541 research outputs found

    Canonical extension and canonicity via DCPO presentations

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    The canonical extension of a lattice is in an essential way a two-sided completion. Domain theory, on the contrary, is primarily concerned with one-sided completeness. In this paper, we show two things. Firstly, that the canonical extension of a lattice can be given an asymmetric description in two stages: a free co-directed meet completion, followed by a completion by \emph{selected} directed joins. Secondly, we show that the general techniques for dcpo presentations of dcpo algebras used in the second stage of the construction immediately give us the well-known canonicity result for bounded lattices with operators.Comment: 17 pages. Definition 5 was revised slightly, without changing any of the result

    Canonical extensions and ultraproducts of polarities

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    J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra with operators has evolved into an extensive theory of canonical extensions of lattice-based algebras. After reviewing this evolution we make two contributions. First it is shown that the failure of a variety of algebras to be closed under canonical extensions is witnessed by a particular one of its free algebras. The size of the set of generators of this algebra can be made a function of a collection of varieties and is a kind of Hanf number for canonical closure. Secondly we study the complete lattice of stable subsets of a polarity structure, and show that if a class of polarities is closed under ultraproducts, then its stable set lattices generate a variety that is closed under canonical extensions. This generalises an earlier result of the author about generation of canonically closed varieties of Boolean algebras with operators, which was in turn an abstraction of the result that a first-order definable class of Kripke frames determines a modal logic that is valid in its so-called canonical frames

    A view of canonical extension

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    This is a short survey illustrating some of the essential aspects of the theory of canonical extensions. In addition some topological results about canonical extensions of lattices with additional operations in finitely generated varieties are given. In particular, they are doubly algebraic lattices and their interval topologies agree with their double Scott topologies and make them Priestley topological algebras.Comment: 24 pages, 2 figures. Presented at the Eighth International Tbilisi Symposium on Language, Logic and Computation Bakuriani, Georgia, September 21-25 200

    Duality and canonical extensions for stably compact spaces

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    We construct a canonical extension for strong proximity lattices in order to give an algebraic, point-free description of a finitary duality for stably compact spaces. In this setting not only morphisms, but also objects may have distinct pi- and sigma-extensions.Comment: 29 pages, 1 figur

    Topological Representation of Canonicity for Varieties of Modal Algebras

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    Thesis (Ph.D.) - Indiana University, Mathematics, 2010The main subject of this dissertation is to approach the question of countable canonicity of varieties of modal algebras from a topological and categorical point of view. The category of coalgebras of the Vietoris functor on the category of Stone spaces provides a class of frames we call sv-frames. We show that the semantic of this frames is equivalent to that of modal algebras so long as we are limited to certain valuations called sv-valuations. We show that the canonical frame of any normal modal logic which is directly constructed based on the logic is an sv-frame. We then define the notion of canonicity of a logic in terms of varieties and their dual classes. We will then prove that any morphism on the category of coalgebras of the Vietoris functor whose codomain is the canonical frame of the minimal normal modal logic are exactly the ones that are invoked by sv-valuations. We will then proceed to reformulate canonicity of a variety of modal algebras determined by a logic in terms of properties of the class of sv-frames that correspond to that logic. We define ultrafilter extension as an operator on the category of sv-frames, prove a coproduct preservation result followed by some equivalent forms of canonicity. Using Stone duality the notion of co-variety of sv-frames is defined. The notion of validity of a logic on a frame is presented in terms of ranges of theory maps whose domain is the given frame. Partial equivalent results on co-varieties of sv-frames are proved. We classify theory maps which are maps invoked by a valuation on a Kripke frame using the classification of sv-theory maps and properties of ultrafilter extension. A negative categorical result concerning the existence of an adjoint functor for ultrafilter extension is also proved

    Canonical functions: a proof via topological dynamics

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    Canonical functions are a powerful concept with numerous applications in the study of groups, monoids, and clones on countable structures with Ramsey-type properties. In this short note, we present a proof of the existence of canonical functions in certain sets using topological dynamics, providing a shorter alternative to the original combinatorial argument. We moreover present equivalent algebraic characterisations of canonicity

    Priestley duality for MV-algebras and beyond

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    We provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators. Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order conditions on dual spaces. In particular, we specialize our general results to the variety of MV-algebras, obtaining a duality for these in which the equations axiomatizing MV-algebras are dualized as first-order conditions
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