Computing finite semigroups

TL;DR

This paper introduces a method to compute properties of subsemigroups within finite regular semigroups efficiently, leveraging algebraic relations to avoid exhaustive enumeration and enabling various structural analyses.

Contribution

It presents a novel approach using Schreier's Theorem and Green's relations to reduce complex subsemigroup computations to subgroup problems, applicable to many important classes of semigroups.

Findings

Efficient computation of subsemigroup size and Green's relations.

Ability to test membership and factorize elements without full enumeration.

Methods to find generated semigroups and analyze their structure.

Abstract

Using a variant of Schreier's Theorem, and the theory of Green's relations, we show how to reduce the computation of an arbitrary subsemigroup of a finite regular semigroup to that of certain associated subgroups. Examples of semigroups to which these results apply include many important classes: transformation semigroups, partial permutation semigroups and inverse semigroups, partition monoids, matrix semigroups, and subsemigroups of finite regular Rees matrix and $0$-matrix semigroups over groups. For any subsemigroup of such a semigroup, it is possible to, among other things, efficiently compute its size and Green's relations, test membership, factorize elements over the generators, find the semigroup generated by the given subsemigroup and any collection of additional elements, calculate the partial order of the $\mathscr{D}$-classes, test regularity, and determine the idempotents. This is achieved by representing the given subsemigroup without exhaustively enumerating its elements. It is also possible to compute the Green's classes of an element of such a subsemigroup without determining the global structure of the semigroup.