Quadratic semigroups on affine spaces

As a generalization of some aspects of the theories of algebraic groups and finite dimensional algebras one can study polynomially defined, associative operations on affine spaces. In this paper we study the quadratic case—i.e., the case when the polynomials are of degree at most two. We call the resulting semigroup a quadratic semigroup. We obtain the complete structure for one class of quadratic semigroups and a representation theorem for another class.