396 KANSAS UNIVERSITY SCIENCE BULLETIN. 



Obviously (??, r)o(m, q) = (m, q)o (n, r) . Thus 

 we have an abelian semigroup and the distributive 

 principle follows by taking equals with equals. 



59. Definition. The symbols of G shall be ordered 

 according to the convention that ( m , q ) shall precede 

 or follow {n, q) according as m precedes or follows n 

 and by ^ 3. 



60. Scholium. If a, h denote two natural numbers 

 that have no common factor other than unity, the set 

 of symbols (a, />) is simply ordered by the above defi- 

 nition. This set, which we will denote by R, may be 

 taken as representative of G since every element of G 

 is equal to some member of R. 



61. Definition. The symbols constituting the set R 

 are called absolute, rational numbers. 



62. Proposition XXIV. Every two unequal ele- 

 ments of G are connected by a relation h = g ox where 

 h, g denote the given elements, g being that which 

 precedes. 



For g = (m, q) = (mr, qr) and h = (n, r) = (nq, qr) 

 and mr precedes nq hy hypothesis. Therefore h = gox 

 is satisfied by a? == (d, qr) where nq = mr od. 



63. Corollary. The set R is simply ordered accord- 

 ing to the lower rule denoted by o . 



64. Proposition XXV. If we exclude the modulus 

 u the remaining symbols of R form a set which is sim- 

 ply ordered according to the higher rule. For let x 

 denote any element of R which precedes and y any one 

 which follows u . 



Then gx precedes and gy follows g'. That is, the 

 trios u,g,gx and u,g,gy are ordered whether g 

 precedes or follows u . 



65. Corollary. The modulus u separates the re- 

 maining symbols of R into two simply ordered classes, 

 neither of which contains either a first or last element. 



