The free $F$-restriction semigroups
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909963106910208 |
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| author | Kudryavtseva, Ganna Furlani, Ajda Lemut |
| author_facet | Kudryavtseva, Ganna Furlani, Ajda Lemut |
| contents | We provide a geometric model for the free $X$-generated $F$-restriction semigroup in the extended signature $(\cdot\,, ^+, ^m,λ)$, where the unary operation $^m$ maps an element $a$ to the maximum element $a^m$ of its $σ$-class, and the constant $λ$ 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 $σ$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13332 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The free $F$-restriction semigroups Kudryavtseva, Ganna Furlani, Ajda Lemut Rings and Algebras 20M05, 08B20, 20M07, 20M10 We provide a geometric model for the free $X$-generated $F$-restriction semigroup in the extended signature $(\cdot\,, ^+, ^m,λ)$, where the unary operation $^m$ maps an element $a$ to the maximum element $a^m$ of its $σ$-class, and the constant $λ$ 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 $σ$-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. |
| title | The free $F$-restriction semigroups |
| topic | Rings and Algebras 20M05, 08B20, 20M07, 20M10 |
| url | https://arxiv.org/abs/2512.13332 |