| Title: | $F$-birestriction monoids in enriched signature |
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| Authors: | ID Kudryavtseva, Ganna (Author)
ID Lemut Furlani, Ajda (Author) |
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| Files: |  PDF - Presentation file, download (656,63 KB)
MD5: 1276F81699CBEBA88FD67077756B8E83
 URL - Source URL, visit https://link.springer.com/article/10.1007/s40840-025-01995-2
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | 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 $\sigma$-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) $\sigma$-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. |
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| Keywords: | birestriction monoid, F-birestriction monoid, free F-birestriction monoid, inverse monoid, F-inverse monoid, Margolis-Meakin expansion, Schützenberger graph, partial action, partial action product |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.11.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | 36 str. |
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| Numbering: | Vol. 48, iss. 6, article no. 212 |
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| PID: | 20.500.12556/DiRROS-23980  |
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| UDC: | 512 |
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| ISSN on article: | 0126-6705 |
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| DOI: | 10.1007/s40840-025-01995-2 |
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| COBISS.SI-ID: | 255627779 |
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| Note: |
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| Publication date in DiRROS: | 03.11.2025 |
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| Views: | 163 |
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| Downloads: | 84 |
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| Metadata: |    |
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