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Baer–Levi semigroups of linear transformations

Published online by Cambridge University Press:  12 July 2007

Suzana Mendes-Gonçalves  and
R. P. Sullivan
Suzana Mendes-Gonçalves
Affiliation:
Centro de Matematica, Universidade do Minho, 4710 Braga, Portugal
R. P. Sullivan
Affiliation:
School of Mathematics and Statistics, University of Western Australia, Nedlands 6009, Australia
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Abstract

Given an infinite-dimensional vector space V, we consider the semigroup GS (m, n) consisting of all injective linear α: V → V for which codim ran α = n, where dim V = m ≥ n ≥ ℵ0. This is a linear version of the well-known Baer–Levi semigroup BL (p, q) defined on an infinite set X, where |X| = p ≥ q ≥ ℵ0. We show that, although the basic properties of GS (m, n) are the same as those of BL (p, q), the two semigroups are never isomorphic. We also determine all left ideals of GS (m, n) and some of its maximal subsemigroups; in this, we follow previous work on BL (p, q) by Sutov and Sullivan as well as Levi and Wood.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 134 , Issue 3 , June 2004 , pp. 477 - 499
Copyright
Copyright © Royal Society of Edinburgh 2004

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