On DR-semigroups satisfying the ample conditions

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 $$xD(y)=D(xD(y))xandR(y)x=xR(R(y)x)\mathrm{.}$$ 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 generalized ample conditions, but we give other examples using strongly order-preserving functions on a quasiordered set as well as so-called double demonic" composition on binary relations. 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 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.