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

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.

arXiv admin note: text overlap with arXiv:2401.12904, arXiv:2212.06753, arXiv:2112.07271, arXiv:2012.08400