400 KANSAS UNIVERSITY SCIENCE BULLETIN. 



Also in like manner v + v = v = vv . 



In the set K|h, v| the associative, commutative 

 and distributive lav^s and the first group property are 

 all preserved. The law of equals with unequals is 

 violated for multiplication but still holds for addition, 

 for which v is a modulus. 



78. Since the introduction of infinity would destroy 

 both semigroups while zero destroys only that on mul- 

 tiplication, it seems preferable to admit zero to our 

 arithmetic and exclude infinity. But this removes the 

 only objection to enlarging the addition semigroup into 

 a group. This is effected at a stroke by applying Prop. 

 XX to the set X and thus building an abelian group 

 with reference to addition. In this group we then de- 

 fine a higher rule by the formulas 



aob — ah = a&T> and aeT) = al>. 



This completes the system of real numbers, forming 

 an abelian group on addition and, when we exclude its 

 modulus, an abelian group on multiplication, which is 

 distributive over addition. The system of real numbers 

 can be simply ordered according to the lower rule, but 

 the abelian group on the higher rule can not be simi- 

 larly treated. 



79. We might have proceeded by first applying XX 

 to the natural numbers as a semigroup on addition. 

 This yields the whole set of integers, positive, negative 

 and zero. Omitting zero and defining by the law of 

 signs we should have an abelian semigroup on multi- 

 plication. Applying XX to it, there results an abelian 

 group composed of all rational numbers except zero. 

 Then the introduction of the limits would supply the 

 whole set of irrational numbers and close by reintro- 

 ducing zero. 



80. As long as we postulate two rules of combina- 

 tion, one distributive over the other, and demand semi- 



