394 KANSAS UNIVERSITY SCIENCE BULLETIN. 



{n',r') = (m,q)& (n,r)y or equals by equals give equals 

 and denotes a C-rule. 



3) The fundamental group property is secured by 

 the definition. To prove the associative law we have 

 (Mj n) Q \ (p^ q) Q {r, s) \ = {mopor,noqos) = (m,n)Q 

 ip,q)(^(r,s). Suppose that (m',q')Q{n,r) = {m,q)G 

 (n,r). Then {m' oriyq' or') = {mon,qor) by defi- 

 nition. Hence {m' on) o {qor) = {mon) o {q' or) 

 and {m' oq)o{nor) = {moq')o{nor). Therefore 

 m' oq = moq' hy %1L That is (m', g') = (m, g) and 

 similarly for the other form of this property. Thus C 

 is a semigroup. But the element {m,m) is a modulus 

 and to every in,q) corresponds an inverse (q,n). 

 Therefore by XIX, C is a group, w^hich v^e v^ill denote 

 byG. 



Finally {n,r)Q{m,q) = {nom,roq) 

 — {mon,qor) — {m,q)o{n,r). 

 Therefore by § 16, G is an abelian group. 



52. The semigroup C and group G are connected by 

 Proposition XXL If v^e declare (m o q, q) = m then 

 {moq, q)Q(nor, r) = mon. For {moq, q)Q(nor,r) 

 = {moqonor,qor) = monoqor,qor)=mon. 



53. Scholium. The group G contains a semigroup 

 K holoedrically isomorphic with C in such manner that 

 the rule © becomes identical with the rule o for K. 



54. Obviously Prop. XX may be applied to the set 

 of natural numbers in two different ways according as 

 we use the semigroup on addition or that on multipli- 

 cation. An essential distinction between these two 

 procedures is furnished by 



55. Proposition XXII. Given two rules of combi- 

 nation of which one is distributive over the other, no 

 set of symbols can form a group with respect to both 

 rules. For such a set would contain a modulus v for 

 the lower rule (XVI). Then by the distributive law 



