0-Minimal Quasi-ideals of Generalized Linear Transformation Semigroups
Yupaporn Kemprasit, Ronnason Chinram · Communications in Algebra · 2003
For vector spaces V and W over a field F, L F (V, W) denotes the set of all linear transformations α : V → W, and for a cardinal number k > 0, let L F (V, W, k) be the set of all α ∈ L F (V, W) of rank less than k. For θ ∈ L F (W, V), let (L F (V, W, k), θ) denote the semigroup L F (V, W, k) under the operation ∗ defined by α ∗ β = αθβ for all α, β ∈ L F (V, W, k). In this paper, all 0-minimal quasi-ideals of the semigroup (L F (V, W, k), θ) are completely characterized. It is also shown from this characterization that every nonzero semigroup (L F (V, W, k), θ) always has a 0-minimal quasi-ideal.