Closed inverse subsemigroups of graph inverse semigroups

TL;DR

This paper extends Lawson's classification of closed inverse submonoids from polycyclic monoids to graph inverse semigroups and analyzes their cosets, indices, and conjugacy properties.

Contribution

It generalizes Lawson's description to graph inverse semigroups and applies Schein's coset theory to determine conditions for finite index and compute index values.

Findings

Extended Lawson's classification to graph inverse semigroups.

Provided necessary and sufficient conditions for finite index of subsemigroups.

Determined the index values when finite.

Abstract

As part of his study of representations of the polycylic monoids, M.V. Lawson described all the closed inverse submonoids of a polycyclic monoid $P_n$ and classified them up to conjugacy. We show that Lawson's description can be extended to closed inverse subsemigroups of graph inverse semigroups. We then apply B. Schein's theory of cosets in inverse semigroups to the closed inverse subsemigroups of graph inverse semigroups: we give necessary and sufficient conditions for a closed inverse subsemigroup of a graph inverse semigroup to have finite index, and determine the value of the index when it is finite.