386 KANSAS UNIVERSITY SCIENCE BULLETIN. 



elements of K [i^ n ] . Hence the totality of symbols 

 forms a well ordered set. 



13. Proposition 11. A rule of combination is asso- 

 ciative throughout a given K-class if 



ao{hoc)=aohoc 

 for every a , 6 , c in K. 



14. Definition. A K-class is said to form a semi- 

 group with regard to a C-rule associative throughout K 

 if by this rule unequals with equals always give une- 

 quals or if from the relation a'oh = aoh, resp. aob'= 

 a oh we can always infer a' = a resp. h' = h. 



15. Proposition III. No semigroup contains more 

 than one modulus for its defining C-rule. For if u and 

 u' not = n were both moduli, we should have aou' = 

 a = aou contrary to § 14. 



16. Definition. An abelian K-class with reference 

 to a C-rule is one for which the rule is without excep- 

 tion commutative. 



17. Proposition IV. Sufficient conditions of an 

 abelian class are the relations 



ao(boc) = aohoc and boa = aoh 

 for every a, b, c in the class. 



18. Proposition V. A C-rule is distributive over 

 another in a given K-class if the relations 



{aoh)c = acobc and a{boc) =^aboac 

 are satisfied by every a, 6, c in K. 



19. Proposition VI. A necessary and sufficient con- 

 dition of the generating rule of an e-set being associ- 

 ative for the set is the inductive formula of definition. 



eoaob = eo(aob). 

 Proof. If the relation of ao(boc) = aob oc has been 

 established for a certain a and every b , c of the set, 

 then by hypothesis eoao(boc) = eo\ao{b oc)\ = 

 eo{aoboc\ = eo(aob)oc = eoaoboc. 



