Finitely generated algebras defined by homogeneous quadratic monomial relations and their underlying monoids

We consider algebras over a field K   with generators x1,x2,…,xnx1,x2,…,xn subject to (n2) quadratic relations of the form xixj=xkxlxixj=xkxl with (i,j)≠(k,l)(i,j)≠(k,l) and, moreover, every monomial xixjxixj appears at most once in one of the defining relations. If these relations are non-degenerate then it is shown that the algebra is left and right Noetherian, satisfies a polynomial identity and has Gelfand–Kirillov dimension at most n. In case the defining relations are square-free this was already established by Gateva-Ivanova, Jespers and Okniński. To prove these results we investigate the structure of the underlying monoid S, defined by the same presentation. It is called a quadratic monoid. We show that there is a strong link with the divisibility monoids and monoids of I-type (also referred to as YB-monoids). Monoids of I-type are examples of non-degenerate quadratic monoids. They are a monoid interpretation of non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation and have been studied quite intensively in recent years, by Gateva-Ivanova and Van den Bergh, Etingof, Schedler and Soloviev, Jespers and Okniński. Divisibility monoids have been introduced by Kuske. These are cancellative monoids that include the class of monoids of I  -type. We show that they have a presentation with at most (n2) relations and if they have precisely (n2) defining relations then they are monoids of I-type.