Natural Dualities in Partnership

Traditionally in natural duality theory the algebras carry no topology and the objects on the dual side are structured Boolean spaces. Given a duality, one may ask when the topology can be swapped to the other side to yield a partner duality (or, better, a dual equivalence) between a category of topological algebras and a category of structures. A prototype for this procedure is provided by the passage from Priestley duality for bounded distributive lattices to Banaschewski duality for ordered sets. Moreover, the partnership between these two dualities yields as a spinoff a factorisation of the functor sending a bounded distributive lattice to its natural extension, alias, in this case, the canonical extension or profinite completion. The main theorem of this paper validates topology swapping as a uniform way to create new dual adjunctions and dual equivalences: we prove that, for every finite algebra of finite type, each dualising alter ego gives rise to a partner duality. We illustrate the theorem via a variety of natural dualities, some classic and some less familiar. Dedicated to the 75th birthday of Professor Tibor Katriňák. The first author wishes to thank the Research Institute of M. Bel University in Banská Bystrica for its hospitality while working on this paper. The second author acknowledges support from Slovak grant VEGA 1/0485/09. This work was partially supported by the Agency of the Slovak Ministry of Education for the Structural Funds of the EU, under project ITMS:26220120007. B. A. Davey (B) Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia e-mail: [email protected] M. Haviar Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia e-mail: [email protected] H. A. Priestley Mathematical Institute, University of Oxford, 24/29 St Giles, Oxford OX1 3LB, UK e-mail: [email protected] 584 B.A. Davey et al. For lattice-based algebras this leads immediately, as in the Priestley–Banaschewski example, to a concrete description of canonical extensions.