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 gr...