388 KANSAS UNIVERSITY SCIENCE BULLETIN. 



given Mb form together an infinite, well ordered 

 K-class as regards the rule o . For the set of b's : 



Mb, Mhoiv^^\ ,Mbow^^"''\ .... 

 forms a well ordered K-class between consecutive mem- 

 bers of which the b-sets are interpolated. 



25. Proposition X. The totality of the b-sets forms 

 an infinite, well ordered K-class as regards the rule o . 

 For the set of Mb belonging to a given -6 is a well or- 

 dered K-class and so is the whole set of 6's, the latter 

 being simply K[u'n]. 



26. New combinations of symbols already defined 

 are defined inductively in accordance with the formulas 



b o a = a o b and 

 a o (b o c) = a o b o c. 



This secures the group property for the whole set and 

 the reasom'figof F/and F// establishes 



Proposition XL The set of symbols generated ac- 

 cording to two rules, a higher and a lower, from a single 

 arbitrary symbol iv by aid of the postulate of order, 

 the restricted form of the group property and the above 

 inducted formulas of definition, is a well ordered, infi- 

 nite set constituting an abelian semigroup with regard 

 to the lower rule. 



27. Scholium. The higher rule remains unre- 

 stricted and has not been defined beyond the set 

 K [ w; n ] , therefore is not yet a C-rule for the K-class 

 of § 26. 



28. Definition. The higher rule shall be defined 

 inductively throughout K [ 'w; n ] by the formula 



w ab = w a ab, 



where a, b are any members of K [ t<; n ] for which a b 

 has already been defined. 



29. Proposition XII. The class K [ w u ] forms an 

 abelian semigroup with regard to its generating rule. 

 Proof by Propositions VI and VII, 



