Simple solutions of the Yang-Baxter equation of cardinality $p^n$

TL;DR

This paper constructs new simple and non-simple set-theoretic solutions to the Yang-Baxter equation for specific cardinalities, addressing open questions and expanding known solution classes.

Contribution

It introduces explicit constructions of solutions for prime power and composite cardinalities, including answers to existing open problems.

Findings

Constructed simple solutions for prime power cardinalities.

Developed non-simple solutions for certain composite cardinalities.

Answered affirmatively to Castelli's question on singular solutions.

Abstract

For every prime number p and integer $n>1$, a simple, involutive, non-degenerate set-theoretic solution $(X,r$) of the Yang-Baxter equation of cardinality $|X| = p^n$ is constructed. Furthermore, for every non-(square-free) positive integer m which is not the square of a prime number, a non-simple, indecomposable, irretractable, involutive, non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation of cardinality $|X| = m$ is constructed. A recent question of Castelli on the existence of singular solutions of certain type is also answered affirmatively.