384 KANSAS UNIVERSITY SCIENCE BULLETIN. 



a ) the result of comparison be uniquely determined 



b ) the statements b = a and a = b shall be interchange- 



able 



c ) from a = b and b = c must follow a = c 



d ) a = a if K contains no other symbol equal to a. 



2. Definition. An infinite set of symbols is one 

 which contains parts that can be put into one corre- 

 spondence with the whole. A finite set is one that is 

 not infinite. 



3. Definition. An ordered set is one in which we 

 are always able to say either that a precedes b (a < 6 ) 

 or that a follows b {a>b) subject only to the restric- 

 tions that 



a ) the result of comparison is uniquely determined 



b ) a < 6 shall exclude a = b but involve b>a 



c ) it a<b and b<c, then a<c 



d) if a' — a and b' — b, then a' < b' or a' > b' according 



as a < 6 or a > 6. 



4. Definition. An ordered set is said to be well 

 ordered when it has a first element and every subset 

 beginning with the first element has an immediate suc- 

 cessor in the given set. 



5. Definition. A procedure whereby to every a, b 

 of a K-class is assigned a definite symbol c= aob will 

 be called a C-rule or rule of combination provided that 

 by this assignment equals with equals give equals. 



6. Definition. A K-class is said to possess the 

 fundamental group property with respect to a C-rule 

 when a o 6 also belongs to K. 



7. Definition. Modulus of a K-class with regard to 

 a C-rule is a symbol uotK such that aou = a = uoa 

 for every a in K. 



8. Proposition I. An ordered set defined by the 

 requirements : a ) it shall contain a given symbol e ; 

 b ) it shall possess the group property for every combi- 



