Regular Elements of Some Semigroups of Linear Transformations and Matrices

Let V be a vector space over a field F and LF (V) the semigroup, under composition, of all linear transformations α: V → V and Mn(F) the multiplicative semigroup of all n × n matrices over F. It is well-known that LF (V) and Mn(F) are regular semigroups. If W is a subspace of V, let IF (V, W) and KF (V, W) be the subsemigroups of LF (V) defined by IF (V, W) = {α ∈ LF (V) | im α ⊆ W} and KF (V, W) = {α ∈ LF (V) | W ⊆ kerα}. The purpose of this paper is to characterize the regular elements of the semigroups IF (V, W) and KF (V, W). These characterizations are then applied to determine respectively the regular elements of the subsemigroups Cn(F, k) and Rn(F, k) of Mn(F) where Cn(F, k) = {A ∈ Mn(F) | Aij = 0 for all i, j ∈ {1,..., n} and j> k} and Rn(F, k) = {A ∈ Mn(F) | Aij = 0 for all i, j ∈ {1,..., n} and i> k}