On commutative set-theoretic solutions of the Pentagon Equation

TL;DR

This paper extends the theory of set-theoretic solutions to the Pentagon Equation, introduces a permutation group framework for commutative solutions, and provides a classification of irretractable solutions, especially on left-zero semigroups.

Contribution

It develops a new machinery for constructing and classifying commutative non-degenerate solutions of the Pentagon Equation, including irretractable cases.

Findings

Explicit classification of irretractable solutions

Analysis of solutions on left-zero semigroups

Characterization of solutions with cyclic permutation groups

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.