  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 $\varphi_x$ on $X$, for
$x\in X$. This structure turns out to be a group precisely when the associated
structure monoid is cancellative and all the maps $\varphi_x$ are equal to an
automorphism of this group. Equivalently, the structure algebra $K[M(X,r)]$ is
right Noetherian, or in characteristic zero it has to be semiprime. The
structure algebra always is a left Noetherian representable algebra of
Gelfand--Kirillov dimension one. To prove these results it is shown that the
structure semigroup $S(X,r)$ 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 $X$.
