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

Abstract

© 2018 Springer Science+Business Media, LLC, part of Springer Nature 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): (Formula presented.)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.