The Sch\"utzenberger category of a semigroup

TL;DR

This paper introduces the Schützenberger category of a semigroup, linking it to existing structures like the Karoubi envelope and Schützenberger groups, and clarifies properties of Green's relations.

Contribution

It defines the Schützenberger category for semigroups, connecting it to known concepts and providing new insights into Green's relations and local divisors.

Findings

Objects correspond to semigroup elements and are classified by Green's relations.

Endomorphism monoids are local divisors, automorphism groups are Schützenberger groups.

Establishes technical results used in symbolic dynamical systems research.

Abstract

In this paper we introduce the Sch\"utzenberger category $\mathbb D(S)$ of a semigroup $S$. It stands in relation to the Karoubi envelope (or Cauchy completion) of $S$ in the same way that Sch\"utzenberger groups do to maximal subgroups and that the local divisors of Diekert do to the local monoids $eSe$ of $S$ with $e\in E(S)$. In particular, the objects of $\mathbb D(S)$ are the elements of $S$, two objects of $\mathbb D(S)$ are isomorphic if and only if the corresponding semigroup elements are $\mathscr D$-equivalent, the endomorphism monoid at $s$ is the local divisor in the sense of Diekert and the automorphism group at $s$ is the Sch\"utzenberger group of the $\mathscr H$-class of $S$. This makes transparent many well-known properties of Green's relations. The paper also establishes a number of technical results about the Karoubi envelope and Sch\"utzenberger category that were used by the authors in a companion paper on syntactic invariants of flow equivalence of symbolic dynamical systems.