Subspace Semigroups

For a finite dimensional algebra A over an infinite field K, the subspace semigroup (A) consists of all subspaces of A with operation V * W = linK(VW). We describe the structure of (A), showing in particular that, similarly to any algebraic linear semigroup, (A) is strongly π-regular; we describe its regular elements and regular -classes. As the key intermediate step, for an arbitrary connected algebraic monoid M, we study the semigroup (M) consisting of all irreducible closed subsets with operation X • Y = , and we transfer the information to (A) via the natural onto homomorphism (A) → (A). With potential applications in mind, our primary focus in this paper is on the case of the full matrix algebra A = Mn(K).