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TitleThe ideal structure of semigroups of linear transformations with upper bounds on their nullity or defect
Author(s)Gonçalves, Suzana Mendes
Sullivan, R. P.
Linear transformation semigroup
Maximal regular
Maximal right simple
Issue date2009
PublisherTaylor and Francis
JournalCommunications in Algebra
Citation"Communications in Algebra". ISSN 0092-7872 . 37:7 (2009) 2522-2539..
Abstract(s)Suppose $V$ is a vector space with ${\rm dim} V=p\geq q\geq\aleph_0$, and let $T(V)$ denote the semigroup (under composition) of all linear transformations of $V$. For each $\alpha\in T(V)$, let ${\rm ker}\alpha$ and ${\rm ran}\alpha$ denote the `kernel' and the `range' of $\alpha$, and write $n(\alpha)={\rm dim}{\rm ker}\alpha$ and $d(\alpha)={\rm codim}{\rm ran}\alpha$. In this paper, we study the semigroups $AM(p,q) =\{\alpha\in T(V):n(\alpha)<q\}$ and $AE(p,q) =\{\alpha\in T(V):d(\alpha)<q\}$. 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 $AM(p,q)$.
Publisher version
AccessOpen access
Appears in Collections:CMAT - Artigos em revistas com arbitragem / Papers in peer review journals

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