392 KANSAS UNIVERSITY SCIENCE BULLETIN. 



43. Proposition XV. The set of symbols defined 

 by the above seven postulates constitutes an abelian 

 semigroup for each rule and has a modulus for the 

 higher rule, which is distributive over the lower. 



Pivof. The abelian semigroup on the lower rule is 

 established by Prop. VII, the distributive law follows 

 from XIII and the other semigroup by § 35. It re- 

 mains to prove the existence of a modulus. 



Let e = uu. By postulate g, e must occur in the set 

 K[?io] : II, u' , u" , . . . 



Suppose e = u" . Then every member of K [eo] will 

 be found in K[?/^o] but beyond n' . That is, u and u' 

 are not in K[eo], contrary to g. Therefore the only 

 possibility of satisfying g is e=^u. Since, then, 

 un = u, it follows by the distributive law that u is a 

 modulus. 



44. Postulates a . . . g define the natural num- 

 bers as ordinal symbols and by XV contain all the laws 

 of their arithmetic. The cardinal numbers may be de- 

 fined in a manner now familiar as names of classes, 

 e. g., ''five" is the name given to the class of all well 

 ordered sets which are ordinally similar to the set u , 

 u', u" , u'" , u^\ 



45. Definition. A group is a semigroup which con- 

 tains, corresponding to every a, l>, in it, symbols p 

 and q such that aop = h = qoa. 



46. Proposition XVI. Every group contains a 

 modulus with respect to its defining C-rule. 



47. Definition. If a class which has a mod- 

 ulus u for its defining C-rule contains symbols a , a 

 such that aoa = u = aoa then a , a are said to be 

 each the inverse of the other. 



48. Proposition XVII. No member of a semigroup 

 can have more than one inverse in the semigroup. For 

 otherwise equals with unequals would give equals. 



