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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2301.09944 |
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| _version_ | 1866909148367552512 |
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| author | Mazzotta, Marzia Pérez-Calabuig, Vicent Stefanelli, Paola |
| author_facet | Mazzotta, Marzia Pérez-Calabuig, Vicent Stefanelli, Paola |
| contents | Given a set-theoretical solution of the pentagon equation $s:S\times S\to S\times S$ on a set $S$ and writing $s(a, b)=(a\cdot b,\, θ_a(b))$, with $\cdot$ a binary operation on $S$ and $θ_a$ a map from $S$ into itself, for every $a\in S$, one naturally obtains that $\left(S,\,\cdot\right)$ is a semigroup. In this paper, we focus on solutions on Clifford semigroups $\left(S,\,\cdot\right)$ satisfying special properties on the set of the idempotents $E(S)$. Into the specific, we provide a complete description of idempotent-invariant solutions, namely, those solutions for which $θ_a$ remains invariant in $E(S)$, for every $a\in S$. Moreover, considering $(S,\,\cdot)$ as a disjoint union of groups, we construct a family of idempotent-fixed solutions, i.e., those solutions for which $θ_a$ fixes every element in $E(S)$, for every $a\in S$, starting from a solution on each group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_09944 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Set-theoretical solutions of the pentagon equation on Clifford semigroups Mazzotta, Marzia Pérez-Calabuig, Vicent Stefanelli, Paola Group Theory 16T25, 81R50, 20M18 Given a set-theoretical solution of the pentagon equation $s:S\times S\to S\times S$ on a set $S$ and writing $s(a, b)=(a\cdot b,\, θ_a(b))$, with $\cdot$ a binary operation on $S$ and $θ_a$ a map from $S$ into itself, for every $a\in S$, one naturally obtains that $\left(S,\,\cdot\right)$ is a semigroup. In this paper, we focus on solutions on Clifford semigroups $\left(S,\,\cdot\right)$ satisfying special properties on the set of the idempotents $E(S)$. Into the specific, we provide a complete description of idempotent-invariant solutions, namely, those solutions for which $θ_a$ remains invariant in $E(S)$, for every $a\in S$. Moreover, considering $(S,\,\cdot)$ as a disjoint union of groups, we construct a family of idempotent-fixed solutions, i.e., those solutions for which $θ_a$ fixes every element in $E(S)$, for every $a\in S$, starting from a solution on each group. |
| title | Set-theoretical solutions of the pentagon equation on Clifford semigroups |
| topic | Group Theory 16T25, 81R50, 20M18 |
| url | https://arxiv.org/abs/2301.09944 |