Union and Symmetry Preserving Homomorphisms of Semigroups of Closed Relations

Abstract

The symbol βX denotes the semigroup of all binary relations on a nonempty set X under composition which is defined by αoβ={(x,y)} ∈ X×X: (x,z) ∈ β and (z,y) ∈ α for some z∈X} for all α,β ∈ βx . In a recent paper [1, Theorem 3, p. 310], A. H. Clifford and D. D. Miller initiated a study of the endomorphisms of βX when they completely determined those which preserve unions and take symmetric relations into symmetric relations. The purpose here is to place the theorem of Clifford and Miller in a topological setting and to discuss some of the problems which then arise naturally. The full results will appear in [9].