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arXiv · 2512.13332

The free $F$-restriction semigroups

Ganna Kudryavtseva·Ajda Lemut Furlani
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Abstract

We provide a geometric model for the free $X$-generated $F$-restriction semigroup in the extended signature $(\cdot\,, ^+,\mx{},\lambda)$, where the unary operation $\mx{}$ maps an element $a$ to the maximum element $\mx{a}$ 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.

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Ganna Kudryavtseva, Ajda Lemut Furlani. 2025-12-15. The free $F$-restriction semigroups. https://arxiv.org/abs/2512.13332

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