Locality and Centrality: The Variety ZG - Archive ouverte HAL
Journal Articles Logical Methods in Computer Science Year : 2023

Locality and Centrality: The Variety ZG

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.

Dates and versions

hal-04589443 , version 1 (27-05-2024)

Identifiers

Cite

Antoine Amarilli, Charles Paperman. Locality and Centrality: The Variety ZG. Logical Methods in Computer Science, 2023, 19 (4), ⟨10.46298/lmcs-19(4:4)2023⟩. ⟨hal-04589443⟩
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