Abstract
Let
{r\colon 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\text{ 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 then prove some decompositions results extending those of Catino, Colazzo and Stefanelli.
The Bicategory-Theoretic Solution of Recursive Domain Equations
