The free $F$-restriction semigroups

Abstract

We provide a geometric model for the free $X$-generated $F$-restriction semigroup in the extended signature $(\cdot\,, ^+,\mx{},λ)$, where the unary operation $\mx{}$ maps an element $a$ to the maximum element $\mx{a}$ of its $\ddot{I}ƒ$-class, and the constant $\hat{I}»$ is the unique left identity. This model is based on a certain quotient of the Cayley graph expansion of the free monoid $X^{\ast }$ with respect to the extended set of generators $X\cup \overline{X^{\ast }}$, where the generators from $\overline{X^{\ast }}$ are in a bijection with the free monoid $X^{\ast }$ and serve to capture the maximum elements of $\ddot{I}ƒ$-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.

Revised version