Publication at Faculty of Mathematics and Physics |
2021
Abstract
Idempotent left nondegenerate solutions of the Yang-Baxter equation are in one-to-one correspondence with twisted Ward left quasigroups, which are left quasigroups satisfying the identity (x * y) * (x * z) = (y * y) * (y * z). Using combinatorial properties of the Cayley kernel and the squaring mapping, we prove that a twisted Ward left quasigroup of prime order is either permutational or a quasigroup.
Up to isomorphism, all twisted Ward quasigroups (X, *) are obtained by twisting the left division operation in groups (that is, they are of the form x * y = (sic)(x(-1) y) for a group (X, .) and its automorphism (sic)), and they correspond to idempotent Latin solutions. We solve the isomorphism problem for idempotent Latin solutions.