Two-sided wreath product done right

TL;DR

This paper introduces a refined semigroup construction related to the two-sided wreath product, integrating concepts from semigroup theory, category theory, and ordered structures, and offering a nuanced decomposition in the Krohn-Rhodes Theorem.

Contribution

It presents a new semigroup construction that generalizes known structures and refines the Krohn-Rhodes decomposition using a two-element semilattice.

Findings

Provides a new semigroup construction combining wreath products and category theory

Offers a finer decomposition in the Krohn-Rhodes Theorem

Establishes foundational links between semigroup theory, category theory, and ordered structures

Abstract

We investigate a semigroup construction related to the two-sided wreath product. It encompasses a range of known constructions and gives a slightly finer version of the decomposition in the Krohn-Rhodes Theorem, in which the three-element flip-flop is replaced by the two-element semilattice. We develop foundations of the theory of our construction, showing in the process that it naturally combines ideas from semigroup theory (wreath products), category theory (Grothendieck construction), and ordered structures (residuated lattices).