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Set-theoretical solutions to the pentagon equation: a survey

Marzia Mazzotta

TL;DR

This survey tackles the set-theoretical pentagon equation by cataloging and organizing PE solutions on algebraic structures. It develops a framework for semigroup-based PE solutions, highlighting key constructions (group factorizations, Clifford semigroups with $E(X)$-invariance/fixedness, and congruence-pair methods), and then focuses on specialized classes: involutive, idempotent, and (co)commutative solutions, including retract/extension schemes for involutive cases. The work connects PE solutions to broader algebraic structures such as Hopf algebras and the Yang–Baxter/Tetrahedron equations, and raises open questions about full classifications and extensions to new semigroup classes. Overall, it provides a structured atlas of methods to generate and classify PE solutions, with explicit descriptions and counting results for key families.

Abstract

This survey aims to collect the main results of the theory of the set-theoretical solutions to the pentagon equation obtained up to now in the literature. In particular, we present some classes of solutions and raise some questions.

Set-theoretical solutions to the pentagon equation: a survey

TL;DR

This survey tackles the set-theoretical pentagon equation by cataloging and organizing PE solutions on algebraic structures. It develops a framework for semigroup-based PE solutions, highlighting key constructions (group factorizations, Clifford semigroups with -invariance/fixedness, and congruence-pair methods), and then focuses on specialized classes: involutive, idempotent, and (co)commutative solutions, including retract/extension schemes for involutive cases. The work connects PE solutions to broader algebraic structures such as Hopf algebras and the Yang–Baxter/Tetrahedron equations, and raises open questions about full classifications and extensions to new semigroup classes. Overall, it provides a structured atlas of methods to generate and classify PE solutions, with explicit descriptions and counting results for key families.

Abstract

This survey aims to collect the main results of the theory of the set-theoretical solutions to the pentagon equation obtained up to now in the literature. In particular, we present some classes of solutions and raise some questions.
Paper Structure (5 sections, 17 theorems, 26 equations)

This paper contains 5 sections, 17 theorems, 26 equations.

Table of Contents

  1. Definitions and examples
  2. PE solutions on Clifford semigroups
  3. Description of the involutive PE solutions
  4. Idempotent PE solutions
  5. Commutative and cocommutative PE solutions

Key Result

proposition 1

Let $G$ be a group, $X$ a subsemigroup of $G$, and $\lambda, \mu : X \to G$ maps such that, for all $x, y \in X$, Then, the map $s: X \times X \to X \times X$ given by $s(x,y)=(x y, x \ast y)$ is a PE solution on $X$.

Theorems & Definitions (22)

  • definition 1
  • proposition 1
  • theorem 1
  • lemma 1
  • definition 2
  • theorem 2
  • definition 3
  • theorem 3: cf. Theorem 5.3.3 in Ho95
  • theorem 4
  • proposition 2
  • ...and 12 more