7 found
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  1.  3
    A General Framework for Product Representations: Bilattices and Beyond.L. M. Cabrer & H. A. Priestley - 2015 - Logic Journal of the IGPL 23 (5):816-841.
  2. 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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  3.  15
    Natural Dualities Through Product Representations: Bilattices and Beyond.L. M. Cabrer & H. A. Priestley - 2016 - Studia Logica 104 (3):567-592.
    This paper focuses on natural dualities for varieties of bilattice-based algebras. Such varieties have been widely studied as semantic models in situations where information is incomplete or inconsistent. The most popular tool for studying bilattices-based algebras is product representation. The authors recently set up a widely applicable algebraic framework which enabled product representations over a base variety to be derived in a uniform and categorical manner. By combining this methodology with that of natural duality theory, we demonstrate how to build (...)
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  4.  34
    Natural Dualities for Varieties Ofn-Valued Łukasiewicz Algebras.H. A. Priestley - 1995 - Studia Logica 54 (3):333 - 370.
    Natural dualities are developed for varieties ofn-valued ukasiewicz algebras with and without negation. These dualities are based on hom-functors, and parallel Stone duality for Boolean algebras. A translation is described which relates the natural dualities to the corresponding restricted Priestley dualities. This enables a unified approach to free algebras to be presented, whence R. Cignoli's characterisations of the finitely generated free algebras are elucidated and new descriptions of arbitrary free algebras obtained. Finally it is shown how dualities for subvarieties encode (...)
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  5.  21
    Duality for Double Quasioperator Algebras Via Their Canonical Extensions.M. Gehrke & H. A. Priestley - 2007 - Studia Logica 86 (1):31-68.
    This paper is a study of duality in the absence of canonicity. Specifically it concerns double quasioperator algebras, a class of distributive lattice expansions in which, coordinatewise, each operation either preserves both join and meet or reverses them. A variety of DQAs need not be canonical, but as has been shown in a companion paper, it is canonical in a generalized sense and an algebraic correspondence theorem is available. For very many varieties, canonicity (as traditionally defined) and correspondence lead on (...)
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  6.  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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  7.  29
    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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