Finite dimensional semigroup quadratic algebras with the minimal number of relations

A quadratic semigroup algebra is an algebra over a field given by the generators x 1, . . . , x n and a finite set of quadratic relations each of which either has the shape x j x k = 0 or the shape x j x k = x l x m . We prove that a quadratic semigroup algebra given by n generators and $${d\leq \frac{n^2+n}{4}}$$ relations is always infinite dimensional. This strengthens the Golod–Shafarevich estimate for the above class of algebras. Our main result however is that for every n, there is a finite dimensional quadratic semigroup algebra with n generators and δ n relations, where δ n is the first integer greater than $${\frac{n^2+n}{4}}$$ . That is, the above Golod–Shafarevich-type estimate for semigroup algebras is sharp.