Finite Idempotent Set-Theoretic Solutions of the Yang–Baxter Equation

It is proven that finite idempotent left non-degenerate set-theoretic solutions (X,r) of the Yang–Baxter equation on a set X are determined by a left simple semigroup structure on X (in particular, a finite union of isomorphic copies of a group) and some maps q and φx on X⁠, for x∈X⁠. This structure turns out to be a group precisely when the associated Yang–Baxter monoid M(X,r) is cancellative and all the maps φx are equal to an automorphism of this group. Equivalently, the Yang–Baxter algebra K[M(X,r)] is right Noetherian, or in characteristic zero it has to be semiprime. The Yang–Baxter algebra is always a left Noetherian representable algebra of Gelfand–Kirillov dimension one. To prove these results, it is shown that the Yang–Baxter semigroup S(X,r) has a decomposition in finitely many cancellative semigroups Su indexed by the diagonal, each Su has a group of quotients Gu that is finite-by-(infinite cyclic) and the union of these groups carries the structure of a left simple semigroup. The case that X equals the diagonal is fully described by a single permutation on X⁠.