$F$-birestriction monoids in enriched signature

Kudryavtseva, Ganna (author), Lemut Furlani, Ajda (author)

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.

• Language: English
• 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
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.11.2025
• Year: 2025
• Number of pages: 36 str.
• Numbering: Vol. 48, iss. 6, article no. 212
• PID: 20.500.12556/RUL-175543
• UDC: 512
• ISSN on article: 0126-6705
• DOI: 10.1007/s40840-025-01995-2
• COBISS.SI-ID: 255627779
• Publication date in RUL: 03.11.2025

Record is a part of a journal

• Title: Bulletin of the Malaysian Mathematical Sciences Society
• Shortened title: Bull. Malays. Math. Sci. Soc.
• Publisher: Malaysian Mathematical Society, Springer
• ISSN: 0126-6705
• COBISS.SI-ID: 515781657

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: biomejitveni monoid, F-biomejitveni monoid, prosti F-biomejitveni monoid, inverzni monoid, F-inverzni monoid, Margolis-Meakinova razširitev, Schützenbergerjev graf, parcialno delovanje, produkt glede na parcialno delovanje

Projects

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: P1-0288
• Name: Algebra in njena uporaba

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: J1-60025
• Name: Interakcija aritmetičnih lastnosti in algebraične strukture v nekomutativnih kolobarjih