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 Sch\"utzenberger 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.
Categories: math.RA math.GR
