On DR-semigroups satisfying the ample conditions

Abstract

A DR-semigroup S (also known as a reduced E-semiabundant or reduced E-Fountain semigroup) is here viewed as a semigroup equipped with two unary operations D, R satisfying finitely many equational laws. Examples include DRC-semigroups (hence Ehresmann semigroups), which also satisfy the congruence conditions. The ample conditions on DR-semigroups are studied here and are defined by the laws (Formula presented.) Two natural partial orders may be defined on a DR-semigroup and we show that the ample conditions hold if and only if the two orders are equal and the projections (elements of the form D(x)) commute with one-another. Restriction semigroups satisfy the ample conditions, but we give non-restriction examples using closure operators on sets, strongly order-preserving functions on a quasiordered set, and certain subsets of partial categories. Following the work of Stein, we show how to construct a certain partial algebra C(S) from any DR-semigroup, which is a category if S satisfies the congruence conditions, but is “almost" a category if the ample conditions hold. We then characterise the ample conditions in terms of the equality of two natural partial orders, and in terms of a converse of the condition on S ensuring that C(S) is a category. Our main result is an ESN-style theorem for DR-semigroups satisfying the ample conditions, based on the C(S) construction. We also obtain an embedding theorem, generalizing a result for restriction semigroups due to Lawson.