The free $F$-restriction semigroups

We provide a geometric model for the free $X$-generated $F$-restriction semigroup in the extended signature $(\cdot\,, ^+, ^m,\lambda)$, where the unary operation $^m$ maps an element $a$ to the maximum element $a^m$ of its $\sigma$-class, and the constant $\lambda$ is the unique left identity. This model is based on a certain quotient of the Cayley graph expansion of the free monoid $X^*$ with respect to the extended set of generators $X\cup \overline{X^*}$, where the generators from $\overline{X^*}$ are in a bijection with the free monoid $X^*$ and serve to capture the maximum elements of $\sigma$-classes of the quotient. We also provide models for the free $X$-generated strong and perfect $F$-restriction semigroups in the same extended signature. The constructed models enable us to solve the word problems for all the free objects under consideration.