Ordered groupoid quotients and congruences on inverse semigroups

TL;DR

This paper introduces a new preorder and equivalence relation on inverse semigroups, leading to a natural ordered groupoid structure on quotients, which helps classify congruences and factorize homomorphisms.

Contribution

It presents a novel preorder and equivalence relation on inverse semigroups, enabling a structured approach to quotienting and classification of congruences.

Findings

Constructed a preorder between natural partial order and Green's J-relation.

Defined an equivalence relation that yields an ordered groupoid upon quotient.

Provided a factorization of inverse semigroup homomorphisms into quotient and star-injective functors.

Abstract

We introduce a preorder on an inverse semigroup $S$ associated to any normal inverse subsemigroup $N$, that lies between the natural partial order and Green's ${\mathscr J}$-relation. The corresponding equivalence relation $\simeq_N$ is not necessarily a congruence on $S$, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on $S$. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation.