On multiplicative systems defined by generators and relations

Abstract

It is the purpose of this paper to study the properties of multiplicative systems, for which the associative law is not assumed, when these systems are given in terms of generators and relations. We confine ourselves mainly to loop theory, although the general theory holds also for groupoids, groupoids with division on one side, and quasigroups. Throughout the paper we are guided by two main considerations, to discover how far the concepts and results of group theory carry over to the non-associative case, and to exhibit a specific example of some of the fundamental concepts of abstract algebra. In many ways, in the general theory, we are able to obtain more complete results than in group theory. There remain, however, many interesting analogues of group theoretical concepts. It is hoped to deal with some of these in a later paper.