A Ternary Analogue of Abelian Groups

Introduction. A class K with an operation called multiplication applied to pairs of elements of K is an abstract group provided certain postulates are satisfied. Unfortunately the name group does not suggest the binary character of multiplication. The entities with which this note is concerned are similar to groups, the class K being subjected to a ternary operation however. For want of a more descriptive name we have called them triplexes. The need for their consideration arose in an attempt to obtain solutions of a pair of functional equations, but their investigation would seem justified by intrinsic interest especially when compared with Abelian groups. The system of postulates on which we base our investigation is modelled after Hurwitz's t system for Abelian groups, and accordingly differs somewhat from the definitions of a group found in most treatises on group theory. In this way we reduce the proofs of several theorems to a minimum and have at the same time a more perfect system from a strictly logical. point of view. The role that Abelian groups play in this theory is described in ? 3. The rest of the paper deals with finite triplexes and concepts analogous to the fundamental notions of the theory of Abelian groups such as order and inverse of an element, sub-group, cyclic group, quotient group, etc. Two notions however are conspicuous by their absence, the unit and the basis. Other facts such as the existence of triplexes with no subtriplex stand out as being different from what one might expect from the theory of Abelian groups.