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<p>We provide a notion of <i>retract relation</i> for commutative non-degenerate set-theoretic solutions of the Pentagon Equation, which is consistent with the one given by Colazzo, Jespers, and Kubat (Set-theoretic solutions of the pentagon equation, Commun. Math. Phys. <b>380</b>(2), 1003–1024 (2020)) for involutive solutions. Moreover, we develop a machinery involving a permutation group, which we will call <i>associated permutation group</i>, useful to construct all these solutions and we will provide a description of the irretractable ones. Finally, non-degenerate solutions on left-zero semigroup are studied in detail, with an emphasis on the ones with cyclic associated permutation group and on the ones having small size.</p>

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On commutative set-theoretic solutions of the Pentagon Equation

  • 期刊论文

  • 发表日期：

    2026年1月21日

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  • Marco Castelli

摘要

We provide a notion of retract relation for commutative non-degenerate set-theoretic solutions of the Pentagon Equation, which is consistent with the one given by Colazzo, Jespers, and Kubat (Set-theoretic solutions of the pentagon equation, Commun. Math. Phys. 380(2), 1003–1024 (2020)) for involutive solutions. Moreover, we develop a machinery involving a permutation group, which we will call associated permutation group, useful to construct all these solutions and we will provide a description of the irretractable ones. Finally, non-degenerate solutions on left-zero semigroup are studied in detail, with an emphasis on the ones with cyclic associated permutation group and on the ones having small size.