The Ideal Structure of Semigroups of Linear Transformations with Upper Bounds on Their Nullity or Defect |
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Authors: | Suzana Mendes-Gonçalves R. P. Sullivan |
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Affiliation: | 1. Centro de Matemática , Universidade do Minho , Braga , Portugal smendes@math.uminho.pt;3. School of Mathematics and Statistics , University of Western Australia , Nedlands , Australia |
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Abstract: | Suppose V is a vector space with dim V = p ≥ q ≥ ?0, and let T(V) denote the semigroup (under composition) of all linear transformations of V. For α ∈ T (V), let ker α and ran α denote the “kernel” and the “range” of α, and write n(α) = dim ker α and d(α) = codim ran α. In this article, we study the semigroups AM(p, q) = {α ∈ T(V):n(α) < q} and AE(p, q) = {α ∈ T(V):d(α) < q}. First, we determine whether they belong to the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide. Then, for each semigroup, we describe its maximal regular subsemigroup, and we characterise its Green's relations and (two-sided) ideals. As a precursor to further work in this area,, we also determine all the maximal right simple subsemigroups of AM(p, q). |
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Keywords: | Bi-ideal Linear transformation semigroup Maximal regular Maximal right simple Quasi-ideal |
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