Normal submonoids and congruences on a monoid

A notion of {\em normal submonoid} of a monoid \(M\) is introduced that generalizes the normal subgroups of a group. When ordered by inclusion, the set \(\mathsf{NorSub}(M)\) of normal submonoids of \(M\) is a complete lattice. Joins are explicitly described, and the lattice is computed for the fini...

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Bibliographic Details
Published in:arXiv.org 2022-10
Main Author: Elgueta, Josep
Format: Article
Language:English
Subjects:
Computation
Congruences
Lattices
Monoids
Subgroups
Online Access:Get full text
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Summary:A notion of {\em normal submonoid} of a monoid \(M\) is introduced that generalizes the normal subgroups of a group. When ordered by inclusion, the set \(\mathsf{NorSub}(M)\) of normal submonoids of \(M\) is a complete lattice. Joins are explicitly described, and the lattice is computed for the finite full transformation monoids \(T_n\), \(n\geq 1\). It is also shown that \(\mathsf{NorSub}(M)\) is modular for a specific family of commutative monoids, including all Krull monoids, and that, as a join semilattice, embeds isomorphically onto a join subsemilattice of the lattice \(\mathsf{Cong}(M)\) of congruences on \(M\). This leads to a new strategy for computing \(\mathsf{Cong}(M)\) consisting of computing \(\mathsf{NorSub}(M)\), and the lattices of the so called unital congruences on the quotients of \(M\) modulo its normal submonoids. This provides a new perspective on Malcev computation of the congruences on \(T_n\).
ISSN:2331-8422
DOI:10.48550/arxiv.2210.08546