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THE IDEAL STRUCTURE OF SEMIGROUPS OF TRANSFORMATIONS WITH RESTRICTED RANGE

Part of: Semigroups

Published online by Cambridge University Press:  09 December 2010

SUZANA MENDES-GONÇALVES  and
R. P. SULLIVAN
SUZANA MENDES-GONÇALVES*
Affiliation:
Centro de Matemática, Universidade do Minho, 4710 Braga, Portugal (email: [email protected])
R. P. SULLIVAN
Affiliation:
School of Mathematics and Statistics, University of Western Australia, Nedlands, 6009, Australia (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Let Y be a fixed nonempty subset of a set X and let T(X,Y ) denote the semigroup of all total transformations from X into Y. In 1975, Symons described the automorphisms of T(X,Y ). Three decades later, Nenthein, Youngkhong and Kemprasit determined its regular elements, and more recently Sanwong, Singha and Sullivan characterized all maximal and minimal congruences on T(X,Y ). In 2008, Sanwong and Sommanee determined the largest regular subsemigroup of T(X,Y ) when |Y |≠1 and Y ≠ X; and using this, they described the Green’s relations on T(X,Y ) . Here, we use their work to describe the ideal structure of T(X,Y ) . We also correct the proof of the corresponding result for a linear analogue of T(X,Y ) .

Keywords

transformation semigroupsidealsmaximalreductive

MSC classification

Secondary: 20M20: Semigroups of transformations, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 2 , April 2011 , pp. 289 - 300
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The authors acknowledge the support of the Portuguese ‘Fundação para a Ciência e a Tecnologia’ through its Multi-Year Funding Program for ‘Centro de Matemática’ at the University of Minho, Braga, Portugal.

References

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