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Inverse semigroups generated by linear transformations

Published online by Cambridge University Press:  17 April 2009

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 WA 6009, Australia
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Suppose X is a set with |X| = p ≥ q ≥ ℵ0 and let B = BL(p, q) denote the Baer-Levi semigroup defined on X. In 1984, Howie and Marques-Smith showed that, if p = q, then BB−1 = I(X), the symmetric inverse semigroup on X, and they described the subsemigroup of I(X) generated by B−1B. In 1994, Lima extended that work to ‘independence algebras’, and thus also to vector spaces. In this paper, we answer the natural question: what happens when p > q? We also show that, in this case, the analogues BB−1 for sets and GG−1 for vector spaces are never isomorphic, despite their apparent similarities.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 71 , Issue 2 , April 2005 , pp. 205 - 213
Copyright
Copyright © Australian Mathematical Society 2005

References

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[5]Mendes-Gonçalves, S. and Sullivan, R.P., ‘Baer-Levi semigroups of linear transformations’, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 477–499.CrossRefGoogle Scholar
[6]Mendes-Gonçalves, S. and Sullivan, R.P., ‘Maximal inverse subsemigroups of the symmetric inverse semigroup on a finite-dimensional vector space’, Comm. Algebra (to appear).Google Scholar