The set-theoretic Yang-Baxter equation, Kimura semigroups and functional graphs

TL;DR

This paper establishes an equivalence between solutions of a specific set-theoretic Yang-Baxter equation and Kimura semigroups, classifies various solutions, and analyzes their automorphism groups using group theory.

Contribution

It introduces a categorical equivalence linking Yang-Baxter solutions to Kimura semigroups and provides classifications and automorphism group descriptions for these solutions.

Findings

Classified involutive, idempotent, nondegenerate, surjective, finite order, unitary, indecomposable solutions.

Derived formulas for the number of isomorphism classes of solutions based on set size.

Described automorphism groups as compositions of cyclic and symmetric groups.

Abstract

We prove that the category of solutions of the set-theoretic Yang-Baxter equation of Frobenius-Separability (FS) type is equivalent to the category of pointed Kimura semigroups. As applications, all involutive, idempotent, nondegenerate, surjective, finite order, unitary or indecomposable solutions of FS type are classified. For instance, if $|X| = n$, then the number of isomorphism classes of all such solutions on $X$ that are (a) left non-degenerate, (b) bijective, (c) unitary or (d) indecomposable and left-nondegenerate is: (a) the Davis number $d(n)$, (b) $\sum_{m|n} \, p(m)$, where $p(m)$ is the Euler partition number, (c) $\tau(n) + \sum_{d|n}\left\lfloor \frac d2\right\rfloor$, where $\tau(n)$ is the number of divisors of $n$, or (d) the Harary number $\mathfrak{c} (n)$. The automorphism groups of such solutions can also be recovered as automorphism groups $\mathrm{Aut}(f)$ of sets $X$ equipped with a single endo-function $f:X\to X$. We describe all groups of the form $\mathrm{Aut}(f)$ as iterations of direct and (possibly infinite) wreath products of cyclic or full symmetric groups, characterize the abelian ones as products of cyclic groups, and produce examples of symmetry groups of FS solutions not of the form $\mathrm{Aut}(f)$.