J Austral Math Soc Ser A 58 pp248--280, 1995.

The Quest for Strong Dualities

David M. Clark and Brian A. Davey

Abstract

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, Priestly 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.

1991 AMS Subject Classification: 08C05, 08C15, 18A40

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Authors

David M. Clark

Mathematics and Computer Science, SUNY, College at New Paultz, NY 12561, USA.

Brian A. Davey

Mathematcs, La Trobe UNiversity, Bundoora, Victoria 3083, Australia.