Towards Commutator Theory for Relations

In a general algebraic setting, we state some properties of commutators of reflexive admissible relations. After commutator theory in Universal Algebra has been discovered about thirty years ago [12], many important results and applications have been found. An introduction to commutator theory for congruence modular varieties can be found in [2] and [3]. Shortly after, results valid for larger classes of varieties have been obtained in [4] and [8]. More recent results, as well as further references, can be found, among others, in [1, 6, 10, 11]. Present-day theory deals with commutators of congruences. However, the possibility of a commutator theory for compatible reflexive relations has been voiced already in [8, p. 186]. Indeed, as noticed in [9] (in part independently in [5]), some notions from classical commutator theory can be extended to relations. If A is any algebra, and R,S are compatible and reflexive relations, define M(R,S) to be the set of all matrices of the form