Computing maximal subsemigroups of a finite semigroup

TL;DR

This paper introduces an algorithm to compute maximal subsemigroups of finite semigroups using Green's structure, enabling practical analysis of semigroups generated by transformations, permutations, or matrices.

Contribution

The paper presents a novel algorithm that efficiently determines maximal subsemigroups of finite semigroups based on their Green's structure, filling a gap in computational semigroup theory.

Findings

Algorithm effectively computes maximal subsemigroups from Green's structure.

Computational complexity is comparable to existing Green's structure calculations.

Reduces to known problems like maximal clique detection and subgroup computations.

Abstract

A proper subsemigroup of a semigroup is maximal if it is not contained in any other proper subsemigroup. A maximal subsemigroup of a finite semigroup has one of a small number of forms, as described in a paper of Graham, Graham, and Rhodes. Determining which of these forms arise in a given finite semigroup is difficult, and no practical mechanism for doing so appears in the literature. We present an algorithm for computing the maximal subsemigroups of a finite semigroup given knowledge of its Green's structure, and the ability to determine maximal subgroups of certain subgroups. For a finite semigroup $S$ represented by a generating set $X$, in many examples, if it is practical to compute the Green's structure of $S$ from $X$, then it is also practical to find the maximal subsemigroups of $S$ using the algorithm we present. The generating set $X$ for $S$ may consist, for example, of transformations, or partial permutations, of a finite set, or of matrices over a semiring. In such examples, the time taken to determine the Green's structure of $S$ is comparable to that taken to find the maximal subsemigroups. Certain aspects of the problem of finding maximal subsemigroups reduce to other well-known computational problems, such as finding all maximal cliques in a graph and computing the maximal subgroups in a group. The algorithm presented comprises two parts. One part relates to computing the maximal subsemigroups of a special class of semigroups, known as Rees 0-matrix semigroups. The other part involves a careful analysis of certain graphs associated to the semigroup $S$, which, roughly speaking, capture the essential information about the action of $S$ on its $\mathscr{J}$-classes.