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The quest for strong dualities

Part of: General theory of categories and functors Other classes of algebra

Published online by Cambridge University Press:  09 April 2009

David M. Clark  and
Brian A. Davey
David M. Clark
Affiliation:
Mathematics and Computer Science, SUNY, College at New Paltz, NY 12561, USA
Brian A. Davey
Affiliation:
Mathematics, La Trobe University, Bundoora, Victorial 3083, Australia
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Abstract

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We give a revised and updated exposition of the theory of full dualities initiated by Clark, Davey, Krauss and Werner, introducing the (stronger) notion of a strong duality. All known full dualities turn out to be strong. A series of theorems which provide necessary and sufficient conditions for a strong duality to exist is proved. All full dualities in the literature can be obtained from these results and many new strong dualities can be derived. In particular, we show that within congruence distributive varieties every duality can be upgraded to a strong duality. Amongst the new strong dualities are the dualities of Davey, Priestley and Werner for the varieties of pseudocomplemented distributive lattices which are either strong as they stand or can easily be made strong by the addition of partial operations to the dual structures.

Keywords

duality theoryfull dualitystrong dualitycongruence distributivitynear unanimity.

MSC classification

Secondary: 08C05: Categories of algebras 08C15: Quasivarieties 18A40: Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 58 , Issue 2 , April 1995 , pp. 248 - 280
Copyright
Copyright © Australian Mathematical Society 1995

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