PÅonka bi-magmas and the set-theoretic Yang--Baxter equation
arXiv:2305.14138
The paper introduces Plonka bi‑magmas, a new class of non‑associative algebras, to describe and classify set‑theoretic solutions of the Yang‑Baxter equation of Baaj‑Long‑Skandalis type, and studies special classes such as bi‑connected and simple solutions, showing they correspond to odometer transformations and establishing categorical adjunctions.
Abstract
We introduce a new variety of non-associative algebras, called \emph{PÅonka 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 PÅonka 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 is a finite set with , then the number of isomorphism classes of all ideal-simple YB-solutions on is equal to the sum of all divisors of .
v3 constitutes the first part of a series of papers into which v2 will be divided, with any subsequent parts to be posted as separate papers; first author joins; title changed; 26 pages + references