Left semi-braces and solutions of the Yang–Baxter equation

Let r : X 2 → X 2 {r X²→ X²} 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 ⁢ 〈 x ∈ X ∣ x ⁢ y = u ⁢ v ⁢ if ⁢ r ⁢ ( x , y ) = ( u , v ) 〉 {K x∈ X xy=uv if r(x,y)=(u,v)} 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 {rB} 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.