Antoine Amarilli ; Charles Paperman - Locality and Centrality: The Variety ZG

lmcs:11555 - Logical Methods in Computer Science, October 18, 2023, Volume 19, Issue 4 - https://doi.org/10.46298/lmcs-19(4:4)2023
Locality and Centrality: The Variety ZGArticle

Authors: Antoine Amarilli ; Charles Paperman

    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.


    Volume: Volume 19, Issue 4
    Published on: October 18, 2023
    Accepted on: July 7, 2023
    Submitted on: July 7, 2023
    Keywords: Computer Science - Formal Languages and Automata Theory

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