Set-theoretical solutions of the pentagon equation on Clifford semigroups

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2024年3月27日

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Marzia Mazzotta,
Vicent Pérez-Calabuig,
Paola Stefanelli

摘要

Given a set-theoretical solution of the pentagon equation $s:S\times S\rightarrow S\times S$ $s : S \times S \to S \times S$ on a set S and writing $s(a, b)=(a\cdot b,\, \theta _a(b))$ $s (a, b) = (a \cdot b, θ_{a} (b))$ , with $\cdot $ $\cdot$ a binary operation on S and $\theta _a$ $θ_{a}$ a map from S into itself, for every $a\in S$ $a \in S$ , one naturally obtains that $\left( S,\,\cdot \right) $ $(S, \cdot)$ is a semigroup. In this paper, we focus on solutions defined in Clifford semigroups $\left( S,\,\cdot \right) $ $(S, \cdot)$ satisfying special properties on the set of all idempotents ${{\,\textrm{E}\,}}(S)$ $E (S)$ . Into the specific, we provide a complete description of idempotent-invariant solutions, namely, those solutions for which $\theta _a$ $θ_{a}$ remains invariant in ${{\,\textrm{E}\,}}(S)$ $E (S)$ , for every $a\in S$ $a \in S$ . Moreover, we construct a family of idempotent-fixed solutions, i.e., those solutions for which $\theta _a$ $θ_{a}$ fixes every element in ${{\,\textrm{E}\,}}(S)$ $E (S)$ for every $a\in S$ $a \in S$ , from solutions given on each maximal subgroup of S.