$F$-birestriction monoids in enriched signature
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915599266873344 |
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| author | Kudryavtseva, Ganna Furlani, Ajda Lemut |
| author_facet | Kudryavtseva, Ganna Furlani, Ajda Lemut |
| contents | Motivated by recent interest to $F$-inverse monoids, on the one hand, and to restriction and birestriction monoids, on the other hand, we initiate the study of $F$-birestriction monoids as algebraic structures in the enriched signature $(\cdot, \, ^*, \,^+, \, ^{\mathfrak{m}},1)$ where the unary operation $(\cdot)^{\mathfrak{m}}$ maps each element to the maximum element of its $σ$-class.
We find a presentation of the free $F$-birestriction monoid ${\mathsf{FFBR}}(X)$ as a birestriction monoid ${\mathcal F}$ over the extended set of generators $X\cup\overline{X^+}$ where $\overline{X^+}$ is a set in a bijection with the free semigroup $X^+$ and encodes the maximum elements of (non-projection) $σ$-classes. This enables us to show that ${\mathsf{FFBR}}(X)$ decomposes as the partial action product $E({\mathcal I})\rtimes X^*$ of the idempotent semilattice of the universal inverse monoid ${\mathcal I}$ of ${\mathcal F}$ partially acted upon by the free monoid $X^*$. Invoking Schützenberger graphs, we prove that the word problem for ${\mathsf{FFBR}}(X)$ and its strong and perfect analogues is decidable. Furthermore, we show that ${\mathsf{FFBR}}(X)$ does not admit a geometric model based on a quotient of the Margolis-Meakin expansion $M({\mathsf{FG}}(X), X\cup \overline{X^+})$ over the free group ${\mathsf{FG}}(X)$, but the free perfect $X$-generated $F$-birestriction monoid admits such a model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12082 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $F$-birestriction monoids in enriched signature Kudryavtseva, Ganna Furlani, Ajda Lemut Rings and Algebras Group Theory 20M05, 08B20, 20M18, 05E18, 20M10 Motivated by recent interest to $F$-inverse monoids, on the one hand, and to restriction and birestriction monoids, on the other hand, we initiate the study of $F$-birestriction monoids as algebraic structures in the enriched signature $(\cdot, \, ^*, \,^+, \, ^{\mathfrak{m}},1)$ where the unary operation $(\cdot)^{\mathfrak{m}}$ maps each element to the maximum element of its $σ$-class. We find a presentation of the free $F$-birestriction monoid ${\mathsf{FFBR}}(X)$ as a birestriction monoid ${\mathcal F}$ over the extended set of generators $X\cup\overline{X^+}$ where $\overline{X^+}$ is a set in a bijection with the free semigroup $X^+$ and encodes the maximum elements of (non-projection) $σ$-classes. This enables us to show that ${\mathsf{FFBR}}(X)$ decomposes as the partial action product $E({\mathcal I})\rtimes X^*$ of the idempotent semilattice of the universal inverse monoid ${\mathcal I}$ of ${\mathcal F}$ partially acted upon by the free monoid $X^*$. Invoking Schützenberger graphs, we prove that the word problem for ${\mathsf{FFBR}}(X)$ and its strong and perfect analogues is decidable. Furthermore, we show that ${\mathsf{FFBR}}(X)$ does not admit a geometric model based on a quotient of the Margolis-Meakin expansion $M({\mathsf{FG}}(X), X\cup \overline{X^+})$ over the free group ${\mathsf{FG}}(X)$, but the free perfect $X$-generated $F$-birestriction monoid admits such a model. |
| title | $F$-birestriction monoids in enriched signature |
| topic | Rings and Algebras Group Theory 20M05, 08B20, 20M18, 05E18, 20M10 |
| url | https://arxiv.org/abs/2412.12082 |