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 $$\begin{aligned} xD(y)=D(xD(y))x{ \text{ and } }R(y)x=xR(R(y)x). \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>x</mml:mi> <mml:mi>D</mml:mi> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>D</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mi>D</mml:mi> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> <mml:mi>x</mml:mi> <mml:mrow> <mml:mspace/> <mml:mtext>and</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mi>R</mml:mi> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi>x</mml:mi> <mml:mi>R</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> 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.