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 phi(x) on X, for x is an element of 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 phi(x) are equal to an automorphism of this group. Equivalently, the Yang-Baxter algebra 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 S(X, r) semigroup has a decomposition in finitely many cancellative semigroups S-u indexed by the diagonal, each S-u has a group of quotients G(u) 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.