$F$-birestriction monoids in enriched signature

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

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.

• Jezik: Angleški jezik
• Ključne besede: birestriction monoid, F-birestriction monoid, free F-birestriction monoid, inverse monoid, F-inverse monoid, Margolis-Meakin expansion, Schützenberger graph, partial action, partial action product
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Datum objave: 01.11.2025
• Leto izida: 2025
• Št. strani: 36 str.
• Številčenje: Vol. 48, iss. 6, article no. 212
• PID: 20.500.12556/RUL-175543
• UDK: 512
• ISSN pri članku: 0126-6705
• DOI: 10.1007/s40840-025-01995-2
• COBISS.SI-ID: 255627779
• Datum objave v RUL: 03.11.2025

Gradivo je del revije

• Naslov: Bulletin of the Malaysian Mathematical Sciences Society
• Skrajšan naslov: Bull. Malays. Math. Sci. Soc.
• Založnik: Malaysian Mathematical Society, Springer
• ISSN: 0126-6705
• COBISS.SI-ID: 515781657

Licence

• Licenca: CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by/4.0/deed.sl
• Opis: To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.

Sekundarni jezik

• Jezik: Slovenski jezik
• Ključne besede: 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

Projekti

• Financer: ARIS - Javna agencija za znanstvenoraziskovalno in inovacijsko dejavnost Republike Slovenije
• Številka projekta: P1-0288
• Naslov: Algebra in njena uporaba

• Financer: ARIS - Javna agencija za znanstvenoraziskovalno in inovacijsko dejavnost Republike Slovenije
• Številka projekta: J1-60025
• Naslov: Interakcija aritmetičnih lastnosti in algebraične strukture v nekomutativnih kolobarjih