Locality and Centrality: The Variety ZG
Antoine Amarilli, Charles Paperman
TL;DR
This paper proves locality for the variety $\mathbf{ZG}$ of monoids by showing $\mathbf{ZG}_p * \mathbf{D} = \mathbf{LZG}_p$ for every $p>0$, and consequently $\mathbf{LZG} = \mathbf{ZG} * \mathbf{D}$. The authors develop a robust framework based on Straubing's delay theorem, introducing $n,p$-congruences on the arrows of the category of idempotents and proving that any $\mathbf{ZG}_p$-congruence is refined by such a congruence. They derive a concrete characterization of $\mathbf{ZG}$-languages as finite unions of disjoint shuffles of singleton languages with regular commutative languages, and show $\mathbf{ZG} = \mathbf{MNil} \lor \mathbf{Com}$. The work leverages loop insertion and prefix substitution lemmas to manage path manipulations in the idempotent-category, coupled with a distant rare-frequent threshold to enable pumping, ultimately establishing the main locality theorem with detailed inductive arguments. The results illuminate the interplay between semidirect products, locality, and language-theoretic characterizations in a central class of semigroups, with implications for dynamic language membership and complexity boundaries.
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.