  For a finite involutive non-degenerate solution $(X,r)$ of the Yang--Baxter
equation it is known that the structure monoid $M(X,r)$ is a monoid of I-type,
and the structure algebra $K[M(X,r)]$ over a field $K$ share many properties
with commutative polynomial algebras, in particular, it is a Noetherian
PI-domain that has finite Gelfand--Kirillov dimension. In this paper we deal
with arbitrary finite (left) non-degenerate solutions. Although the structure
of both the monoid $M(X,r)$ and the algebra $K[M(X,r)]$ is much more
complicated than in the involutive case, we provide some deep insights.
  In this general context, using a realization of Lebed and Vendramin of
$M(X,r)$ as a regular submonoid in the semidirect product
$A(X,r)\rtimes\mathrm{Sym}(X)$, where $A(X,r)$ is the structure monoid of the
rack solution associated to $(X,r)$, we prove that $K[M(X,r)]$ is a module
finite normal extension of a commutative affine subalgebra. In particular,
$K[M(X,r)]$ is a Noetherian PI-algebra of finite Gelfand--Kirillov dimension
bounded by $|X|$. We also characterize, in ring-theoretical terms of
$K[M(X,r)]$, when $(X,r)$ is an involutive solution. This characterization
provides, in particular, a positive answer to the Gateva-Ivanova conjecture
concerning cancellativity of $M(X,r)$.
  These results allow us to control the prime spectrum of the algebra
$K[M(X,r)]$ and to describe the Jacobson radical and prime radical of
$K[M(X,r)]$. Finally, we give a matrix-type representation of the algebra
$K[M(X,r)]/P$ for each prime ideal $P$ of $K[M(X,r)]$. As a consequence, we
show that if $K[M(X,r)]$ is semiprime then there exist finitely many finitely
generated abelian-by-finite groups, $G_1,\dotsc,G_m$, each being the group of
quotients of a cancellative subsemigroup of $M(X,r)$ such that the algebra
$K[M(X,r)]$ embeds into $\mathrm{M}_{v_1}(K[G_1])\times\dotsb\times
\mathrm{M}_{v_m}(K[G_m])$.
