$F$-birestriction monoids in enriched signature

TL;DR

The paper develops a comprehensive theory of $F$-birestriction monoids in the enriched signature, providing a canonical coordinatization of the free object ${\mathsf{FFBR}}(X)$ as $E({\mathcal I}) \rtimes X^*$ with ${\mathcal I}$ the universal inverse monoid of an enriched presentation. It proves a precise structure theorem via a partial action product, shows that the word problem is decidable through Schützenberger graphs, and analyzes geometric models and Margolis–Meakin-type expansions, including positive results for the free perfect case ${\mathsf{FFBR}}_p(X)$. The work defines and situates several varieties ${\bf FBR}, {\bf FBR_{ls}}, {\bf FBR_{rs}}, {\bf FBR_s}, {\bf FBR_p}$ in the enriched setting, clarifying how max-elements $a^{\mathfrak{m}}$ interact with projections and premorphisms. Altogether, it links algebraic, combinatorial, and geometric methods to understand free and structured $F$-birestriction monoids and their decidability properties, with explicit geometric models for certain subcases.

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 $σ$-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.

Figures (5)

• Figure 1: Illustration of Case 1.
• Figure 2: Illustration of Case 2.
• Figure 3: Illustration of a $P$-expansion: the case of ${\mathcal{I}_{ls}}$.
• Figure 4: The Schützenberger graph of $a$.
• Figure 5: The Schützenberger graph of $\overline{x}\overline{x^2}^{-1}\overline{x}$.

Theorems & Definitions (101)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Definition 2.9