FRIZELL: FOUNDATIONS OF ARITHMETIC. 391 



39. Scholium. M is a well ordered set with no 

 modulus. 



40. We postulated two rules of combination for the 

 symbol w , but only one for the lower symbol u . If 

 we should postulate a higher rule for u by the same 

 requirements as for iv the resulting set of symbols 

 would not differ as regards any group property from 

 the ^{;-set ; the new abelian semigroups would be re- 

 spectively holoedrically isomorphic with the former. 

 If we postulate no higher rule the set K[iio] still 

 forms an abelian semigroup holoedrically isomorphic 

 with K[i(;o]. 



41. Instead let us set up postulates for the u class 

 of symbols differing from those of § 38 only in adding 

 another postulate H. The symbol u shall be a modulus 

 for the higher rule, with the restriction on A-G that 

 they shall not conflict with H. Then the class K [^i^] 

 reduces to the single element u , the abelian semigroup 

 on the two rules reduces to the set K[^to] and postu- 

 late B drops out, since every combination by the higher 

 rule is found among those made according to the lower 

 rule. This new set of postulates may be replaced by 

 the equivalent set. 



42. a. There shall be a higher rule defined from a 

 lower by the inductive formulas {uoa)h = uhoah and 

 a(uoh) = auoah. 



b. There shall be a lower rule defined inductively by 

 the formula uoaob = uo{aoh). 



c. There shall be a set M containing the symbol u . 



d. The set M shall possess the group property for 

 every combination uoa. 



e. M shall be an ordered set. 



f . There shall be a set M ^ built up by postulates c, d 

 and e on the symbol uu . 



g. The sets M' and M shall be identical. 



