The Sch\"utzenberger category of a semigroup

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.