$F$-birestriction monoids in enriched signature

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 $(\cdot ,{}^{\ast },{}^{+},{}^{\mathfrak{m}},1)$ 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 $E(\mathcal{I})⋊X^{\ast }$ of the idempotent semilattice of the universal inverse monoid $\mathcal{I}$ of $\mathcal{F}$ partially acted upon by the free monoid $X^{\ast }$. 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.

34 pages, revised version

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