ESN-style theorems beyond congruence conditions (cat-semigroup case)

Develop an Ehresmann–Schein–Nambooripad-style representation and characterization theorem for DR-semigroups that satisfy the cat-semigroup condition (i.e., R(x) = D(y) implies D(xy) = D(x) and R(xy) = R(y)) but may fail the congruence conditions, by identifying an appropriate categorical or partial categorical structure that fully captures their semigroup operation and order-theoretic features.

Background

The paper surveys ESN-style characterizations for classes of biunary semigroups where both congruence conditions hold (Ehresmann and DRC-semigroups). It then discusses extending ESN-style results to broader classes that do not satisfy these congruence conditions.

A natural intermediate assumption is the cat-semigroup condition, which ensures that the cat-product yields a category. However, the authors note that even under this condition, obtaining ESN-style theorems in full generality remains unresolved. The paper proceeds to develop ESN-style results for the subclass satisfying the ample conditions, leaving the more general cat-semigroup case open.

References

However, so far no ESN-style theorems of this type have been developed for classes of DR-semigroups not satisfying at least one of the two congruence conditions (generally, both are required, but see [10] where only one is). Thus, in seeking ESN theorems for DR-semigroups that are not congruence, one approach would be to work with those at least satisfying the cat-semigroup condition, since then we get a category from the cat-product; however, it is not clear how to then obtain ESN-style theorems in general.

— On DR-semigroups satisfying the ample conditions  (2504.20397 - Stokes, 29 Apr 2025) in Section 4.2 (The (cat,trace)-product and partial categories)