Computing Embeddings and Isomorphisms of Finite Semigroups

TL;DR

This paper explores algorithms for computing embeddings and isomorphisms of finite semigroups, addressing algebraic problems related to state transition systems and computational models, with new computational results on transformation and diagram semigroups.

Contribution

It introduces backtrack search algorithms exploiting semigroup properties and reports new computational findings on transformation and diagram semigroups.

Findings

Extended algebraic knowledge through computational experiments

New results on minimal degree representations

Insights into embedding into 2-generated subsemigroups

Abstract

Semigroup theory is a branch of abstract algebra, and it provides mathematical tools for the theory of computation. Finite semigroups can describe state transition systems and thus they model physically realizable computers. Engineering questions like `What is the minimal number of states to realize a particular computation?' and `Which type of computation is more capable?' translate into the algebraic tasks of constructing isomorphisms and embeddings between semigroups of different representations. The underlying problem is (sub)graph isomorphism, which is computationally difficult in general. We describe variations of backtrack search algorithms that exploit the algebraic properties of semigroups, and we carry out computational experiments to extend our algebraic knowledge. In particular, we report new computational results on transformation semigroups and on the more general family of diagram semigroups. We study the minimal degree representation problem, count distinct embeddings and work on an open problem of embedding into 2-generated subsemigroups.