2024/12/16 by Ganna Kudryavtseva, Ajda Lemut Furlani, Kudryavtseva, Ganna +1
Computer Science · Mathematics · #05E18 #08B20 #20M05 #20M10 #20M18 #Advanced Algebra and Logic #FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #semigroups and automata theory
paper · doi:10.48550/arxiv.2412.12082
openalex publication_date 2024/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03
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 (⋅, ^*, +, ^\mathfrakm,1) where the unary operation (⋅)^\mathfrakm maps each element to the maximum element of its σ-class. We find a presentation of the free F-birestriction monoid FFBR(X) as a birestriction monoid \mathcal F over the extended set of generators X∪X+ where 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 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 FFBR(X) and its strong and perfect analogues is decidable. Furthermore, we show that FFBR(X) does not admit a geometric model based on a quotient of the Margolis-Meakin expansion M(FG(X), X∪ X+) over the free group FG(X), but the free perfect X-generated F-birestriction monoid admits such a model.