Idempotent set-theoretical solutions of the pentagon equation (Q6564821)

For a set \(X\), a map \(s: X^2 \to X^2\); \((x,y) \mapsto (xy, \theta_x(y))\) is called a \textit{(set-theoretic) solution} of the \textit{pentagon equation} if \(s\) satisfies the equation of maps \(X^3 \to X^3\)\N\[\Ns_{23}s_{13}s_{12} = s_{12}s_{23},\N\]\Nwhere the map \(s_{ij}: X^3 \to X^3\) acts as \(s\) on the \(i\)-th and \(j\)-th coordinate and as identity on the others. Similar to the Yang-Baxter equation, the set-theoretic pentagon equation is the combinatorial version of a linear operator equation that appears, for example, in representation theory and the theory of Hopf algebras.\N\NIt can be shown that the first coordinate of \(s\), which is \((x,y) \mapsto xy\), defines a semigroup structure on \(X\). Therefore, the pentagon equation is intimately connected with semigroup theory.\N\NA natural approach to solutions of the pentagon equation is to consider the semigroup structure defined by \(s\). In the case when \(X\) is a group, a full classification has been provided by Catino, Mazzotta and Miccoli. In this manuscript, the author generalizes the group setting to semigroups and monoids with central idempotents. The author investigates properties of \(s\) for general solutions and gives a full classification of idempotent solutions on these semigroups. Furthermore, some general results on (co-)commutative and idempotent solutions are proven.\N\NThe results in this article are original, highly non-trivial and appear to be correct. Also, the references are correct, and the article is well written.