2024/05/30 by Colazzo, I., Okniński, J., Van Antwerpen, A. · 1 citation
#18M20 #20E22 #20M99 #81R50 #FOS: Mathematics #Group Theory (math.GR) #Primary: 81R12 #Quantum Algebra (math.QA) #Secondary: 16T25
paper · doi:10.48550/arxiv.2405.20406
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.