Set-theoretical solutions to the pentagon equation: a survey (Q6623956)

In this survey the author collects the main results of the theory of the set-theoretical solutions of the pentagon equation in the literature.\N\NAfter recalling the basics on the pentagon equation solutions together with examples and constructions, the author focuses on solutions defined on Clifford semigroups discussed in [\textit{M. Mazzotta} et al., Semigroup Forum 108, No. 2, 413--431 (2024; Zbl 1547.16031)]. Then she discusses involutive solutions obtained by \textit{I. Colazzo} et al. [Commun. Math. Phys. 380, No. 2, 1003--1024 (2020; Zbl 1482.16059)] and idempotent solutions with emphasis in a description of solutions on monoids having central idempotents obtained in her article [Boll. Unione Mat. Ital. 17, No. 2, 457--469 (2024; Zbl 07873922)]. A further classes of solutions discussed in the survey are the commutative and cocommutative ones, introduced in her joint work [\textit{F. Catino} et al., Commun. Algebra 48, No. 1, 83--92 (2020; Zbl 1447.16034)]. The survey also presents some classes of solutions and raises several questions.