + C54& + CZ54Z ,
irn^ =:lm = cjc + CZ45Z, w? = 645^ + c^k + cZgZ + ^45771 ,
=: m^ =: ^45 , lim =zpz=: c^Jmi = =: mli =: CZ54C41& = CZ54C41 ,
=: miia = cZ54Zm =: CZ54C45 , =: m*Z = cZ54mZ = (£54.*
There are two cases :
[231*], when C41 does not vanish ;
[2312], when c^ vanishes.
[231*]. The defining equation of this case is reducible to
Zi=: A;.
There are two cases :
[231^], when C45 does not vanish ;
[231*2], when c^ vanishes.
[231^]. The defining equation of this case can be reduced to
Zrn = ifc,
which gives
Tffi = & + d^^ + d^^ = A; + cigAj , (^ = cZgi + cZg*! ,
m' = A; + d^^ + d^d^^ = cZgJfc, cZ|i = — 1 ;
* To these equations are to be added the f oUowing, which is taken for granted below : mX = mim :
e^^dsift. [C. a P.]
\
Peirge : Idnear Associative Algebra.
and if x is one of the imaginary cube roots of — 1 , there are two cases :
[231*], when(i5i = r;
[23P2], when 0^51 = — 1.
[231*]. The defining equation of this case is
63
which gives
i{m
(m
{m
cj)i =
ly l{m -
= 0, (m
=y+[c.-
^61 = ?,
— CriO l=Xh,
-C5i(l + r)]* + (2r — 1)?;
so that the substitution of w — CgjZ for m is the same as to make
Cr, = .
There are two cases :
'61
[231*], when c^ does not vanish ;
[231*2], when c^ vanishes.
[231*^]. The defining equation of this case can be reduced to
Cb= 1.
There is then a quintuple algebra which may be called (^5), its multiplication
table being *
* The author has overlooked the circumstance that (t^) and (u,) are forms of the same algebra. If in
(fj) weput»i=:»— r*Ji ji=j—2v^ky &!=*,;, = — t*Aj + Z, mi = — tV+w, we get (u^). The struc-
ture of this algebra may be shown by putting t'x = t* , A = r V ^ Aj^ =: — A? , ij = r V— ' ti , wii == tt — m ,
when we have this multiplication table (where the subscripts are dropped):
(t*») t J
m
I
m
•
k
Xk
xl
k
k
I
k
X'k
In relative form, %=iA :B + A:C+ B:E+ C: D + E:G, J= A:D + A: E+B :G, k = A:G,
IzzvAiE+CiOy m = x^A:B+A:F+xC:E+D:Q"F:G. [C. S. P.]
64
Peibce : Linmr Associative Algebra.
{U)
Te
I
m
Te
I
•
k
I
k
k
k
j + rl
{l+x)k
Vk
{2v-l)l
m
[231^2]. The defining equation of this case is
There is then a quintuple algebra which may be called (u^), its multiplication
table being
(M5) * j
k
I
m
k
I
•
J
k
I
k
k
k
J+vl
{l+V)k
r*
J +
{2t-l)l
•
m
[231*2]. The defining equation of this case is
^1 = — 1 ,
which gives
dg = , i{m — C51Z) = 7 , l{m — c^il) = k ,
{m — c^il)i =zj — /, {m — C5iZ)Z = — k, {m — CgiZ), =j + c^]
so that the substitution ofm — c^il * for m is the same as to make
C51 = .
* The original text has m — Cnk throughout these equations, but it is plain that m^Cul is meant.
[C. S. P.]
Pbibcb : Linear Associative Algebra.
65
There are two cases :
[23P21], when c^ does not vanish;
[231^2*], when Cg vanishes.
[231^21]. The defining equation of this case can be reduced to
There is a quintuple algebra which may be called (vj), its multiplication table
being *
(^5) i J
h
I
m
I
Jc
I
•
J
k
I
k
k
k
J -I
k
j + k
m
[231^2*]. The defining equation of this case is
C5=0.
This gives a quintuple algebra which may be called (w^), its multiplication table
being *
11 2
* The algebra (Vg) reduces to (w^) on eubstitating I'l = t + ^j + s ' 1 ii =i+ Af,Afi=A?,Zi=gfc + Z,
Wi = gj + 5- Z + m . To exhibit the structure of this algebra, we may put p and p' for imaginary cube
roots of 1, and substitute in (105)11 =i + p'm, Ji = (1 — p)j+fc+^---8i» fci=8A, ii = (l — p')i+*~
^ — 8Z , mizzi+pni. Then, dropping the subscripts, we have this multiplication table.
m
I
m
J
k
I
m
•
f
k
•
J
•
J
k
k
I
k
m
I
k
1
In relative form, i = p'A:B + p'C:F+9pD : E, j=.SpA : C + Bp'D : F, fc=8il:D, l^Sp'AiE
+8pB:F. m = pA:D + Bp'B:C+pE:F. [O. S. P.]
66
Peibce : Linear AssodcUive Algebra.
(w) i
Jc I m
k
I
m
•
J
k
I
k
k
k
j-l
k
j+k
[231*2], The defining equation of this case is
fon = 0,
which gives
mZ=0, m* = 05^ + ^5/, mH:=d^=^ [l+d^i)k, d^=l + df^,
and Cgi may be made to vanish without loss of generality.
There are three cases :
[231*21], when neither d^i nor c^i + 1 vanishes ;
[231*2*], when ^51 + 1 vanishes ;
[231*23], when d^i vanishes.
[231*21]. The defining formulae of this case are
There are two cases :
d5itO» ^i+-l
[231*21*], when Cg does not vanish ;
[231*212], when Cj vanishes.
[231*21*]. The defining equation of this case can always be reduced to
C5= 1.
This gives a quintuple algebra which may be called (xs), its multiplication table
being *
*lti relative form, 1=1 A:B + A:E+B:D + D:F, j= A: D-hB :F. k=zA:F, l=:A:D.
m=(l + a)A:B + A:C+A:E+B:D+C:D + D:F-{-E:F. [C. S. P.]
Pbibcb : Linear Associative Algebra,
67
(^5)
h
I
m
h
I
1
1
/
Ic
h
j+al
(1 + a)k
(1 + a)l
m
[231*212]. The defining equation of this case is
This gives a quintuple algebra which may be called (y^), its multiplication t3.ble
being *
h
I
m
k
I
m
J
k
I
k
Q
k
*
(1 + a)l
j-\-al
(1 + a)k'
[231*2*]. The defining equation of this case is
^1 = — 1 ,
which gives
mi=:j — /, m/=0, m^ = c^k.
There are two cases:
[23P2*1], when c^ does not vanish ;
[231*2^], when Cg vanishes.
* The relatiye form is the same as that of (079) ; omitting from m the terms A : E and E : F. [0. S. P.]
68
Peircb: Linear Associative Algebra.
[231*2*1]. The defining equation of this case can be reduced to
771* = h.
This gives a quintuple algebra which may be called (zj), its multiplication table
being *
k
I m
k
I
m
m
J
h
1
I
Tc
h
j-l
I h
1
[231*2']. The defining equation of this case is
7n* = 0.
This gives a quintuple algebra which may be called (oag), its multiplication
table being f
(adg) i j k I m
•
•
Jc
.0
I
•
J
h
k
I
h
m
j-l
*In relative form, i = ^:BH-B:C+C:JD, i=^:C+B:D, k=iA:D, l — A:C, mzuBiC
+ A:E+E:D. [C. S. K]
t In relative form, the same as (z^)^ except that mz=:B:C, [C. S. P.]
Peirce : Linear AssodcUive Algebra, 69
[231*23]. The defining equation of this case is
mi=zj\
which gives
= (/ — j)i := (m — i)i ;
so that, by the substitution of 1 — j for / and m — i for m, this case would
virtually be reduced to [232].
[2312]. The defining equation of this case is
which gives
mj = , mim = ml = d^ilm , d^^ = , ^54 = ^1045 ,
mH = d^ifnl = C45A; , C45 = d^iC^ , m^ = d^lTn = d'^ml , d^{c^ — C45) = .
There are two cases :
[23121], when d^ does not vanish ;
[2312*], when d^ vanishes.
[23121]. The defining equation of this case can be reduced to
which gives
% ^^ ^64 5
and it may be assumed without loss of generality that
05 = 0.*
There are two cases :
[23121*], when C45 does not vanish ;
[231212], when C45 vanishes.
[23121*]. The defining equation of this case can be reduced to
Zm = wZ = k,
which gives
^1 = 1.
There are two cases :
[23121®], when c^i does not vanish ;
[23121*2], when % vanishes.
[23121®]. The defining equation of this case can be reduced to
C51 = 1 .
* Namely, by putting Z, = c^k -|- Z , mj = m — cj, [C. 8. P.]
70
Pbircb : 1/inear Associative Algebra.
This gives a quintuple algebra which may be called (aftg), its multiplication
table being *
(065) i j k I m
•
•
■ k
1
1
I
•
k
1
Tt
I
k
m
k + l
k
9 + 1
[23121*2]. The defining equation of this case is
Cbi = 0.
This gives a quintuple algebra which may be called (ocj), its multiplication
table being f
*The structure of this algebra is best seen on making the foUowing substitutions: Let l^i and 1^,
represent the two roots of the equation x^=ix+l. That is, ^ j = ^ (1 + ^ 6) and ^^ = o" ^^ ""'^ ^^ '
Then substitute t,== ^T*(f+*im), ii =kM (2+*i)i+l^i*+ U + 8^,)Zf, AjiZiJa:, Z^
+ ^a* + ( 1 + Sia) Z f , mi = 5a Y (t + JaWi). Then, we have the multiplication table :
i j k I m
J
k
I
m
•
k
k
0.
k
I
k
Ml*
9
In relative form, t=4:B + B:C+C:i>+iji^:(?+ if :2),i=^ :C+B:/>, A;=:A:Z>, l:=iA\F
+ E:D, m = A:E+E:F+F:D+^i2A:H+G:D, [C. S. P.]
t On making the same substitutions for i and m as in the last note, this algebra falls apart into two
algebras of the form (&,). [C. S. P.]
\
Peibce : Zmear Associative Algebra,
71
{aot) i
k
I m
k
I
•
J
k
■
I
k
,
1
1 •
k
I
'
1
k
J+l
m
[231212]. The defining equation of this case is
7W? = Zm = .
There are two cases :
[2312121], when c^i does not vanish ;
[231212*], when C51 vanishes.
[2312121]. The defining equation of this case can be reduced to
C51 = 1 .
This gives a quintuple algebra which may be called (ad^), its multiplication
table being*
•
J
k
I
•
k
•
•
k
I
m
k-\-dl
I
* In relative form, i = ^ : B+ B : C'\-C \ D+ E . F+aF:Q . jzn A:C+ B\D + oE'.O , k — A.D,
l=zE:Q, m = A:C+E:F+F:0. [C. S. P.]
72 Pbiecb: Lmear Associative Algebra.
[231212*]. The defining equation of this case is
Cbi = 0.
This gives a quintuple algebra which may be called (065), its multiplication
table being *
(0^5) i j Je I m
m
%
•
k
I
•
Te
le
I
m
(d
I
[2312*]. The defining equation of this case is
d^ = (i.
There are two cases :
[2312*1], when C45 does not vanish;
[2312^], when % vanishes.
[2312*1], The defining equation of this case can be reduced to
?m = ^,
which gives
There are two cases :
[2312*1*J, whend5i = l;
[2312*12], when d5i= — l.
[2312*1*]. The defining equation of this case is
^1 = 1,
which gives
C54 = 1 , ml = h .
^
* In relative form, the same as (adj,) except that m = ^ : F+ F:0, [C. S. P.]
I
Peibce : Linear Associative Algebra.
73
There are two cases :
[2312*1'], when c^i does not vanish ;
[2312'1»2], when c,! vanishes.
[2312*1*]. The defining equation of this case can be reduced to
Cji = 1 .
This gives a quintuple algebra which may be called (o/j), its multiplication
table being*
Tc
I
m
Tc
I
m
•
Ic
I
h
A;
*+Z
A;
j-\-ck
* To show the construction of this algebra, we may substitute I'l = t + m , j^ = !^*+ (a + 1) & + 22 ,
fti=4^, li'=.2j+(a—\)h—%l^ mi=:i— m. This gives the following multiplication table :
h I
3
m
k
I
J
k
4 ^
k
k
4 '^
k
I
This algebra thus strongly resembles (065). In relative form, i=iA :B + B : C+C:D+A : O
^^-^0:D,j = A:C+B:D-^!-^A:D, k = A:D,lzzA : F+E iD--^^ A:D, mznAiE
+ E:F'¥F:D+A:G'-'^^G:D. [C. S. P.]
74
Peibge : Linear Associaiive Algebra.
[2312*1*2] . The defining equation of this case is
There are two cases :
cu = 0.
[2312*1*21], when Cj does not vanish;
[2312*1*2*], when c^ vanishes.
[2312*1*21]. The defining equation of this case can be reduced to
C5= 1.
This gives a quintuple algebra which may be called (ag^), its multiplication table
being *
{ag^) i j k I m
*
•
J
h
I
•
J
h
Jc
I
Ic
m
I
h
j + Tc
[2312*1*2*]. The defining equation of this case is
This gives a quintuple algebra which may be called (oAg), its multiplication
table being f
* On substituting tj = i + iy + wi , mj = t + ^ /— m , this algebra falls apart into two of the form
(5,). [cap.]
t On substituting ii = t+ m , wij =i — wi, J^ =y+Z , Zj zzj — l , this algebra falls apart into two of
the form (ft,). [C. S. P.]
Peibce : Linear Associative Algebra.
76
(dij) ♦
"k I m
l6
I
fH
•
J
k
I
Je
k
I
k
•
J
[2312*12]. The defining equation of this case is
^1 ='— 1 ,
C54 = — 1 , ml = — k .
which gives
There are two cases :
[2312*121], when c^i does not vanish;
[2312*12*], when C51 vanishes. •
[2312*121]. The defining equation of this case can be reduced to
C51 = 1 .
This gives a quintuple algebra which may be called (atg), its multiplication
table being *
(aig) i j h I m
m
J
k
I
•
J
k
k
•
I
k
m
k—l
k
j+ck
- ♦In relative fonn, %=:A:C—B:F+C:E + D:0 + E:G, j=iA:E+C:G, k=:A:G,
LzzA.F-^B.Q, m-=zA\B+A\D'^B\E+C:F'\'aD\Q+F:Q, [O. S. P.]
76 Feirce : Linear Associative Algebra.
[2312*12*]. The defining equation of this case is
tni ^ — Z.
There are two cases :
[2312*12*1], when Cj does not vanish ;
[2312*12^], when c^ vanishes.
[2312*12*1]. The defining equation of this case can be reduced to
Cb= 1.
This gives' a quintuple algebra which may be called (a/g), its multiplication
table being*
(o/s) i j k I m
•
t
•
J
k
I
•
J
h
k
I
k
m
— I
— *
j + k
[2312*12']. The defining equation of this case is
m^=j.
This gives a quintuple algebra which may be called (oAtj), its multiplication
table being f
* In relative form, i = A: C+C:E+ E:0 — B :F, j=A:E+C:G, k
m = A:B + B:E+C:F+F:0 + A:D+D:0, [C. S. P.]
t In relative form ,i = A:C+C:D+D:F—B:E,j = A:D+C:F,k
m=iA:B + B:D+C:E+E:F. [C. S. P.]
= A:(?, l = A:F-'B:Q,
=:A:F, l=A:E—B:F,
Peikce : Linear Associative Algebra.
77
{ak^ i
k
I
m
k
I
•
J
k
I
k
k
— I
— k
9
J
m
[2312^. The defining equations of this case are
mZ = ?m =: , m^ =^ c^k.
There are two cases :
[2312^1], when d^i is not unity;
[2312*], when d^i is unity.
[2312^1]. The defining equation of this case is
which gives
i[{l — d^i)m — c^J'] = {l — d^^)l—c^Jc, i[{l — d^i)l—c^Jc'] = 0,
[(1 — rf^i) l—c^iJc]i = , [(1 — rfn) m—c^J]i = d^^ [{l—d^)l—Ciik'] ,
[{l — d^i)l— c^ik] [(1 — dgi) m — Cji/ ] = ,
[(1 — dpi) m — C5J] [(1 — dji) I— c^Jc] = ,
[(1 — ^1) ^ — (Hijy = (1 — ^1) V ;
so that the substitution of (1 — d^i)m — c^ij for m, and of (1 — d^i)l — c^ik for
I, is the same as to make
Cbi = .
There are now two cases :
[2312'P], when c^ does not vanish ;
[2312^12], when c^ vanishes.
[2312'P]. The defining equation of this case can be reduced to
w* =: A.
78
Peirge: Linear Associative Algebra.
This gives a quintuple algebra which may be called (aZj), its multiplication
table being*
(al^ i j k I m
•
•
J
k
I
•
k
Ic
I
m
dl
k
I
[2312^12]. The defining equation of this case is
This gives a quintuple algebra which may be called (a^Wj), its multiplication
table being
(amj) i j k I m
•
•
J
k
I
•
J
k
k
I
m
dl
*Jn Tel&Uve form, i = A:B']-B:C+C:D + dE:F, j:=-A:C+B:D, k = A:D, l=iA:F,
m:=:^A:E+B:F+E:D. [C. S. PJ
Peibce : Linear Associative Algebra.
[2312*]. The defining equation of this case is
79
There are two cases :
^51= 1.
[2312*1], when c^i does not vanish ;
[2312^^], when Cgi vanishes.
[2312*1]. The defining equation of this case is easily reduced to
There are two cases :
C5i= 1.
[2312*1*], when Cj does not vanish ;
[2312*12], when Cj vanishes.
[2312*1*]. The defining equation of this case is easily reduced to
m* = A;.
This gives a quintuple algebra which may be called {an^}, its multiplication
table being *
(an^) i j Tc I m
•
•
Ic
I
•
h
h
I
m
l-\-Tc
0.
h
[2312*12]. The defining equation of this case is
7n* = 0.
This gives a quintuple algebra which may be called (005), its multiplication
table being f
»In relative form, %z:^A:E+ AiB + B\C-\- C\D+ E\F, j=:A:C+B:D+A:F, k = A:D,
l = A:F, m=.A:C+A:E+E:D. [C. S. P.]
tin relative form, %=iA:B + B:C+C:D + E:F, j=A:C+B:D, k=zA:D, l=:A:F,
m — A:C+A:E + B:F. [C. S. P.]
80
Peibce : Linear Associative Algebra.
(aoj) i
h
I m
k
I
•
J
Je
I
Je
l + lc
m
[2312^]. The defining equation of this case is
mi ^ I,
There are two cases :
[2312^2], when Cg does not vanish ;
[231 2*] , when c^ vanishes.
[2312^1]. The defining equation of this case can be reduced to
7n* = A;.
This gives a quintuple algebra which may be called (opg), its multiplication
table being*
(ops) i j Tc I m
•
m
J
Te
I
•
Te
Te
I
m
I
•
Te
m =
*In relative form, i = ^ :B + B :C+C: D+ J?: JP, ; = u4:C+B:D, h
A:E+B:F+E:D. [C. S. P.]
=:A:D, l=zA:F,
Peiboe: Linear Associative Algebra.
81
[2312']. The defining equation of this case is
m* = 0.
This gives a quintuple algebra which may be called (og^g), its multiplication
table being
(«S^b) ^ j
Jc I
m
I
h
I
m
•
J
k
I
k
I
[232]. The defining equation of this case is
im = ,*
= jm = hm ,
Zi=0.
which gives
and it may be assumed that
This gives
Ij =zlJcz=0 =^iP =^ Pi = ihn = iml = mli = im
There are two cases :
[2321], when mi = 1;
[232*], when mi = 0.
[2321], The defining equation of this case is
which gives
=
lmi=:
mH=:
mi = Z,
mj = mk, lm = cjc + ^45? + e^^^m ,
P = ej, = 1^ = ej? = e^B = ?*, m* = cjc + d4+ e^m ,
ml=:ej J = mH = e^mH = ^5 = m J ; = Im^ = d4j>m
= d
'4S
* What is meant is that every quantity not inyolving powers of t is nilf aoiend with reference to i.
Hence, fZ = , also. [C. S. P.]
82
Peirce : Linear Associative Algebra.
There are two cases :
[2321*], when C45 does not vanish;
[23212], when c^^ vanishes.
[2321*]. The defining equation of this case can be reduced to
Zm = A;,*
which gives
771* = C5A;, (m — c^iy = ,
so that the substitution of ?n — c^l for m is the same as to make
This gives a quintuple algebra which may be called {ar^)y of which the multipli-
cation table is
(arj) i j k I m
•
•
J
k
•
J
k
k
I
k
m
I
[23212]. The defining equation of this case is
fon = 0.
There are two cases :
[232121], when d^ does not vanish ;
[23212*], when d^ vanishes.
[232121]. The defining equation of this case can be reduced to
^5 = 1.
There are two cases :
[232121*], when c^ does not vanish ;
[23212I2], when C5 vanishes.
[232121*]. The defining equation of this case can be reduced to
c« = 1.
* But = <m = mim zz Im . Thus, this case disappears, and the algebra (car^) is inoorrect. [G. S. P.]
Peibge : Linear Associative Algebra,
83
This gives a quintuple algebra which can be called (0^5), its multiplication
table being*
(0*5) i j k I m
•
•
J
Te
•
h
Ic
•
I
m
I
Ic + l
[2321212]. The defining equation of this case is
C5=0.
This gives a quintuple algebra which may be called {at^, its multiplication
table being
Te
I m
h
I
•
Te
Tc
I
I
m
[23212*]. The* defining equation of this case is
m* = CgA; .
There are two cases :
[23212*1], when c^ does not vanish ;
[23212^], when Cg vanishes.
*In relative form, i:=zA:B + B'. C+C \D + E\F, j=A:C+B:D, k = A:D, l = A:F,
m = A:E+E:F+E:D. Omitting the last term of m , weh&Ye {att)- [C. S. P.]
84
Peirge : Li/near Assodatwe Algebra.
[23212*1]. The defining equation of this case can be reduced to
This gives a quintuple algebra which may be called (a%), its multiplication
table being*
Jc
I m
k
I
m
•
J
Je
h
I
[23212^]. The defining equation of this case is
m» = 0.
This gives a qiiintuple algebra which may be called (aug), its multiplication
table being
(av^) i j Tc I m
m
•
k
•
J
«
k
.
k
•
I
m
I
*In relative form, tzz A :B-h B :C+ C:D, i=A : C+B\D, k = A:D, l = E:D, m = E:C
+ A : J^+ F:D. The omission of the last two terms of m gives (avg). [C. S. P.]
Peirge: Linear Associative Algebra. 85
[232*]. The defining equation of this case is
mi = ,
which gives
= my = mk = Imi = mH ,
and there is no pure algebra in this case.
[24]. The defining equation of this case is
and by § 59 ,
ir=J, vf=ji=f=0.
There are three cases:
[241], when ik = I, il = m ;
[242], when ik = Z, il = tm = ;
[243], when ik = il =zim=^0.
[241]. The defining equations of this case are
ik=^l, il=:m,
which give
jk = m, im ^=zjl =ijm = , = iml = mP = e^mP = e^f jk:=m,
iP:=ml=^ b^\ P = b^i -+- b^J + e^m, = ? = b^m + e^ml^=-b^ = mZ,
im* = , m* ^ bij + e^m , = m® = e^rf? = e^ ,
imi = , mi = b^ij + e^^m , mj = e^ymi , mi^ = = Cgj , "
Hi = mi = &5iy , li = b^ii + 641/ + e^m ,
ZiZ =zlm=^ 651m , =: Pm =zb^^ = lm = mi = mi7 = m*, (?i)i = Z; ,
ik? z=lk=^ Og/ + Cg? + dgm , i7A; = mk = c^m , lik = P = Ogjm ,
= mk? := c|m = 03 = mk := i*m ,
fry = ki? = asj H- cigiZi = agi (1 t d^)j + d^im ,
kil = frm = a3i(l + ^i)m , = k?m = agi(l + d3i)frm = 031(1 + d^i) ^ km ,
kf z= c^m , = fr^ = OgZ H- 63m + d^k = 03 = 63 + ^3 = 63^;' + d^ ,
A? = Osi? + (631 + ^^8^1) w , = fr?& = agi?fr = c^gOgi = IM = o^P = 031 := ?,
*0 = fr? + iki + i?k = {d^i + d^^+l)m, dg^ = >^1 = r ,
= i*i + frfA; + i** = *8i + ^(l + 2c4i)» *(* + FO=^+i?/» *(^ + JK/) = ^»
(fr + pi)i = bsj + dsil + e^m +pj = (631 + p—pd^^j + ^i(? -^pj) + %w,
(Z + 2») i = c?gim,
* This line and the first equation of the next can be derived from = (t + J^) '• [C. S. P.]
86
Peirge: Imear Associative Algebra.
so that if p satisfies the equation
the substitution oi k+pi for h and of 1 + pj for I is the same as to make
= 681 = ^8 = 63.
There are four cases :
[241*], when neither % nor e^ vanishes ;
[2412], when % does not vanish but e^ vanishes;
[2413], when % vanishes and not e^ ;
[2414], when e^ and e^ both vanish.
[241*]. The defining equations of this case can be reduced, without loss of
generality, to
We thus obtain a quintuple algebra which may be called {aw^\ its multiplication
table being*
(o^b) i j k I m
•
•
J
I
m
•
m
k
xl+m
fm
m
I
tm
•
m
[2412], The defining equations of this case can be reduced to
*In relative formi==^:B + B:I)+tO:^+tjG?:-F+Gf:F,j==^: 2) + t'C'.JP, h—A\C+B:E
•■\-D:F+A',0+QiF, l=zA:E+B:F, m — A\F. To obtain (oaja), omit the last term of fc. To
obtain (ay^)^ omit, instead, the last term of t. To obtain (0^5), omit both these last terms. [C. S. P.]
Feirge: Lmear Associative Algebra.
87
We thus obtain a quintuple algebra which may be called (ax^), its multiplication
table being
{cuc^) i j Je I m
•
•
J
I
m
•
J
m
k
xl-\-m
f*OT
I
Xm
m
[2413]. The defining equations of this case can be reduced to
We thus obtain a quintuple algebra which may be called (ay^), its multiplication
table being
Je I
m
Jc
I
m
a
J
I
m,
m
tl
m
fm
m
xm
[2414]. The defining equations of this case are
We thus obtain a quintuple algebra which may be called (azj), its multiplication
table being
88
Peibce : Imear Associaiive Algebra.
{azt) i
Je
I
m
1e
I
m
m
J
I
m
m
tl
fm
tm
[242]. The defining equations of this case are
whidi give
K
mj
tktn
mik
iJc=zlj i7 =: im = ,
ild = d^j + CgiZ, = Zt^ = Cgi = li^=lj\
d^jk =:lik = P=:^ikl=:^a^=^c^,
Uc = a^j + Cg7 , =1 ik? = W=: c^k = c^ ,
imi = aji = Cji , mi^ =: tw; = d^Ji + e^iini , = mji =
i^fc + iki + ki^=i 681^51 = 2agi + Ojidgi + 681651 ,
— (hijy = Ji?j=a^ = Jg'=U = ^sAi,
imfc =z Ojg =: C53 , = mk? = 653 ,
ml = Ogdji/ , A^'Aj =,U=. (ogc^i + e^^j + ejidjs? » ^
A:^Z = e^d^ i=lki^= e^ilm = 631035 := Ji?m = 635.
%
= 0,
There are two cases :
[2421], when 631 does not vanish ;
[242*], when 631 vanishes.
[2421]. The defining equation of this case can be reduced to
ki^ vn J
Peiecb: Linear Assodaiive Algebra.
89
which, by the aid of the above equations, gives
= ?ni = kil :=^ml^=. him = m* , a^j = il^
b^J = kik =zkl=i mk , = iJ{? + kik + A*i
= A;^ = ag = &53 = A^ = >fc7w = mk = ml ;
and if p is determined by the equation
= Zm,
Aj + ^i, I + pj\ and m+^y can be respectively substituted for A;, ? and m,
which is the same thing as to make
63=0.
There are three cases :
[2421*], when neither d^ nor e^ vanishes ;
[24212], when d^ vanishes and not e^ ;
[24213], when d^ and Cg both vanish.
[2421*]. The defining equation of this case can be reduced to
d^=zl.
This gives a quintuple algebra which may be called (ftag), its multiplication
table being*
(ftaj) i j Je I m
m
t
I
•
J
h
m
l-\-em
I
m
*In relative form, % = A :B + B :C+ A:E , j=A:C, k=D:B+ E:F + D :G + eQ:C+A:E,
l=.A:F^m=.D:C. By omittiiig the last term of k and putting e = 1 we get (b&s), and by omitting the
last two terms of k we get (be,). [C. S. P.]
90
Peibge : Lmear ABSociative Algebra.
[2421*]. The defining equation of this case can be reduced to
This gives a quintuple algebra which may be called (bb^), its multiplication
table being
Jc
I m
Je
I
m
•
I
f
1
1
t
I
m
1
i
1
■ ■ ■ ■ '
1
1
1
1
1
1
[24213]. The defining equation of this case is
A? = 0.
This gives a quintuple algebra which may be called (Jcj), its multiplication
table being
(ftcg) i j k I m
•
J
I
•
J
h
m
I
m
1
1
Peibge : Linear Associative Algebra. 91
[242*]. The defining equation of this case is
681 = 0.
There are two cases :
[242*1], when e^ does not vanish ;
[242^], when 6g vanishes.
[242*1]. The defining equation of this case can be reduced to
A? = aji + w ,
which gives
Idle = kl = a^ij , il(? = ZA =: a^' , J^i = a^j + mi = d^kl = a^^^J,
= ^i + ik + kik = a^{€l^ + ^i + 1) , fJ^i = (h{<^^ — !)«/ > ^' = ^^^ = ^ ,
= A? = OgZ + mA; = Os^ri + km ,
TwA; = — a^I , A»7i = — <h^zij — cc^il , Z?n = .
There are two cases :
[242*P], when Oj does not vanish;
[242*12], when Oj vanishes.
[242*1*]. The defining equation of this case can be reduced to
A? = i + m ,
which gives
d3i = >v^l = f, lk=j\ mk=> — ?,
ki=. — A?m = h^ij + rZ, mi = (t* — l)y , m* = — ?/ .
There are two cases :
[242*1^], when 631 does not vanish ;
[242*1*2], when 631 vanishes.
[242*1^]. The defining equation of this case can be reduced to
ki'=-j + rZ.
This gives a quintuple algebra which may be called (icZg), its multiplication
table being *
♦In relative form, t = il : D + 2>:J?'+B:^+C:jF',y= A rJ?*, A: = rA:B + rB:C+i): J7— -D:F
+ ^:F, ^.^=,A\E-'^- A:F+B:F, m = x*A:C—A:D — B:E—C:F. [O.S.P.]
t
92
Peibce : lAnear Associative Algebra.
bd.
) i
•
J
k
I
m
•
■
I
•
J
k
j+xl
i + m
•
^i
j tl
I
m
if-l)J
-/
[242*1*2], The defining equation of this case is
This gives a quintuple algebra which may be called (ftcg), its multiplication
table being *
(be^) i j k I m
•
•
I
•
k
xl
i + m
tj
xl
I
•
J
m
(r'-i)y
— I
-tj
[242*12]. The defining equation of this case is
/»,* = tn ,
which gives
0:=kl=lk=: km = mk zn m^ = J(?i=^ mi .
There are two cases :
[242*121], when 631 does not vanish;
[242*12*], when b^ vanishes.
* On adding to the expression for k in the last note the term — A:C, we have this algebra in relative
form. [C. 8. P.]
Peirce : Linear Associative Algebra.
93
[242*121]. The defining equation of this case can be reduced to
This gives a quintuple algebra which may be called (6/5), its multiplication
table being *
{tfs) i j Ic I m
•
1
•
J
I
1
•
J
Je
j+di
tn
I
m
1
[242*12*]. The defining equation of this case is
This gives a quintuple algebra which may be called (bg^), its multiplication
table being f
(%) i J *
I
m
•
•
J
1
I
•
k
dl
m
I
m
1
1
)
*In relative form, i = -4 : B+ B : C+i) : JE?, i = ^:C, k=.A \B+dA \ D+ B :E + B \F,
l^A'.E, mzziAiE-^A-.F. [C. S. P.]
tin relative form, i — A :B+B:C+D:E, j=A : C, k = dA :D + B:E+B':F, l — A:E,
mznAiF, The algebra (car^ ) is what this becomes when d = . [C. S. P.]
94 Peibce : Linear Associative Algebra.
[242^]. The defining equation of this case is
(38= 0,
which gives *
* It is not easy to see how the author proves that a, = . But it can be proved thus. = ft* =
The algebras of the case [242'] are those quintuple systems in which every product containing j or I
as a factor vanishes, while every product which does not vanish is a linear function of j and I, Any
multiplication table conforming to these conditions is self -consistent, but it is a matter of some trouble
to exclude every case of a mixed algebra. An algebra of the class in question is separable, if all
products are similar. But this case requires no special attention ; and the only other is when two
dissimilar expressions U and V can be found, such that both being linear functions of t , ft and m ,
r7F= 717=0. It wHl be convenient to consider separately, first, the conditions under which
ITF— "n7=0, and, secondly, those under which UV+VU=-0. To bring the subjects under a
familiar form, we may conceive of i , ft , m as three vectors not coplanar, so that, writing
U'zzict + yft + jCT», V=x'i + y'k + z'm^
we have a;, y, «, and a;' , yf ^ fi ^ the Cartesian coordinates of two points in space. [We might
imagine the space to be of the hyperbolic kind, and take the coefficients of j and I as coordinates of a
point on the quadric surface at infinity. But this would not further the purpose with which we now
introduce geometric conceptions.] But since we are to consider only such proi>erties of XJ and V as
belong equally to all their numerical multiples, we may assume that they always lie in any plane
^^ + 5^+08=1,
not passing through the origin ; and then a; , |/ , z^ and x' , {^ , ^ ^ will be the homogeneous coordinates
of the two points TJ and V in that i^ane. Let it be remembered that, although t , ft , m are vectors, yet
their multiplication does not at all follow the rule of quaternions, but that
t» = 6jj + d^ , fft = biJ-t- diaZ, »m = &i&; + d^Ji ,
*» = 6aii + d8i'i ft> = 6J+dJ, ftm = d8j+ 4,6^1
mt = d6ii+ d^^l , mft = 6g J+ d^^JL , to» =6J+ djZ .
The condition that TJV— VU=- is expressed by the equations
(bi?-b3i)(«^-a?'y) + (&i.--&M)(a^-«'«) + (&.5--&6.)(!/^--y'«) = 0,
The first equation evidently signifies that for every value of CT, Fmust be on a straight line, that this
line passes through U*, and that it also passes through the point
The second equation expresses that the line between ]7and F contains the point
e=(^86 -^68)*+ (^51 — di5)ft+(di8 — d,i)m.
The two equations together signify, therefore, that Usjid Fmay be any two points on the line between
the fixed points P and Q. Linear transformations of J and I may shift P and Q to any other situations
on the line joining them, but cannot turn the line nor bring the two points into coincidence.
The condition that UV+ VU:=z is expressed by the equations
26iaa/+ 26,j^+ 265^'+ (6i, + 6.i)(«i^+ «V) + (^5 + &5i)(a»'+ a^^^
2dia»'+2d8OT('+2d5«af'+ (di, + d.i){a^+a?V)+ (di5 + d50(aa^+a?'2j) + (d,8^
The first of these evidently signifies that for any position of V the locus of U' is a line ; that U being fixed
at any point on that line, Fmay be carried to any position on a line passing through its original position ;
and tiiat further, if O* is at one of the two points where its line cuts the conic
4
1
Peibce : Linear Associative Algebra.
95
= A*i = a^j ^:^a^=iUc=:ml = kl=im^ = e^=^ d^a^ = T^m = e^ ,
= hmk = a^^l =z a^=ilm .
then V may be at an infinitely neighboring point on the same conic, so that tangents to the conic from
Vcut the locus of rZat their points of tangency. The second equation shows that the i)oint8 U and V
have the same relation to the conic
These conies are the loci of points whose squares contain respectively no term in j and no term in I .
Their four intersections represent expressions whose squares vanish. Hence, linear transformations of
j and I will change these conies to any others of the sheaf passing through these four fixed points. The
two equations together, then, signify that through the four fixed points, two conies can be drawn
tangent at 17 and Vto the line joining these last points.
Uniting the conditions of UV— VU= and UV+ V77=0 , they signify that U and V are on the
line joining P and Q at those points at which this line is tangent to conies through the four fixed points
whose squares vanish. But if the algebra is pure, it is impossible to find two such points ; so that the
line between P and Q must pass through one of the four fixed points. In other words, the necessary
condition of the algebra being pure is that one and only one nilpotent expression in i , ft , m , should be
a linear function of P and Q .
The two points P and Q together with the two conies completely determine all the constants of the
multiplication table. . Let S and T be the points at which the two conies separately intersect the line
between P and Q . A linear transformation of /will move P to the point pP+ (1 — p)Q and will move
S to the point j>S^+ (1 ~ j>) T , and a linear transformation of I will move Q and T in a similar way. The
points P and S may thus be brought into coincidence, and the point Q may be brought to the common
point of intersection of the two conies with the line from P to Q . The geometrical figure determining
the algebra is thus reduced to a first and a second conic and a straight line having one common intersec-
tion. This figure will have special varieties due to the coincidence of different intersections, etc.
There are six cases : [1], there is a line of quantities whose squares vanish and one quantity out of
the line ; [2], there are four dissimilar quantities whose squares vanish ; [8], two of these four quantities
coincide ; [4], two pairs of the four quantities coincide ; [5], three of the four quantities coincide : [6], all
the quantities coincide.
We may, in every case, suppose the equation of the plane tobej; + 2^ + 2 = l-
[1]. In this case, the line common to the two conies may be taken as {^ = , and the separate lines of
the conies as ;s = and a; =: , respectively. We may also assume 2Pz=.x + y and 2Qzzx+z. We
thus obtain the following multiplication table, where the rows and columns having j and I as their
arguments are omitted :
t km
m
8Z
m
—3
— z
3i+i
■
J
l-j
[2]. In this case, we may take k as the common intersection of the two conies and the line, i , m ,
and i^k-\'mwA the other intersections of the conies. We have Q = ft , and we may write
P=:5=in+(1— 1> — g)ft + qw, r=:rP+(l— r)g = rpi+(l — rp -rg)ft + fvm.
We thus obtain the following multiplication table :
96
Peirge: Linear Associative Algebra.
There are two cases :
[242^1], when d^ does not vanish ;
[242*], when d^ vanishes.
[242*1]. The defining equation of this case can be reduced to
which gives
and if
i (^ + b^i+pm) = I + hj = ?n* ;
•
i
k m
•
f
■
,i,+ ^)J+r,ir,-l)l ^.f 1 P,[P, %\%^;^'%i
k
■
3(«-8)j+r3(rg-l)l
_p(p_8)y_rp(rp-l)J
m
[3 -p(p +!) + «(«- 8)]i+
[2-rp(rp-\) + rq(rq-\)-\l
1
1
-i>(p+l)i-rp(rp-l)l .
[3]. Let A; be the double point common to the two conies, and let t and m be their other intersections.
Then all expressions of the form ku + vk are similar. The line between P and Q cannot pass through
k , because in that case all products would be similar. We may therefore assume that it passes through
i. Then, we have Q = i, we may assume <S = P = t— &+m, and we may write r=rP+(l — r)Q
= t — rfc + rm. The equation of the conunon tangent to the conies at ^ may be written to + (1 — ^)« = 0.
Then the equations of the two conies are
hacy + a:« + (1 — h)yz = ,
hxy +(h + r — hr)xz+ (1 — h)yz = 0.
We thus obtain the following multiplication table :
i k m
1
{h+l)j+(h + r)l .2j+[h(l-r) + ^]l
i
1
(?i-l)i+(fe-r)Z
1
(3-h)(j+i)
^(1 — r)Z
-hU+i)
*
1
fH
[4]. In this case we may take i and m as the two points of contact of the conies, fc as P, and
i — X; + m as T. Then writing the equations of the two tangents
gy + z = (}. x-{'hy = 0,
the two conies become
gocy + xz + hyzzzO,
{g+h'-l)y\-{-gxy + xz + hyz = 0,
and the multiplication table is as follows :
Peirce : Linear Associative Algebra.
97
the gubstitution of h + h^i -{-pm for h and I + h^j for I is the same as to make
5, = c?,= 0,
This gives
There are two cases :
'5
[242^1*], when 63 does not vanish ;
[242^12], when 63 vanishes.
[242^P]. The defining equation of this case can be reduced to
m
{g + h-l)l
a+{g+i)i
fU
»■+((/- 1)1
hj+(h+i]i
sy
hj+(h-i)i
m
[5]. In this case, we may take h as the point of osculation of the conies and % as their point of inter-
section. The line between P and Q must either, [51], pass through X;, or, [62], pass through t •
[51]. We may, without loss of generality, take
Pzzfc, Q = m,
and the equations of the two conies are
z^ + rxz = , rxy + 2qxz + 2yz = .
Then, the multiplication table is as follows :
m
t
ql
k
rl
I
m
rJ-{-ql
I
•
J
[62]. We have C = *? we may take r=wi, and we may assume P=2i — w and 6,3 + 6,1 = 1
Then, we may write the equations of the two conies,
2z^ + xy + XZ + ryz=. ^
— rxy + (2 — r) xz + r^yz = .
We thus obtain the following multiplication table :
98
Peirce : Linear Associative Algebiu.
This gives a quintuple algebra which can be called (W5), its multiplication
table being *
k
I
m
•
•
J
1
1
I
•
J
1
1
1
Je
nj-\-hl
•
J
cj-\-dl
I
m
+ 1/1
dj
+ m
I
m
— rl
J-(r-Z)l
2j — W
m
(r-2)i
j^(r^2)l
(r-2)j
%• •
m
[6]. The oonicB have but one point in common. This may be taken at k . We have Q = A; , we may
take r= i and 2P=.2S=:i + k. We may also take 6i = — 1 . Then the equations of the two conies
may be written
— x^ +pz^ + 2xy+ Aqxz + 2rifZ z= ,
(A:+pr'^)z--^2xy+A(q + r)xz+2ryzz:L{^,
We thus find this multiplication table :
% k m
m
9
i
i-^i
(2g-l)j + 2(3 + r-p)/
*
3-Vl
(r+lb-+W
(2«+i) + 2(a+»-+p)i
('■ — i)y+»"i
l>J+(4+i>r»)/
If this analysis is correct, only three indeterminate coefficients are required for the multiplication
tables of this class of algebras. [C. S. P.]
* See last note. I do not give relative forms for this class of algebrsuB, owing to the extreme ease
with which they may be found. [C. S. P.]
/
Peirce : Luiear Associatioe Algebra,
[242^12]. The defining equation of this case is .
99
There are two cases :
= 0.
[242^121], when igj does not vanish ;
[242^12*], when 631 vanishes.
[242''*121]. The defining equation of this case can be reduced to
631 = 1 .
This gives a quintuple algebra which may be called (6^), its multiplication
table being
{h%) i
Tc
I m
k
I
m
•
I
j-\-al
hj+cl
a'j
-If VI
dj
-\-d!l
I
[242^12*]. The defining equation of this case is
let — — • Cvoi V ,
There are two cases :
81*
[242^12*1], when 651 does not vanish ;
[242^12^], when b^i vanishes.
[242^12*1]. The defining equation of this case can be reduced to
&5i = l.
This gives a quintuple algebra which may be called (A/g), its multiplication
table being
100
Peirce : Linear Associative Algebra.
m .i
Je
I m
k
I
m
•
J
1
I
al
bj+cl
j+a'l
b'j+il
I
[242*12^]. The definiDg equation of this case is
mi := d^J, ;
which can always, in the case of a pure algebra, be reduced to
mi = ?.
This gives a quintuple algebra which may be called {hh^j its multiplication
table being
•
•
I
•
J
Te
al
bj+cl
I
m
I
a'j
+ 111
I
[242*]. The defining equation of this case is
»n* = bJ,
K
'Peirce : Linear Associative Algebra,
101
and it can be reduced to [242^1] unless
^1 = ^3 = 0, J(^ = hj\ ^31 = — !, ^33 = — cijs;
whence it may be assumed that
and since
when
(k + bif = ,
/ + ^; J31 + 63 = ,
7^ = 0.
it may also be assumed that
There are two cases :
[242*1], when b^i does not vanish;
[242^], when b^i vanishes.
[242*1]. The defining equation of this case can be reduced to
bsi = l.
This gives a quintuple algebra which may be called (64)» its multiplication
table being
(JZ5) i j Jc I m
•
%
•
J
I
•
Tc
J-l
aj+hl
I
m
•
J
aj+ hi
9
[242*^]. The defining equation of this case is
There are two cases :
Id = — I.
[242^1], when 635 does not vanish ;
[242*], when 635 vanishes.
102
Peibce : Linear Associative Algebra.
[242*1]. The defining equation of this case can be reduced to
fts8 = l.
This gives a quintuple algebra which may be called (img), its multiplication
table being*
. {bnif) i
Jc
I
m
k
I
m
•
J
I
.
I
j+ai
•
J
bf—al
^
[242*]. The defining equation of this case is
b^ =
There are two cases :
'85
[242*1], when 653 does not vanish ;
[242'], when 653 vanishes.
[242*1] . The defining equation of this case can be reduced to
653= 1.
This gives a quintuple algebra which may be called (6715), its multiplication
table being f
*Thi8 algebra it) mixed. Namely, if &4:1, it separates on substituting t^ = (1 — &)i + A;,
ki = (1 — 6) t+ [a(l — 6) + 1] A;— (1 — 6)«m ; but if 6 = 1 , it separates on substituting ii = at — (a* + a
+ c)k + m, kizzcd + qk+m. [C. S. P.]
t Substitute t'l = f — k^ /^^ = ofc + f7i , and the algebra separates. [C. S. P.]
Peirce: Linear Associative Algebra.
103
(bih) i
Je
I m
Jc
I
m
•
I
*
I
al
J
j al
9
[242^]. The defining equation of this case is
&53 = 0.
This gives a quintuple algebra which may be called (ioj), its multiplication
table being *
h
I m
le
I
m
•
I
I
al
•
J
— al
9
[243]. The defining equations of this case are
which give
= ik = il = im ,
0=jIc=Jl=Jm.
*' Substitute for m , at + wi , and the algebra separates. [C. S. P.]
104
Peirce: Linear Associative AJgehra.
There are two cases :
[2431], when hi = /, K=^m, wi = ;
[2432], when Jci = 7, K = mi = 0.
[2431]. The defining equations of this case are
ki = l, K=mf ?ni=0,
which give
b':
= m,
Ij = mj =
— Uc = mic
= ?
0:
= i7i? =
= iM = ihn =
= «3 — «34 =
'«35>
m
= H =
= CgZ + efgWi ,
Mi
'=.hm
= CgW , =
^m = Cg =
Ajyw ,
= *3 =
= b^m + c^TW =
= *3 + ^.
=zJm=^ ml = 7?r,
There are two cases :
[2431*], when e^ does not vanish;
[24312], when eg vanishes.
[2431*]. The defining equation of this case can be reduced to
eg = 1 .
This gives a quintuple algebra which may be called {hp^^ its multiplication
table being*
*The structure of this algebra may be exhibited by putting fci =t + a"V — a"'A?, li=:j—a 7,
mi = — a~^fn , when the multipUcation table becomes
i i k I m
3
k
I
m
•
•
3
I
m
m
m
Inrelativeform, t = B:C+C:D,i=B:D, k-^AiB + C'.D, l=zA:C, m=:.4:D. [C. S. P.]
1
Pbircb : Linear AjBsoeiaiive Algebra.
105
(*P5)
»
k
I
m
le
I
m
•
1.
I
1
1
1
I
m
al + m
anh
I
m
1
i
1
1
J
1
1
[24312]. The defining equation of this case is
This gives a quintuple algebra which may be called (ftjj), its multiplication
table being*
Tc
I
m
J
1
*
I
m
— a*j
+ <a
(Mn
»
m
1
1
[2432]. The defining equations of this case are
*0n 8ub6titutiiig /Ci = t— a-*fc, Zi=j — a~*Z, mi = a"*m, this algebra reduces to (bps), in the
form given in the last note. [G. S. P.]
106
Peirce : Linear Associative Algebra,
which give
=z J(y' z= Ij z= mj == Ik == P ==?m == iJi^ == a^,
rCv ^— Cgv J yj — - fvv — • Co — — /Cv J
= ikm = 035= Jcmi = C35 = A^^m = 635 = imk
ml = C53?, = mH = c^=iml=:^ mk? = 653 ;
058
and it may be assumed that
which gives
P =:7n,
= A^ = km = mk =: wi* .
There is then a quintuple algebra which may be called {br^), its multiplication
being *
(Jtb) i j h I
m
k
I
m
t
•
1
I
m
*
1
[25]. The defining equations of this case are
= i^ =y* = )fc* = ? =:i= m* = i}' +ji = ijfc + A?i = iZ + ?i = i^ + w** ,
=yA; + kj ='jl + Ij ^=^jm + mj =:kl + lk=i km + mk = ?m + mZ ;
and it may be assumed that
ij=:k=z — ji , {? = ??» = — liy
*In relAiive form, i=iD:E+E:F, j=:D:F, k= A:B + B :F+ C:E, l=iC:F, m=:A:F.
[C. S. P.]
Peirge : Lmear Associative Algebra. 107
which gives
= tfc = Jd =jlc = A; = im = mi = km = mk =:lm=zmlf
ijk =.kl-=i b^k + d^m = — ilj = — mj ^='jm ,
='j^m = d^jm =: d„ = kl^ = h^l =zb^ = kl=Ik =^Jm = mJ ,
0=fl = a^k=a^,
i{<kij + e^l) = c^k + e^m ,
j{cuj + ^*) = ^(cm* + ^w),
Z(c,4y + e^l) = — Cm(cm* + e^m) ;
80 that it is easy to see that there is no pure algebra in this case.
Sextuple Algebra.
There are two cases :
[1], when there is an idempotent basis;
[2], when the algebra is nilpotent.
[1]. The defining equation of this case is
t •
V = I.
There are 19 cases :
[1*], when all the other units but i are in the first group ;
[12], wheny, k, I, m are in the first and n in the second group ;
[13], wheny, k and I are in the first and m and n in the second group ;
[14], wheny, k and I are in the first* m in the second and n in the third group;
[15], wheny and k are in the first and 7, m and n in the second group ;
[16], when j and k are in the first, I and m in the second and n in the third
group ;
[17], wheny and k are in the first, / in the second, m in the third, and n in the
fourth group ;
[18], wheny is in the first, and k, I, m and n in the second group ;.
[19], wheny is in the first. A;, I and m in the second, and n in the third group ;
[10'], wheny is in the first, k and Zin the second, and m and n in the third group ;
[11'], when y is in the first, k and I in the second, m in the third and n in
the fourth group ;
[12'], when j is in the first, k in the second, / in the third and m and n in the
fourth group ;
108 Peirce: Linear Associative Algebra,
[13'], wheny, k, I, m and n are in the second group ;
[14'], wheny, Aj, I and m are in the second and m in the third group ;
[15'], wheny, k and I are in the second and m and n are in the third group;
[16'], when y, k and I are in the second, m in the third, and n in the fourth
group ;
[17'], wheny and k are in the second, I and m in the third, and n in the fourth
•group ;
[18'], wheny and k are in the second, I in the third, and m and n in the fourth
group ;
[19'], wheny is in the second, k in the third, and I, m and n in the fourth group.
[1*] ; The defining equations of this case are
ij =:ji =:y , ik=:ki^k, ilziz li:=z l^ im := mi =^ m , in=^ni=^nj
and the 54 algebras of this case deduced from (q^) to {br^) may be called (a^) to
[12]. The defining equations of this case are
iJ =:ji =y , ik=^ki = ky il=zli = l^ im=^mi=:^ niy in^riy m = ,
which give
=yn = ny = A:n = Tiifc = &i = wZ = mn = nm = n*,
so that there is no pure algebra in this case.
[13]. The defining equations of this case are
ij ^ji =y , ik=iki=^k, il=:li=il, im:=m, in = n^ mt = m = .
There are four cases, which correspond to relations between the units of the
first group similar to those of the quadruple algebras (a^) , (64) , (c^) or (^4) .
[131]. The defining equations of this case are
f = k, jk = kf = l, jl=k?=zkl=lJ = lk=P = 0',
and, in the result, we obtain
jm = 71, jn=: km = A^ = /m = Zn = .
* The multiplicatdon tables of these algebras, formed from the nilpotent quintuple algebras, in the
same manner in which the first class of quintuple algebras are formed from the nilpotent quadruple
algebras, have been omitted. [C. S. P.]
U
Peiboe : Linear Associative Algebra.
109
This gives a sextuple algebra which may be called (6cj), of which the multipli-
cation table is *
(bot) i j k I m n
I
m
n
t
k
I
m
n
k
I
I
•
I
n
[132]. The defining equations of this case are
/z=fc = p, ^'=ak, jk=jl=kj = J^ = M=lk = 0,
which give
km = foi = 0.
There are two cases :
[1321], when e^ does not vanish ;
[132*], when e^^ vanishes.
[1321]. The defining equation of this case can be reduced to
jn = m,
which gives
=ym = Im .
This gives a sextuple algebra which may be called (bd^), of which the multipli-
cation table is f
♦In relative form, %=:A:A + B:B+C:C+D:D,j=:A:B+B:C+0:D, k = A:C+'B:D,
l = A:D, m:=B:E, n=:A:E, [C. S. P.]
• t This algebra is diBtmgaishable into two, in the same manner as (c, ). Namely, if a = =b 2 , on sub-
stituting li^zl:tj, we have Z*:=0, ;7 = X;, lj:=: — ft, and the multiplication table is otherwise un-
changed. Otherwise, on substituting j\z::l + qf\ Z^ = fc + c^V , where 2c = — a ± -y a* — 4 , we have
y«=Z2— o,iZ=(l — c«)fc, ii=(l-c-»)A;, jn=(6+o)A;, In =: (b + c'^) k , and otherwise the multi-
plication table is unchanged. The following is a relative form for the first variety : i=.A:A + B:B
+ C:C+D:D,j=iA:B + B:D+C:D, k = A:D, l=iA:C—B:D, m=:A:E. n=iB:E+bC:E^
Forthe second variety,t = ^:-4 + B:B+C:C+D:i>,j = A:J5+(l—c«)C:D. k=:A:D. l = A:C
+ (1 — c-«)B:D, m = A:E. n=i(b+e)B :E+ {b+c''')C:E. [C. S. P.]
no
Peircb : Linear Aesodative Algebra.
{bd.) i
I m
n
Je
I
m
n
•
•
J
1
: k
1
I
m
n
•
k
1
'
m
k
1
1
1
I
ak
1
k
hm
[132*]. The defining equation of this case is
yn = o,
and there is no pure algebra in this case.
[13*]. The defining equations of this case are
/ = Aj, lj=h, jk=jl = Jg = ](?=kl=lk = P = 0,
which give
km =zkn=: 0.
There is a sextuple algebra in this case which may be called (ftej), of which the
multiplication table is *
^This algebra may be a little simplified by substituting j — Itorj, In relative form, i'^AiA
+ B:B+C:C+D:D, j = A:D+ B:C, k=zA:C, l = A:B, m=iA:E, n = hB:E+aD:E,
[C. S. P.]
i
Peirge : Linear Associative Algebra.
Ill
(6ce) i
I m
n
I
Tc
I
m
n
i
J
k
I
1
m
1
J
k
am
k
I
*
bm
1
1
1
1
[134]. The defining equations of this case are
There is a sextuple algebra in this case which may be called (6/e), of which the
multiplication table is *
(ft/a) i j * ^ ^ ^
k
I
m
n
i
J
1
k
I
m
n
•
J
k
m
k
III ■ 1
I
— k
a/m
1
1
♦Inrelatdveform,i = il:A + JB:B+C:C+D:Z), j = -A:B — C:D, h = A:D, l:=iA:C+B:D,
m = A:E, nzzB:E+aC:E. [0.8. P.]
112 Feirce : Linear Associative Algebra,
[14]. The defining equations of this case are
ij=:ji=ij, ik=z ki=zkj il=z K= ly im =:z m , ni^^rif mi = w = ,
which give
There are four cases defined as in [13].
[141]. The defining equations of this case are
which give
/ = Aj, jh = hi = h jl=}^=kl = lj=lk = P=0,
mn = d^l .
There is a sextuple algebra which may be called (bg^), of which the multipli-
cation table is*
{J>9>)
I m
n
%
h
I
m
n
1
1
I
m
•
m
J
h
I
k
I
I
1
I
n
I
[142]. The defining equations of this case are the same as in [132],
which give
*In relative form, i = ^ lil + B :B+C:C+/) :D, i = ^ : B + B :C+ CiD, k — A\C+B\D,
l^AiD, mz^AiE, n:=iE:D. [0. S. P.]
i
Peirce: Iaiwolt AseodoHve Algebra,
113
I
I
f
There is a sextuple algebra which may be called (6^), of which the multipli-
cation table is *
(hh^ i j k I m n
•
•
m
J
k
I
m
•
J
•
k
*
k
k
I
I
ak
k
m
k
n
n
[143]. The defining equations of this case are the same as in [13*]. There
is a sextuple algebra which may be called (6i^), of which the multiplication
table is f
{b%) i j Jc I m n
•
^
i 1
1
1
j
k
I
m
•
k
k
k
k
I
m
n
.k
n
1
* This algebra has two varieties, analogous to those of (Cg). The first is, in relative form, %z=.A\A
+ B:B+C:C+D:D,j=iA:B + B:C+A:D, k=A:C, l=i'-'A:B + D:C, m = A:E,n=iE:a
The second in relative form is the same except iih&tj=A : B + h'^D : C, l=iA:D — hB:C. [C. S. P.]
t This algebra may be slightly simplified by putting j — I for/. Then, in relative form, i=zA:A
+ B:B+C:C.j=B:C, k=iA:C,l=:A:B,m = A:D,n = D:C\ [C. S. P.]
114
Peircb : Linear Associative Algebra.
[14^]. The defining equations of this case are the same as in [134]. There
is a sextuple algebra which may be called {bj\), of which the multiplication
table is *
m i
I
m
n
I m
71
m
%
J
k
I
m i
1
m
k
,
1
j 1
k
I
' 1
— Te
.i -_
k
n
,
[15]. The defining equations of this case are
ij = ji = y , ik=^hi^=k, il= I, im = ?n , in := w , liz= mi = ni=^0 y
which give
j^ =1 kj =y^ =1 hf =z J^z=i IJ =z !k=z P =zlm=^ In=^ mj ^= mh =^ ml =^ m^
= mn = nj =z nk=z nl = nrn = n*.
There is a sextuple algebra which may be called {hk^, of which the multipli-
cation table isf
*%-A:A + B\B+C\C\D\D, j=A:B—C:D, k = A:D, l=iA:C+B:D, m = A:E,
n = E:D. [C. S. P.]
tin relative form, i = A:A + B:B+C:C, j=A:B + B:C, k = A:C,l=iC:D,m=:B:D,
n = A:D, [C. S. P.]
I
i
\
Pbirce : Linear Associative Algebra,
115
(bkfi) i j k I
m n
k
I
m
n
•
J
k
I
m
n
•
h
m
n
h
n
[16]. The defining equations of this case are
ij =ji^j\ ik=:ki=^k, ilznl^ im=^m, ni^^rij li=zmi=zin=^0,
which give
j^ =:kj =^jk =jn ==Jy'=zIi?=zhn:=lj=^lk=^P=^lm=^mj==^ mk z=:nl=- rr?
-^z nj '=' rJc '=^ rd -=- nm = tj?.
There is a sextuple algebra which may be called (hl^, of which the multipli-
cation table is *
(JZe) i j k I m n
•
m
t
j
Tc
I
m
•
J
•
J
h
m
h
h
I
Je
m
•
n
n
*In relative form, % = A:A + B :B + C:C, j=A:B + B:C, kzzAiC, l = B:D + A:E,
m — A:D,n=iE:C. [C. S. P.]
- i
116 Peibge: Lmear Associative Algebra.
[17]. The defining equations of this case are
ij z=:ji =y , iJc=:ki=^k^ il = l^ mi=z m , li=^ im = m = m = .
There is no pure algebra in this case.
[18]. The defining equations of this case are
ij :=ji =y , ik = kj il=::lj im=^m, in=^n, Jci=^li= 7ni = ni=^0 .
There is no pure algebra in this case.
[19]. The defining equations of this case are
ij =:zji=zj J ik=zk, ilzzzly im = m, ni^^n, i/n^=^kl=^li=zni=in.
There is no pure algebra in this case.
[l(y]. The defining equations of this case are
ij=jiz=zj\ ik=^k, il=^lj mi^=^m^ ni=^n, im = m = A?i = Zi = 0.
There is no pure algebra in this case.
[11']. The defining equations of this case are
i/=yi=y, ik=^k, iZ = Z, mi = ?n, im:=:K:=^in=^ ni=^0.
There is no pure algebra in this case.
[12']. The defining equations of this case are
ij:=zjiz=ij\ ik=:^k, li = l, il^zim =^ in=^ ki=^ mi=^ni=^0 .
There is no pure algebra in this case.
[13']. The defining equations of this case are
y=y, ik=^kf ilzizl, i7n=^7n, in = 71, ji^izki =:li:=:mi=::ni=^0.
There is no pure algebra in this case.
[14']. The defining equations of this case are
ijz=zj\ ik^='kj il=^l, im = m, ni:=^n, ji=:ki=^li =: mi^:=in=^ .
There is no pure algebra in this case.
[15']. The defining equations of this caee are
^y = y , ik=^kj il=zl^ mi:=my ni=^n, im=^ in =ji = Z?i = Zi =
There is no pure algebra in this case.
i
Peibce: Linear AseodcUive Algebra. 117
[16']. The defining equations of this case are
There is no pure algebra in this case.
[17']. The defining equations of this case are
ij :=y , iJc=::k, K =z I , mi := m ^ ilzzzim^in =^Ji = A?i = ni = .
There is no pure algebra in this case.
[18']. The defining equations of this case are
ij =y , ik=^k, li^=ly ji=zki = il =: im = in = Twi = m = .
There are six cases :
[18'1], when m^ =^ m , mn = n , nm == ,
[18'2], when m* = m , mn = , nm = n ,
[18'3], when m* =^n, mn ^ nm = , n* = m ,
[18'4], when m^ ^=-my mn = nm = w* = ,
[18'5], when m^ ^r-n , m^ = ,
[18'6], when7w*=w* = 0.
[18'1]. The defining equations of this case are
m^ = m, mn = n, nm = {).
There are two cases :
[18'P], whenwZ = 0;
[18'12], when ?nZ=/.
[18'1*]. The defining equation of this case is
7wZ= 0.
There is no pure algebra in this case.
[18'12]. The defining equation of this case is
ml = I ,
There are two cases :
[18'121], whenyrw=y;
. [18'12^], whenym = 0.
[18' 121]. The defining equation of this case is
jm =y.
There is a sextuple algebra which may be called (fewi^), of which the multipli-
cation table is *
Inrelativefonn,i=il:^, j = ^:jB, fcrzArO, l:=.B\A, m — BiB, n = B:C, [C. S. P.]
118
Peirge : Linear Associative Algebra.
{bm^) i
k
I m
n
k
I
m
n
•
•
Ic
1
•
•
1
Te
I
m
n
I
m
n
[18'12*]. The defining equation of this case is
ym = .
There is no pure algebra in this case.
[18'2]. The defining equations of this case are
TW = m , vnn ^ , nvn = n
There are two cases :
[18'21], whenmZ = Z;
[18'2»], when7nZ=0.
[18'21]. The defining equation of this case is
ml=il.
There are two cases :
[18'2P], whenym=y;
[18'212], whenym=0.
[18'21*]. The defining equation of this case is
There is no pure algebra in this case.
[18'212]. The defining equation of this case is
ym = .
There is no pure algebra in this case.
I
f
Peirce : Lmear Associative Algebra. 119
[18'2*]. The defining equation of this case is
There is no pure algebra in this case.
[18'3]. The defining equations of this case are
m* = m , mn = nm = , n^ =zn.
There is no pure algebra in this case.
[18'4]. The defining equations of this case are
m* = w , mn =: nm = w* = .
There are two cases :
[18'41], when ym=y;
[18'42], whenym = 0.
[18'41]. The defining equation of this case is
jm =y.
«
There is no pure algebra in this case.
[18'42]. The defining equation of this case is
y^w = .
There is no pure algebra in this case.
[18'5]. The defining equations of this case are
m* = 71 , m' = . •
There is no pure algebra in this case.
[18'6]. The defining equations of this case are
m* = 71* = .
There is no pure algebra in this case.
[19']. The defining equations of this case are
ij =zj J l-izizJcy ji =zik=z il = im = in = li = mi = m = .
There is no pure algebra in this case.
[2]. The algebras belonging to this case are not investigated, because it is
evident from § 69 that they are rarely of use unless combined with an idempo-
tent basis, so as to give septuple algebras.
Natural Classification.
There are many cases of these algebras which may obviously be combined
into natural classes, but the consideration of this portion of the subject will be
reserved to subsequent researches.
»
120 Peiroe : Linear Associative Algebra.
ADDEISTDA.
I,
On the Uses and Transformations of Lmear Algebra.
By B]E!Njamin Peirce.
[PreseiKted to the American Academy of Arts and Sciences, May 11, 1875.]
Some definite interpretation of a linear algebra would, at first sight, appear
indispensable to its successful application. But on the contrary, it is a singular
fact, and one quite consonant with the principles of sound logic, that its first and
general use is mostly to be expected from its want of significance. The interpre-
tation is a trammel to the use. Symbols are essential to comprehensive argument.
The familiar proposition that all A is 5, and all B is (7, and therefore all A is (7,
is contracted in its domain by the substitution of significant words for the
symbolic letters. The A, B, and G, are subject to no limitation for the purposes
and validity of the proposition ; they may represent not merely the actual, but
also the ideal, the impossible as well as the possible. In Algebra, likewise, the
letters are symbols which, passed through a machinery of argument in accord-
ance with given laws, are developed into symbolic results under the name of
formulas. When the formulas admit of intelligible interpretation, they are
accessions to knowledge ; but independently of their interpretation they are
invaluable as symbolical expressions of thought. But the most noted instance
is the symbol called the impossible or imaginary, known also as the square root
of minus one, and which, from a shadow of meaning attached to it, may be
more definitely distinguished as the symbol of semi-inversion. This symbol is
restricted to a precise signification as the representative of perpendicularity in
quaternions, and this wonderful algebra of space is intimately dependent upon
the special use of the symbol for its symmetry, elegance, and power. The
immortal author of quaternions has shown that there are other significations
which may attach to the symbol in other cases. But the strongest use of the
symbol is to be found in its magical power of doubling the actual universe, and
Peirge : Linear Associative Algebra, 121
placing by its side an ideal universe, its exact counterpart, with which it can be
compared and contrasted, and, by means of curiously connecting fibres, form
with it an organic whole, from which modern analysis has developed her
surpassing geometry. The letters or units of the linear algebras, or to use the
better term proposed by Mr. Charles S. Peirce, the vids of these algebras, are
fitted to perform a similar function each in its peculiar way. This is their
primitive and perhaps will always be their principal use. It does not exclude
the possibility of some special modes of interpretation, but, on the contrary, a
higher philosophy, which believes in the capacity of the material universe for
all expressions of human thought, will find, in the utility of the vids, an indica-
tion of their probable reality of interpretation. Doctor Hermann Hankel's
alternate numbers, with Professor Clifford's, applications to determinants, are a
curious and interesting example of the possible advantage to be obtained from
the new algebras. Doctor Spottiswoode in his fine, generous, and complete
analysis of my own treatise before the London Mathematical Society in Novem-
ber of 1872, has regarded these numbers as quite different from the algebras
discussed in my treatise, because they are neither linear nor limited. But there
is no difficulty in reducing them to a linear form, and, indeed, my algebra (63) is
the simplest case of Hankel's alternate numbers ; and in any other case, in which
n is the number of the Hankel elements employed, the complete number of vids
of the corresponding linear algebra is 2~ — 1 . The limited character of tKe
algebras which I have investigated may be regarded as an accident of the mode
of discussion. There is, however, a large number of unlimited algebras
suggested by the investigations, and HankePs numbers themselves would have
been a natural generalization from the proposition of § 65 of my algebra.*
Another class of unlimited algebras, which would readily occur from the
inspection of those which are given, is that in which all the powers of a vid are
adopted as independent vids, and the highest power may either be zero, or imity,
or the vid itself, and the zero power of the fundamental vid, i. e. unity itself,
may also be retained as a vid. But I desire to draw especial attention to that
class, which is also unlimited, and for which, when it was laid before the math-
ematical society of London in January of 1870, Professor Clifford proposed the
appropriate name of qtjuid/rates.
* This remark is not intended as a foundation for a Ciaim upon the Hankel numbers, which were
published in 1867, three years prior to the publication of my own treatise. — B. P. [They were given
much earlier under the name of clefs by Cauchy, and (substantially) at a still earlier date by Grassmann.
— C. S. P.]
I ■
I
ti
122 Peiboe: Linear Associative Algebra.
m
Qtiadrates.
The best definition of quadrates is that proposed by Mr. Charles S. Peirce.
If the letters A, J?, G, etc., represent absolute quantities, diflfering in quality, I
the vids may represent the relations of these quantities, and may be written in
the form
{A:A){A:B){A:G) . . . {B : A) {B : B) . . . {G:A), etc.
subject to the equations
{A :B){B:G) = {A: G)
{A:B)Ig:D) = 0.
In other words, every product vanishes, in which the second letter of the multi-
plier differs from the first letter of the multiplicand ; and when these ty?'o letters
are identical, both are omitted, and the product is the vid which is compounded
of the remaining letters, which retain their relative position.
Mr. Peirce has shown by a simple logical argument that the quadrate is the
legitimate form of a complete linear algebra, and that all the forms of the
algebras given by me must be imperfect quadrates, and has confirmed this
conclusion by actual investigation and reduction. His investigations do not
however dispense with the analysis by which the independent forms have
been deduced in my treatise, though they seem to throw much light upon their
probable use.
Unity.
The sum of the vids {A : A), {B : B), {G : (7), etc., extended so as to include
all the letters which represent absolute quantities in a given algebra, whether it
be a complete or an incomplete quadrate, has the peculiar character of being
idempotent, and of leaving any factor unchanged with which it is combined as
multiplier or multiplicand. This is the distinguishing property of unity, so that
this combination of the vids can be regarded as unity, and may be introduced
as such and called the vid of miity. There is no other combination which
possesses this property.
But any one of the vids {A : A), {B : B), etc., or the sum of any of these
vids is idempotent. There are many other idempotent combinations, such as
{A:A) + x{A:B), y{A:B) + {B:B),
h{A: A) + h {A: B) + h{B : A) + h{B : B),
which may deserve consideration in making transformations of an algebra
preparatory to its application.
\
Peircb: Linear Associative Algebra. 128
Inversion.
A vid which differs from unity, but of which the square is equal to unity,
may be called a vid of inversion. For such a vid when applied to some other
combination transforms it ; but, whatever the transformation, a repetition of the
application restores the combination to its primitive form. A very general form
of a vid of inversion is
{A yA) ±{B:B)±{C:G)± etc.,
in which each doubtful sign corresponds to two cases, except that at least one of
the signs must be negative. The negative of unity might also be regarded as a
symbol of inversion, but cannot take the place of an independent vid. Besides
the above vids of inversion, others may be formed by adding to either of them
a vid consisting of two different letters, which correspond to two of the one-
lettered vids of different signs ; and this additional vid may have any numerical
coeflBcient whatever. Thus
{A: A) + {B : B) — {G : G) + x{A: G) + y{B : C)
is a vid of inversion.
The new vid which Professor Clifford has introduced into his biquaternions
is a vid of inversion.
Semi' Inversion.
A vid of which the square is a vid of inversion, is a vid of semi4nversion .
A very general form of a vid of semi-inversion is
{A:A)±{B:B)±L J{G : G) ± etc.
in which one or more of the terms {A\ A), {B : B)^ etc., have J for a coeffi-
cient. The combination
{A:A)± J{B: B) + x{A : B) + etc.
is also a vid of semi-inversion. With the exception of unity, all the vids of
Hamilton's quaternions are vids- of semi-inversion.
7%c Use of Gommutative Algebras.
Commutative algebras are especially applicable to the integration of
differential equations of the first degree with constant coefficients. If i, y, /fe,
124
Pkircb : Linear Associative Algebra.
etc., are the vids of such an algebra, while x, y, 2, etc., are independent
variables, it is easy to show that a solution may have the form F{xi + yj + zh
4- etc.), in which i^is an arbitrary function, and i, y, k, etc., are connected by
some simple equation. This solution can be developed into the form
^(a^* + yj + zk + etc.) = Mi + iV;' + PAj + etc.
in which ilf, N, P, etc., will be functions of cc, y, z, etc., and each of them is a
solution of the given equation. Thus in the case of Laplace's equation for the
potential of attracting masses, the vids must satisfy the equation
*^+y* + A? = o.
The algebra (oj) of which the multiplication table is
%
Tc
Je
m
t
•
J
k
•
J
1c
k
may be used for this case. Combinations ii , /i , Aji of these vids can be found
which satisfy the equation
^+yf + At = o,
and if the functional solution
J^{^+yji + zk^)
is developed into the form of the original vids
Mi+NJ+Pk,
M, N, and P will be independent solutions, of such a kind that the surfaces for
which N and P are constant will be perpendicular to that for which M is
constant, which is of great importance in the problems of electricity.
The Use of Mixed Algebras.
It is quite important to know the various kinds of pure algebra in making
a selection for special use, but mixed algebras can also be used with advantage
)
I
Peibge: Linear Associative Algebra. 125
in certain cases. Thus, in Professor Clifford's biquatemions, of which he has
demonstrated the great value, other vids can be substituted for unity and his
new vid, namely their half sum and half difference, and each of the original
vids of the quaternions can be multiplied by these, giving us two sets of vids,
each of which will constitute an independent quadruple algebra of the same
form with quaternions. Thus if i,j\ h, are the primitive quaternion vids and
w the new vid, let
ai = i (1 + t£?) .
ij =1 a^i ,
Then since
•
hj\ = h = —Jih-
a.
= h{l
— w).
h
= Oii.
h
= <hj-
h
= a^.
a| :
= a,.
A-
=yi=
Ji=-
•«!•
i»J%-
=jc=
jth-
k^ii = ^» ^ ijAsg .
in which M^ denotes any combination of the vids of the first algebra, and N^ any
combination of those of the second algebra. It may perhaps be claimed that
these algebras are not independent, because the sum of the vids ai and oj is
absolute unity. This, however, should be regarded as a fact of interpretation
which is not apparent in the defining equations of the algebras.
II.
On the Relative Forms of the Algebras.
By C. S. Peircb.
Given an associative algebra whose letters are i, y, h, Z, etc., and whose
multiplication table is
^ = <hii + buj + Cii^ + etc.*
ij = <hsfi + *i«y + ^ + etc.
ji = ojjii + bjtij + c^ik + etc.,
6X/C., eiX3.
I proceed to explain what I call the relative form of this algebra.
I have used a, i , etc., in place of the Ox , etc.. used by my father in his text.
126 Peibge : Li/near Associative Algebra.
Let us assume a number of new units, A, I, J, K, L, etc., one more in
number, than the letters of the algebra, and every one except the first, A,
corresponding to a particular letter of the algebra. These new units are sus-
ceptible of being multiplied by numerical coefficients and of being added
together ; * but they cannot be multiplied together, and hence are called non-
relative units.
Next, let us assume a number of operations each denoted by bracketing
together two non-relative units separated by a colon. These operations, equal in
number to the square of the number of non-relative units, may be arranged as
follows :
{A :A) {A: I) {A :J) {A: K\ etc.
{I: A) {1:1) {I: J) (/riT), etc.
{J: A) {J: I) {J: J) {J: K), etc.
Any one of these operations performed upon a polynomial in non-relative units,
of which one term is a numerical multiple of the letter following the colon, gives the
same multiple of the letter preceding the colon. Thus, (/ : J) {al+ bJ + cK) = bl.\
These operations are also taken to be susceptible of associative combination.
Hence {I:J){J:K) = {I:K)] foT{J:K) K= /and (/:/)/= /, so that
{I:J){J:K)K=I. And {I:J){K:L) = 0] for {K: L)L= K emd {I:J)K
= (/:«7)(0.f/+Jr)=0./=0. We further assume the application of the
distributive principle to these operations ; so that, for example,
\{I:J) + {K:J) + {K: L)\{aJ+ bL) = aJL+ {a + b)K.
Finally, let us assume a number of complex operations denoted by i', /, Itf,
l\ etc., corresponding to the letters of the algebra and determined by its multi-
plication table in the following manner :
i'= {I:A) + an{I: I) + b,,{J: I) + c^{K
+ a^{I:J) + b^{J: J) + (^{K
+ a^{I: K) + hlJiK) + c^{K
/= {J:A)+ a^,{I: I) + b,,{J: I) + c,,{K
+ a^{I:J) + b^{J:J)^c^{K
+ a^{I: K) + b^{J: K) + (^{K
1(1=^ etc.
/)' + etc.
J) + etc.
K) + etc. + etc.
/) + etc.
J) + etc.
K) + etc. + etc.
Any one of them multiplied by giyes . t If &= , of coarse the result is 0.
\
Peiboe : Linear AssodcUwe Algebra. 127
Any two operations are equal which, being performed on the same operand,
•
invariably give the same result. The ultimate operands in this case are the non-
relative units. But any operations compounded by addition or multiplication
of the operations i\ /, Id, etc., if they give the same result when performed
upon A, will give the same result when performed upon any one of the non-
relative units. For suppose i'fA = TdHA . We have
I'fA = i'J= a^I + bi^ + c^K + etc.
KI!A = Ji!L = a^I+ b^-^- c^K+ etc.
so that aig = a34, b^=:b^, Ci, = C84, etc., and in our original algebra iJ=:kL
Hence, multiplying both sides of the equation into any letter, say m , ijm = Mm .
But
ijm = i {a^i + b^J + Cjjjfc + etc.) = {(hi(hi + (h^h^ +«i8C26 + ^tc.)i
+ (*n««6 + KK + K^b + etc.)y + etc.
But we have equally
i'fm'A = (aii%5 + a^b^ + a^^c^ + etc.)/+ (in««5 + ^lAs + ^la^'w + etc.) J+ etc.
So that Hfm'A = JdHm'A. Hence, i^fM= TdVM. It follows, then, that if i'fA
= VHA , then i*f into any non-relative unit equals TdT into the same unit, so that
iy= Idl'. We thus see that whatever equality subsists between compounds of
the accented letters i', j\ A/, etc., subsists between the same compounds of the
corresponding unaccented letters i,y, h, so that the multiplication tables of the
two algebras are the same.* Thus, what has been proved is that any associ-
tive algebra can be put into relative form, i. e, (see my brochv/re entitled
A brief Description of the Algebra of Relatives) that every such algebra may be
represented by a matrix.
Take, for example, the algebra {bd^. It takes the relative form
i = (/:iL) + (/:/) + (i:jr), y=(/:i),
h={K:A) + {J:I) + x{L:I) + {I:K) + {M:K) + x{J'.L) — {J:M)—x{L:M\
1={L:A) + {J:K), m = {M: A) + {f -I) {J: I) - {L: K) — f{J: M) .
* A brief proof of this theorem, perhaps essentiaUy the same as the above, was published by me in
the Proceedings of the American Academy of Arts and Sciences^ for May 11, 1676.
I
128
Peirgb: Lmear AssocicUive Algebra.
This is the same as to say that the general expression xi + yj + zk + 7il'\- vm
of this algebra has the same laws of multiplication as the matrix
0.
a.
u,
V,
0,
0,
+ (r»-l)»,
0,
Xz,
0,
0, 0, 0,
0, z, 0,
0, u, tz,
0, 0. 0,
0, X — V, 0,
0, 2, 0,
0.
0,
z — X^v,
0,
Vz
0.
Of course, every algebra may be put into relative form in an infinity of
ways ; and simpler ways than that which the rule affords can often be foimd.
Thus, for the above algebra, the form given in the foot-note is simpler, and so is
the following :
i={B:A) + {G:B) + iF:D) + {G:E), j={C:A),
k = {DiA) + {E:D)^'{G:B) + \{F: B) + x{G:Fl
/=(V:il) + ((7:2?), m-{E:A) + {x^—\){G:B)—{B:A) — {F:D)—{G:E).
These different forms will suggest transformations of the algebra. Thus, the
relative form in the foot-note to (6rf^) suggests putting
when we get the following multiplication table, where p is put for f^ :
4;
•
t
•
J
k
I
m
•
•
J
•
h
•
t
•
I
I
m
PV
fl
•
J
I
f
Pbircb: Linear Associative Algebra. 129
Ordinary algebra with imaginaries, considered as a double algebra, is, in
•relative form,
\={X:X) + {Y: F), J = (X: Y) — {Y:X).
This shows how the operation J turns a vector through a right angle in the
plane of X, T. Quaternions in relative form is
\=:{W: W) + {X:X) + {Y: Y)+{Z:Z),
i={X: W) —{W:X) + {Z: Y) — {Y:Z),
j={Y:W) -{Z:X) -{W:Y) + {X:Z),
k={Z:W) +{Y:X) —{X:Y) — {W:Z).
We see that we have here a reference to a space of four dimensions corres-
ponding to X, F, Zy W.
III.
On the Algebras in which Division is Unambiguous,
By C. 8. Peirce.
1. In the Linear Associative Algebra^ the coeflScients are permitted to be
imaginary. In this note they are restricted to being real. It is assumed that
we have to deal with an algebra such that from AB = AG we can infer that
il = or 5 = (7. It is required to find what forms such an algebra may take.
2. lfAB = 0, then either^ = or 5 = 0. For if not, AG = A{B + G),
although A does not vanish and G is unequal to B + G ,
3. The reasoning of § 40 holds, although the coefficients are restricted to
being real. It is true, then, that since there is no expression (in the
algebra under consideration) whose square vanishes, there must be an expression,
i, such that i* = i. .
4. By § 41, it appears that for every expression in the algebra we have
iA =z Ai=z A.
5. By the reasoning of §53, it appears that for every expression A there is
an equation of the form
i;^{a^A^) + U = 0.
But i is virtually arithmetical unity, since iA =^ Ai=: A-, and this equation may
be treated by the ordinary theory of equations. Suppose it has a real root, a ;
then it will be divisible by {A — a) , and calling the quotient B we shall have
{A — ai)B = 0.
I
* The idempotent basis having been shown to be arithmetical unity, we are free to use the letter t to
denote another unit.
130 Peirce : Linear Associative Algebra.
But A — ai is not zero, for A was supposed dissimilar to i . Hence a product of
finites vanishes, which is impossible. Hence the equation cannot have a real
root. But the whole equation can be resolved into quadratic factors, and some
one of these must vanish. Let the irresoluble vanishing factor be
(^ — 5)» + ^ = 0.
Then
or, every expression, upon subtraction of a real number (i. e. a real multiple of i),
can be converted, in one way only, into a quantity whose square is a negative
number. We may express this by saying that every quantity consists of a scalar
and a vector part. A quantity whose square is a negative number we here call
a vector.
6. Our next step is to show that the vector part of the product of two ^
vectors is linearly independent of these vectors and of unity. That is, i and j
being any two vectors,* if
ij = s '\- V
where 5 is a scalar and v a vector, we cannot determine three real scalars
a, b, c^ such that
v = a + bi + cj .
This is proved, if we prove that no scalar subtracted from ij leaves a remainder
bi + cf . If this be true when i and j are any unit vectors whatever, it is true
when these are multiplied by real scalars, and so is true of every pair of vectors.
We will, then, suppose i and y to be unit vectors. Now,
IJ =^ t.
If therefore we had
ij = a + bi + cj ,
we should have
— i=z ij^ =zaj + bij — c=ab — c + Vi + (a + bc)j ;
whence, i andy being dissimilar,
— i = bH,b^ = — l,
and b could not be real.
1
Peircb : Linear Associative Algebra, 131
7. Our next step is to show that, i and j being any two vectors, and
8 being a scalar and v a vector, we have
ji — r(8—v),
where r is a real scalar. It will be obviously sufficient to prove this for the case
in which i and/ are unit vectors. Assuming them such, let us write
ji = s/-^ t/ , W= ^'+ i/' ,
where ^ and «" are scalars, while i/ and i/' are vectors. Then
ij.ji = (6' -|- v) (^+ 9/) = ss^-\- si/-^ dv + i/'+ «".
But we have
V) .ji = ^;^^ := — i^ = 1 .
Hence,
But -y" is the vector of mf, so that by the last paragraph such an equation cannot
subsist unless x/' vanishes. Thus we get
= 1 — ss! — sf' — 8vl — 8fv ,
or
5t/= 1 — se! — e/'- — ^v.
But a quantity can only be separated in one way into a scalar and a vector part ;
so that
That is,
ji=-{s — v). Q.E.D.
8. Our next step is to prove that 5 = ^; so that if i/ = 5 + v then ji =
s — V, It is obviously suflBcient to prove this when i and j are unit vectors. Now
from any quantity a scalar may be subtracted so as to leave a remainder whose
square is a scalar. We do not yet know whether the sum of two vectors is a
vector or not (though we do know that it is not a scalar). Let us then take such
a sum as ai + hj and suppose x to be the scalar which subtracted from it makes
the square of the remainder a scalar. Then, G being a scalar,
{—x + ai-^-bjf^ a.
132 Peiroe: Lmear Associative Algebra.
I. e.
ah
(\ — ^\— G— 7? + a^ + V' — ab8 — ah^ + 2axi + 2bxf.
But V being the vector of ij, by the last paragraph but one the equation must
vanish. Either then i; = Oorl = 0. But if v = , ^y = «, and multiply-
ingintoy,
— i = &y,
which is absurd, i and J being dissimilar. Hence 1 = and
ji = s — v. Q.E.D.
9. The number of independent vectors in the algebra cannot be two. For
the vector of ij is independent of i andy. There may be no vector, and in that
case we have the ordinary algebra of reals ; or there may be only one vector,
and in that case we have the ordinary algebra of imaginaries.
Let i and j be two independent vectors such that
iJ =: 8 + V .
Let us substitute for j
j\ = 6i+j.
Then we have
iji=v, j\i = —v,
Ji^=j\Vi = —Jii = i, VJi = ifi = —i^
iv = i^j\ = —j\ , vi = ij\i = —j\^ =j\ .
Thus we have the algebra of real qvatemions. Suppose we have a fourth unit
vector, k , linearly independent of all the others, and let us write
ki = ^'+ ^'.
Let us substitute for k
h = ^H + s^j\ + k,
and we get
Jih = — «"v + vf, kji = sf'v — t/,
kit = — sfv + t/', ik^zzie/v — t/'.
But developing the square we iiave I
(— X + ai + hjf = 7^ — a* — b^ + abs + absf — 2axi+2bxj + aJb(l — ^^=0', |
1
Peirce : lAnear AssocicUive Algebra. 133
Let us further suppose
iVi) h = ^^^+ t/".
Then, because ij\ is a vector,
k, {y\) = ^^ i/".
But
because both products are vectors.
Hence
i .j\k^ =—%. }c^j\ = — iJc^ .j\ = k^i .j\ = ki . ij\ .
Hence
«'"+ t/''=: 5"'— i/"
or t/"= , and the product of the two unit vectors is a scalar. These vectors
cannot, then, be independent, or k cannot be independent of ij =zv. Thus it is
proved that a fourth independent vector is impossible, and that ordinary real
algebra, ordinary algebra with imaginaries, and real quaternions are the only
associative algebras in which division by finites always yields an unambiguous
quotient.
I
i
1^
I .
4
i
I
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