386 KANSAS UNIVERSITY SCIENCE BULLETIN. 



elements of K[?('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(6oc)= aoboc 

 for every a, b, 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 = aob, resp. aob'= 

 aob we can always infer a ' = a resp. h' = b. 



15. Proposition III. No semigroup contains more 

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

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

 a = aou contrary to i? 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(6oc) = ao6oc and boa = aob 



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 



(aob)c = acobc and a(boc) = aboac 



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



19. Proposition VI. A necessary and suflficient con- 

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

 ative for the set is the inductive formula of definition. 



eoaoh = eo{aoh) . 



Proof. If the relation of ao(hoc) = aoh oc has been 

 established for a certain a and every /> , c of the set, 

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

 eo\aohoc\ = eo{aoh)oc — eoaohoc. 



