FRIZELL: FOUNDATIONS OF ARITHMETIC. 403 



For just as the removal of restrictions on the higher 

 rule carried our series of symbols beyond the finite set 

 of type w so the introduction of a still higher rule, cor- 

 responding to involution, will extend it beyond the set 

 defined in ^ 22 if we impose no restrictions on the new 

 rule except that of order and the special form of the 

 group property. 



87. Let us postulate a rule expressed by a^ higher 

 than the two preceding, that is, the combination a b 

 shall precede a*". To distinguish we will now call ab 

 the lower and a o b the lowest rule. 



The lower rule has been shown to be associative and 

 commutative for the set K [ w n ] , which forms an 

 abelian semigroup with regard to it. A set of symbols 

 shall be generated from w by the higher rule in accord- 

 ance with the postulate of order and the restricted 

 form of the group property for w'', where a is a pre- 

 viously defined element. By Prop. I this set forms a 

 series 10,10"^ = w\ w^' == w" , w'^"w"', . . . and now w' 

 must follow every w^ (N = 2, 3, . . . ) [w^ is not a 

 combination by rule, the N is a mere index | . Then 

 by § 22 we obtain a series holoedrically isomorphic with 

 the set there considered if to the lowest and lower rules 

 respectively we make the lower and higher correspond. 



88. The articulation of the two lower rules in § § 28-35 

 leaves room for much freedom in definition. There we 

 defined the lowest rule first and made the lower de- 

 pend upon it. Here we have already defined the lower 

 rule to the extent implied in the application of § 22. 

 This, however, involves no further restriction than the 

 inductive associative formula of ^ 19. But the defini- 

 tion from the lowest rule carries this property with it. 



It is therefore permissible to define the lowest rule 

 so that the lower shall be distributive over it. Then 

 by the process of § 22 new combinations are made by 

 the higher and lowest rule together, from each series 



