On Regularity of Semigroups of Maps that Preserve a Binary Relation

Abstract

We consider semigroups of full transformations, partial maps, and multi-value maps from a set X to X, where every map preserves a binary relation σ defined on X. We suggest several definitions for the preservation of σ for partial maps and for multi-value maps. We review results about regularity of the semigroups mentioned above in the cases where σ is a partial order, a quasi-order, an equivalence, or one of some special kinds of binary relations. Also we consider the question about regularity of the semigroup of full transformations that preserve a partial order and an equivalence simultaneously.