BRACKET FUNCTIONS ON GROUPOIDS

Abstract

In this paper, we introduce an operation denoted by [Bre], a bracket operation, which maps an arbitrary groupoid (X, ∗) on a set X to another groupoid (X, •) = [Bre](X, ∗) which on groups corresponds to sending a pair of elements (x, y) of X to its commutator xyx−1y−1. When applied to classes such as d-algebras, BCK-algebras, a variety of results is obtained indicating that this construction is more generally useful than merely for groups where it is of fundamental importance.