Bijective solutions to the Pentagon Equation

A complete classification of all finite bijective set-theoretic solutions $(S,s)$ to the Pentagon Equation is obtained. First, it is shown that every such a solution determines a semigroup structure on the set S that is the direct product E× G of a semigroup of left zeros E and a group G. Next, we prove that this leads to a decomposition of the set S as a Cartesian product X× A× G, for some sets $X,A$ and to a discovery of a hidden group structure on A. Then an unexpected structure of a matched product of groups $A,G$ is found such that the solution $(S,s)$ can be explicitly described as a lift of a solution determined on the set A× G by this matched product of groups. Conversely, every matched product of groups leads to a family of solutions arising in this way. Moreover, a simple criterion for the isomorphism of two solutions is obtained. The results provide a far reaching extension of the results of Colazzo, Jespers and Kubat, dealing with the special case of the so called involutive solutions. Connections to the solutions to the Yang--Baxter equation and to the theory of skew braces are derived.