•60 



Algebra g4, i = a S. /3 y ; j = aS.ya{); fc = /3S./3y(); l = (iS.ya{). 

 " hp„i=l^S.{)A.lJj2—l3S.{)A.l,l,l,; j = — l^S.()A.lJ,l,- 

 k = -l,S.( )A. /j/Ji - ^3 S.^ )A. 1,1. J, ; 1=1, S.( )A.Ul,U ; 

 m=—l,S.{ ) A.l,U^. 

 hk„i = l,S.{) A. i,/^/, - /3 S. ( ) A. /./Ji + /3 S. ( ) A. }JJ, ;j = - 

 l, S. ()A.yj, + l,S.{ )A.lJ,l,- k = l,8.( )A.IJ,I,- 

 l = -l,S.()A.l,l,l,; m = -l,S.{ )A.l,I,l,; n = -l,8. 

 ()A.lJ,l,. 

 bme,i = aS.i3}();j = aS./a();A:=aS.a^();/=/?S.,5y();m = 

 ,3S.>'a(); n = /3S.a/3(). 

 These examples can be used to illustrate the general theorems. For example: 

 " Every group of linear vector opei-aiors contains at least one idempotent or one 

 nilpotent expresssioti." 



The group bm^ contains the idempotents 



a S. /?>'(), /3S. >'a(), aS. /?7()+/3S. yaO. 

 The group b p^ contains only nilpotents. 



" When an algebra contains an idempotent expression it may be assKined as the 

 basis and tlie remaining expressions are then divisible into four classes." 



In b mg if we assume a S. /? 7 ( ) as the idempotent then the units are, with 

 reference to the basis, 



idemfaciend, idemfacieiit, a S. /? y ( ) ; 

 nilfaciend, idemfacient, /? S. /? y ( ) ; 

 idemfaciend, nilfacient, a S. y a ( ), and a S. a (3 () ; 

 nilfaciend, nilfacient, /? S. y a ( ), and /? S. a /5 ( ). 



" The fourth class are subject to independent investigation." 



"If the first class comprises any units except the basis, there is, besides the basis, another 

 idempotent expi-ession or a nilpotent expression, and we may free the class from this, when 

 idempotent, by writing for the basis the difference between the two ; in this case expressions 

 may pass from idemfaciend to nilfaciend or from idemfacient to nilfacient, but not the 

 reverse." Thus, if we had taken for our basis in b mg a S. /? y ( ) + Z' S. y n ( ) there 

 would have been only two classes, 



1 : a S. /J y () + /? S. y a ( ) ; /? S. y a ( ) ; a S. y a ( ) ; |3 S. /3 y ( ) ; 



2: «S. a/3(); /? S. « /3 ( ). 



The second idempotent basis is easily seen to be (3 S. y a ( ), and the differei.ce 



is a S. /? y ( ), as before. And making this change of basis, /?S. y « ( ) and (iS.a(3{ ) 



become fourth class, /3 S. j8 y ( ) becomes second class, aS.y a{) becomes third class. 



" When there is no idempotent basis, all expressions are nilpotent, and all poivers of 



each expression that do not vanish are independent. We may take any expression as the 



