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| Formato: | Preprint |
| Publicado: |
2022
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| Acceso en línea: | https://arxiv.org/abs/2210.08546 |
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| _version_ | 1866913349435916288 |
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| author | Elgueta, Josep |
| author_facet | Elgueta, Josep |
| contents | 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 $T_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_08546 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Normal submonoids and congruences on a monoid Elgueta, Josep Group Theory 20M10 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 $T_n$. |
| title | Normal submonoids and congruences on a monoid |
| topic | Group Theory 20M10 |
| url | https://arxiv.org/abs/2210.08546 |