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 \((\cdot , \, ^*, \,^+, \,\phantom {0}^{\mathfrak {m}},1)\) 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 \({\textsf{FFBR}}(X)\) 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 \({\textsf{FFBR}}(X)\) 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 \({\textsf{FFBR}}(X)\) and its strong and perfect analogues is decidable. Furthermore, we show that \({\textsf{FFBR}}(X)\) does not admit a geometric model based on a quotient of the Margolis-Meakin expansion \(M({\textsf{FG}}(X), X\cup \overline{X^+})\) over the free group \({\textsf{FG}}(X)\) , but the free perfect X-generated F-birestriction monoid admits such a model.
