Ranks and isomorphism theorems of semigroups of linear transformations with restricted range

Let P ( V ) be the partial linear transformation semigroup of a vector space V under composition. Given a fixed subspace W of V , define the following subsemigroups of P ( V ): P T ( V , W ) = { α ∈ P ( V ) | V α ⊆ W } , T ( V , W ) = { α ∈ P ( V , W ) | dom α = V } , I ( V , W ) = { α ∈ P ( V , W ) | α is injective } . In this paper, we prove certain isomorphism theorems and compute the ranks of these three semigroups for any proper subspace W of V when V is a finite-dimensional vector space over a finite field. Gaussian binomial coefficients play an essential role in these computations.