2025/04/29 by Stokes, Tim
#20M30 #20M50 #Category Theory (math.CT) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2504.20397
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))x and R(y)x=xR(R(y)x). 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.