390 KANSAS UNIVERSITY SCIENCE BULLETIN. 



For this property has been established for the ele- 

 ments a, 6, .... as members of K[i(; a]. Therefore 

 by the distributive law it holds generally. 



34. Corollary 3. The higher rule is commutative. 

 For this also has been proved in the class K [tf □ ]. 



35. Corollary U- The set of symbols defined in § 22 

 forms an abelian semigroup with reference to the 

 higher rule. 



36. Scholium. Beginning with any a> w we have 

 abelian semigroups with regard to both rules, and so 

 beginning with w for one or the other rule, but no part 

 beginning with w forms a semigroup on both rules. 



37. Proposition XIV. Necessary and sufficient con- 

 ditions that a set M constitute an abelian semigroup 

 with respect to each of two rules of combination, one 

 distributive over the other, that M contain an arbitrary 

 w and that M be as generated an ordered set, are the 

 rules of combination thus far defined. 



38. Corollary. The properties in question may be 

 expressed by the following postulates: 



A. There shall be a higher and a lower rule of com- 

 bination. 



B. Combinations of symbols shall be so ordered that 

 those made by the lower rule precede combinations of 

 the same symbols by the higher. 



C. The higher rule shall be distributive over the 

 lower. 



D. There shall be a set M containing the symbol w . 



E. M shall form an abelian semigroup for the lower 

 rule. 



F. M shall form an abelian semigroup for the higher 

 rule. 



G. Every successive set generated according to re- 

 quirements shall be an ordered set. 



