892 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 lov^er. 



Proof. The abelian semigroup on the lower rule is 

 established by Prop. VII, the distributive law follows 

 from XlII 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[^^o] : ^6, u' , u" , ... 



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

 be found in K[t6o] but beyond u' . 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, 

 uu = u, it follows by the distributive law that i^ 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, h, 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 tt 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. 



