The Lattice of One-Sided Congruences on an Inverse Semigroup

TL;DR

This paper explores the structure of left congruences on inverse semigroups, introducing the inverse kernel concept, and characterizes their lattice structure, including specific cases like the bicyclic monoid.

Contribution

It develops the notion of inverse kernels for left congruences and describes the lattice of left congruences using trace and inverse-kernel maps, extending previous descriptions.

Findings

The lattice of left congruences is a subset of a product of congruence lattices.

Every finitely generated left congruence is a join of two finitely generated types.

Characterizations of left Noetherian inverse semigroups are provided.

Abstract

We build on the description of left congruences on an inverse semigroup in terms of the kernel and trace due to Petrich and Rankin. The notion of an inverse kernel for a left congruence is developed. Various properties of both the trace and inverse-kernel are discussed, in particular that both the trace and inverse-kernel maps are onto $\cap$-homomorphisms. The lattice of left congruences is identified as a subset of the direct product of the lattice of congruences on the idempotents and the lattice of full inverse subsemigroups. We use this to describe the lattice of left congruences on the bicyclic monoid. It is shown that that every finitely generated left congruence is the join of a finitely generated trace minimal left congruence and a finitely generated idempotent separating left congruence. Characterisations of inverse semigroups that are left Noetherian, or such that the universal congruence is finitely generated, are given.