P{\l}onka bi-magmas and the set-theoretic Yang--Baxter equation

Abstract

We introduce a new variety of non-associative algebras, called \emph{P{\l}onka bi-magmas}, and use them to construct novel solutions to the set-theoretic Yang-Baxter (YB) equation. We establish explicit structural and classification results for P{\l}onka bi-magmas, which allow for a detailed study of the induced families of YB-solutions. In particular, we identify and classify several naturally arising classes of YB-solutions from the universal algebra perspective, including the so-called bi-connected and ideal-simple ones. For instance, we prove that there are only countably many isomorphism classes of induced YB-solutions which are ideal-simple, and characterize them in terms of the odometer transformations familiar from ergodic theory. In particular, if $X$ is a finite set with $|X| = n$, then the number of isomorphism classes of all ideal-simple YB-solutions on $X$ is equal to the sum of all divisors of $n$.