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  1.  73
    Optimal Natural Dualities for Varieties of Heyting Algebras.B. A. Davey & H. A. Priestley - 1996 - Studia Logica 56 (1-2):67 - 96.
    The techniques of natural duality theory are applied to certain finitely generated varieties of Heyting algebras to obtain optimal dualities for these varieties, and thereby to address algebraic questions about them. In particular, a complete characterisation is given of the endodualisable finite subdirectly irreducible Heyting algebras. The procedures involved rely heavily on Priestley duality for Heyting algebras.
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  2.  14
    The Syntax and Semantics of Entailment in Duality Theory.B. A. Davey, M. Haviar & H. A. Priestley - 1995 - Journal of Symbolic Logic 60 (4):1087-1114.
    Both syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formulae and a new derivation of the corresponding result in clone theory, viz. the syntactic description of $\operatorname{Inv(Pol}(R))$ for a given set R of finitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given in the form (...)
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  3.  21
    Canonical Extensions and Discrete Dualities for Finitely Generated Varieties of Lattice-Based Algebras.B. A. Davey & H. A. Priestley - 2012 - Studia Logica 100 (1-2):137-161.
    The paper investigates completions in the context of finitely generated lattice-based varieties of algebras. In particular the structure of canonical extensions in such a variety $${\mathcal {A}}$$ is explored, and the role of the natural extension in providing a realisation of the canonical extension is discussed. The completions considered are Boolean topological algebras with respect to the interval topology, and consequences of this feature for their structure are revealed. In addition, we call on recent results from duality theory to show (...)
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  4.  6
    Restricted Priestley Dualities and Discriminator Varieties.B. A. Davey & A. Gair - 2017 - Studia Logica 105 (4):843-872.
    Anyone who has ever worked with a variety \ of algebras with a reduct in the variety of bounded distributive lattices will know a restricted Priestley duality when they meet one—but until now there has been no abstract definition. Here we provide one. After deriving some basic properties of a restricted Priestley dual category \ of such a variety, we give a characterisation, in terms of \, of finitely generated discriminator subvarieties of \. As an application of our characterisation, we (...)
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