  Let $r:X^{2}\rightarrow X^{2}$ be a set-theoretic solution of the Yang-Baxter
equation on a finite set $X$. It was proven by Gateva-Ivanova and Van den Bergh
that if $r$ is non-degenerate and involutive then the algebra $K\langle x \in X
\mid xy =uv \mbox{ if } r(x,y)=(u,v)\rangle$ shares many properties with
commutative polynomial algebras in finitely many variables; in particular this
algebra is Noetherian, satisfies a polynomial identity and has Gelfand-Kirillov
dimension a positive integer. Lebed and Vendramin recently extended this result
to arbitrary non-degenerate bijective solutions. Such solutions are naturally
associated to finite skew left braces. In this paper we will prove an analogue
result for arbitrary solutions $r_B$ that are associated to a left semi-brace
$B$; such solutions can be degenerate or can even be idempotent. In order to do
so we first describe such semi-braces and we prove some decompositions results
extending results of Catino, Colazzo, and Stefanelli.

Group Theory;Quantum Algebra;Rings and Algebras