Aperiodic Flows on Finite Semigroups II: Smallish Monoids Suffice for Complexity 1

TL;DR

The paper demonstrates that finite semigroups can be embedded into smallish monoids, simplifying the analysis of their Krohn-Rhodes complexity 1 through flow theory.

Contribution

It provides a constructive embedding of any finite semigroup into a smallish monoid's evaluation semigroup, linking flow properties to complexity reduction.

Findings

Finite semigroups can be embedded into smallish monoids.

Flow properties in groups correspond to those in smallish monoids.

Krohn-Rhodes complexity 1 analysis reduces to smallish monoids.

Abstract

A smallish monoid M is a monoid that has a unique 0-minimal ideal I(M) that is a 0-simple subsemigroup and such that its regular J -classes are the group of units and the two in I(M). We show constructively how to embed an arbitrary finite semigroup S into the evaluation semigroup of a smallish monoid S^{Ev} . We use the theory of flows to show that a group mapping semigroup S admits an aperiodic flow if and only if S^{Ev} admits one. This reduces the computation of Krohn-Rhodes complexity 1 to the class of smallish monoids.