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Bijective solutions to the Pentagon Equation

I. Colazzo, J. Okniński, A. Van Antwerpen

Abstract

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 solution determines a semigroup structure on the set $S$ that is the direct product $E\times G$ 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 $X\times A\times G$, for some sets $X,A$ and to the 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 $A\times G$ 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. These results significantly extend those of Colazzo, Jespers, and Kubat, in their treatment of the involutive case. Furthermore, connections to solutions to the Yang--Baxter Equation and to the theory of skew braces are uncovered. The latter motivate a further investigation of the relationship between solutions to the Yang--Baxter Equation and the Pentagon Equation at the set-theoretical level.

Bijective solutions to the Pentagon Equation

Abstract

A complete classification of all finite bijective set-theoretic solutions to the Pentagon Equation is obtained. First, it is shown that every such solution determines a semigroup structure on the set that is the direct product of a semigroup of left zeros and a group . Next, we prove that this leads to a decomposition of the set as a Cartesian product , for some sets and to the discovery of a hidden group structure on . Then an unexpected structure of a matched product of groups is found such that the solution can be explicitly described as a lift of a solution determined on the set 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. These results significantly extend those of Colazzo, Jespers, and Kubat, in their treatment of the involutive case. Furthermore, connections to solutions to the Yang--Baxter Equation and to the theory of skew braces are uncovered. The latter motivate a further investigation of the relationship between solutions to the Yang--Baxter Equation and the Pentagon Equation at the set-theoretical level.
Paper Structure (7 sections, 28 theorems, 138 equations)

This paper contains 7 sections, 28 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Set-theoretic bijective solutions to the Pentagon Equation
  3. The retract of a bijective solution
  4. From matched pairs of groups to irretractable solutions
  5. From irretractable solutions to matched pairs of groups
  6. A complete description of finite bijective solutions
  7. Concluding remarks and examples

Key Result

Theorem 1.1

Let $(S,s)$ be a finite bijective set-theoretic solution to the Pentagon Equation. Then there exists a matched pair of groups $(A,G,\sigma, \delta)$, a set $X$ and maps $\varphi_a\in \mathop{\mathrm{Sym}}\nolimits(X)$, for all $a\in A$, such that $S$, as a set, is the Cartesian product $X\times A \t for $\alpha ,\beta \in X, a,b \in A$ and $x,y\in G$. Moreover, for any matched pair of groups $(A,G

Theorems & Definitions (57)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 47 more