The free F-restriction semigroups

Abstract We provide a geometric model for the free X -generated F -restriction semigroup in the extended signature $${(\cdot , ^+,^{\mathfrak {m}},\lambda )}$$ ( · , + , m , λ ) , where the unary operation m maps an element a to the maximum element $$a^{\mathfrak {m}}$$ a m of its $$\sigma $$ σ -class, and the constant $$\lambda $$ λ is the unique left identity. This model is based on a certain quotient of the Cayley graph expansion of the free monoid $$X^*$$ X ∗ with respect to the extended set of generators $$X\cup \overline{X^*}$$ X ∪ X ∗ ¯ , where the generators from $$\overline{X^*}$$ X ∗ ¯ are in a bijection with the free monoid $$X^*$$ X ∗ and serve to capture the maximum elements of $$\sigma $$ σ -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.