Locality and Centrality: The Variety ZG

TL;DR

This paper investigates the variety ZG of monoids where elements in groups are central, proving its locality and characterizing related languages using algebraic and categorical methods.

Contribution

It establishes that ZG is a local variety, proves ZG * D = LZG, and characterizes ZG languages as unions of shuffles of singleton and regular commutative languages.

Findings

ZG is a local variety of monoids.

ZG * D equals LZG, the variety with local monoids in ZG.

ZG languages are unions of shuffles of singleton and regular commutative languages.

Abstract

We study the variety ZG of monoids where the elements that belong to a group are central, i.e., commute with all other elements. We show that ZG is local, that is, the semidirect product ZG * D of ZG by definite semigroups is equal to LZG, the variety of semigroups where all local monoids are in ZG. Our main result is thus: ZG * D = LZG. We prove this result using Straubing's delay theorem, by considering paths in the category of idempotents. In the process, we obtain the characterization ZG = MNil \vee Com, and also characterize the ZG languages, i.e., the languages whose syntactic monoid is in ZG: they are precisely the languages that are finite unions of disjoint shuffles of singleton languages and regular commutative languages.