Płonka bi-magmas and the set-theoretic Yang--Baxter equation

2023/05/23 by A. L. Agore, A. Chirvasitu, Chirvasitu, A. +2
Mathematics · Computer Science · #Advanced Topics in Algebra #Advanced Algebra and Logic #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2305.14138

Abstract

We introduce a new variety of non-associative algebras, called Płonka 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 Płonka 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.

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