Search results for "Heyting algebra"

showing 4 items of 4 documents

Heyting-valued interpretations for Constructive Set Theory

2006

AbstractWe define and investigate Heyting-valued interpretations for Constructive Zermelo–Frankel set theory (CZF). These interpretations provide models for CZF that are analogous to Boolean-valued models for ZF and to Heyting-valued models for IZF. Heyting-valued interpretations are defined here using set-generated frames and formal topologies. As applications of Heyting-valued interpretations, we present a relative consistency result and an independence proof.

Discrete mathematicsLogicConstructive set theoryFormal topologyHeyting-valued modelsConstructive set theoryHeyting algebraConsistency (knowledge bases)ConstructiveAlgebraMathematics::LogicPointfree topologyConstructive set theory Heyting algebras independence proofsMathematics::Category TheoryComputer Science::Logic in Computer ScienceIndependence (mathematical logic)Heyting algebraFrame (artificial intelligence)FrameSet theoryFormal topologyMathematicsAnnals of Pure and Applied Logic
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Modal Consequence Relations Extending S4.3: An Application of Projective Unification

2016

We characterize all finitary consequence relations over $\mathbf{S4.3}$ , both syntactically, by exhibiting so-called (admissible) passive rules that extend the given logic, and semantically, by providing suitable strongly adequate classes of algebras. This is achieved by applying an earlier result stating that a modal logic $L$ extending $\mathbf{S4}$ has projective unification if and only if $L$ contains $\mathbf{S4.3}$ . In particular, we show that these consequence relations enjoy the strong finite model property, and are finitely based. In this way, we extend the known results by Bull and Fine, from logics, to consequence relations. We also show that the lattice of consequence relation…

projective unificationPure mathematicsUnificationLogicFinite model property02 engineering and technology68T15Lattice (discrete subgroup)01 natural sciencesadmissible rulesComputer Science::Logic in Computer Science0202 electrical engineering electronic engineering information engineeringCountable setFinitaryHeyting algebra08C150101 mathematics03B45MathematicsDiscrete mathematics010102 general mathematicsquasivarietiesModal logicstructural completenessconsequence relations03B35Distributive property06E25$\mathbf{S4.3}$S4.3020201 artificial intelligence & image processingNotre Dame Journal of Formal Logic
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The overlap algebra of regular opens

2010

Abstract Overlap algebras are complete lattices enriched with an extra primitive relation, called “overlap”. The new notion of overlap relation satisfies a set of axioms intended to capture, in a positive way, the properties which hold for two elements with non-zero infimum. For each set, its powerset is an example of overlap algebra where two subsets overlap each other when their intersection is inhabited. Moreover, atomic overlap algebras are naturally isomorphic to the powerset of the set of their atoms. Overlap algebras can be seen as particular open (or overt) locales and, from a classical point of view, they essentially coincide with complete Boolean algebras. Contrary to the latter, …

Discrete mathematicsAlgebra and Number Theoryoverlap algebrasNon-associative algebraBoolean algebras canonically definedComplete Boolean algebraconstructive topologyAlgebraQuadratic algebraInterior algebraComplete latticeHeyting algebraNest algebraconstructive topology; overlap algebrasMathematics
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Protoalgebraicity and the Deduction Theorem

2001

This chapter is intended as an introduction to the Deduction Theorem and to applications of this theorem in metalogic.

Pure mathematicsDeduction theoremFundamental theoremComputer Science::Logic in Computer ScienceCompactness theoremHeyting algebraSequent calculusFixed-point theoremGödel's completeness theoremSqueeze theoremMathematics
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