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arXiv · 2404.13758

On commutative set-theoretic solutions of the Pentagon Equation

Marco Castelli
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Abstract

We extend the so-called retract relation given in [6] for involutive set-theoretic solutions of the Pentagon Equation and we introduce the notion of associated permutation group to study the family of the commutative non-degenerate ones. Moreover, we develop a machinery to construct all these solutions and we use it to give a quite explicit classification 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.

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Marco Castelli. 2024-04-21. On commutative set-theoretic solutions of the Pentagon Equation. https://arxiv.org/abs/2404.13758

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