TY - JOUR

T1 - The ideal structure of semigroups of linear transformations with upper bounds on their nullity or defect

AU - Mendes-Goncalves, S.

AU - Sullivan, Robert

PY - 2009

Y1 - 2009

N2 - 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).

AB - 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).

U2 - 10.1080/00927870802622932

DO - 10.1080/00927870802622932

M3 - Article

SN - 0092-7872

VL - 37

SP - 2522

EP - 2539

JO - Communications in Algebra

JF - Communications in Algebra

IS - 7

ER -