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The lattice and semigroup structure of multipermutations

Carvalho, Catarina and Martin, Barnaby (2022) 'The lattice and semigroup structure of multipermutations.', International journal of algebra and computation., 32 (2). pp. 211-235.

Abstract

We study the algebraic properties of binary relations whose underlying digraph is smooth, that is, has no source or sink. Such objects have been studied as surjective hyper-operations (shops) on the corresponding vertex set, and as binary relations that are defined everywhere and whose inverse is also defined everywhere. In the latter formulation, they have been called multipermutations. We study the lattice structure of sets (monoids) of multipermutations over an n-element domain. Through a Galois connection, these monoids form the algebraic counterparts to sets of relations closed under definability in positive first-order logic without equality. We show one side of this Galois connection, and give a simple dichotomy theorem for the evaluation problem of positive first-order logic without equality on the class of structures whose preserving multipermutations form a monoid closed under inverse. These problems turn out either to be in Logspaceor to be Pspace-complete. We go on to study the monoid of all multipermutations on an n-element domain, under usual composition of relations. We characterize its Green relations, regular elements and show that it does not admit a generating set that is polynomial on n.

Item Type:Article
Full text:Publisher-imposed embargo until 10 November 2022.
(AM) Accepted Manuscript
File format - PDF
(543Kb)
Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1142/S0218196722500096
Publisher statement:Electronic version of an article published as International Journal of Algebra and Computation, 30:2,, 2022, 211-235 [10.1142/S0218196722500096] © copyright World Scientific Publishing Company
Date accepted:05 September 2021
Date deposited:08 February 2022
Date of first online publication:10 November 2021
Date first made open access:10 November 2022

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