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

Abstract 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 simple semigroup structure (in particular, union isomorphic copies group) and some maps $q$ $\varphi _{x}$ $X$, for $x\in X$. This turns out to be group precisely when associated monoid $M(X,r)$ cancellative all equal an automorphism this group. Equivalently, algebra $K[M(X,r)]$ right Noetherian, or in characteristic zero it has semiprime. The always Noetherian representable Gelfand–Kirillov dimension one. To prove these results, shown $S(X,r)$ decomposition finitely many semigroups $S_{u}$ indexed diagonal, each quotients $G_{u}$ finite-by-(infinite cyclic) groups carries semigroup. case equals diagonal fully described single permutation $X$.