Merge decompositions, two-sided Krohn-Rhodes, and aperiodic pointlikes

TL;DR

This paper introduces the merge decomposition technique to provide concise proofs of key theorems in finite semigroup theory, simplifying complex proofs and enabling new decompositions of semigroups.

Contribution

The paper presents a novel algebraic technique called merge decomposition, offering shorter proofs of the two-sided Krohn-Rhodes and Henckell's aperiodic pointlike theorems.

Findings

Simplified proofs of fundamental semigroup theorems

A new method for decomposing semigroups into components

Application of merge decomposition to semigroup homomorphisms

Abstract

This paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the two-sided Krohn-Rhodes decomposition theorem and Henckell's aperiodic pointlike theorem, using a new algebraic technique that we call the merge decomposition. A prototypical application of this technique decomposes a semigroup $T$ into a two-sided semidirect product whose components are built from two subsemigroups $T_1,T_2$, which together generate $T$, and the subsemigroup generated by their setwise product $T_1T_2$. In this sense we decompose $T$ by merging the subsemigroups $T_1$ and $T_2$. More generally, our technique merges semigroup homomorphisms from free semigroups.