Quantales

Paseka, Jan; Rosický, Jiřı́

doi:10.1007/978-94-017-1201-9_10

Cited by 20 publications

(9 citation statements)

References 14 publications

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“…All we have to do is, by the previous theorem, prove that for all (y, z) ∈ R T the conditions (10)- (11) are satisfied for all a, b ∈ T ς (L). This has already been done for the pairs of the form (ςy, yy * ) in ¶5.10, so we only have to concern ourselves with the pair (1 L , α).…”

Section: If ♦ and Are Conjugate We Havementioning

confidence: 99%

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An algebraic generalization of Kripke structures

Marcelino¹,

Resende²

2008

Math. Proc. Camb. Phil. Soc.

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The Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4, and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL, and the ramified temporal logic CTL.

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Section: If ♦ and Are Conjugate We Havementioning

confidence: 99%

“…It remains to prove that the pair (y, z) = (α 2 , α) satisfies (10)- (11). The first condition is proved as follows:…”

Section: If ♦ and Are Conjugate We Havementioning

confidence: 99%

An algebraic generalization of Kripke structures

Marcelino¹,

Resende²

2008

Math. Proc. Camb. Phil. Soc.

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“…There is a canonical embedding η : M → W(M ) given by a →↑ a; the mapping is antitone (M, ≤) → (U(M ), ⊆), thus W(M ) = (U(M ), ⊇). See more on quantales in [2,11,17,19,20] and [12,13,21] for free algebras in general. The monoids in the definition are collectively called value monoids.…”

Section: Value Monoids and Generalized Distancesmentioning

confidence: 99%

Topographic spaces over ordered monoids

Pavlík¹

2015

Math. Appl.

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Abstract. A topography on a set is considered to be a collection of features described by two valuations: distance and elevation. Spaces with such structure will be studied on a general level, with generalized metric (gem) and pseudometric (pseudogem). We show many pseudogem distances and characteristics, particularly those related to the elevation function. We focus on paths in the topographic images and cohesiveness (generalized continuity) of their compositions with the elevation function. Special emphasis is also placed on the spaces arising from the digital geometry and, therefore, offering applications in image processing.

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“…For example, standard categories are just categories over Set , 2-categories are defined to be categories over Cat , and quantaloids may be considered as categories enriched in join complete lattices [Borceux and Stubbe 2000]. Quantales then represent the monoidal one-object restrictions [Paseka and Rosický 2000]. We obtain quantale and quantaloid morphisms as the corresponding JCLatt-enriched functors.…”

Section: Category Theorymentioning

confidence: 99%

Operational Galois Adjunctions

Coecke¹,

Moore

2000

Current Research in Operational Quantum Logic

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We present a detailed synthetic overview of the utilisation of categorical techniques in the study of order structures together with their applications in operational quantum theory. First, after reviewing the notion of residuation and its implementation at the level of quantaloids we consider some standard universal constructions and the extension of adjunctions to weak morphisms. Second, we present the categorical formulation of closure operators and introduce a hierarchy of contextual enrichments of the quantaloid of complete join lattices. Third, we briefly survey physical state-property duality and the categorical analysis of derived notions such as causal assignment and the propagation of properties.

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Quantales

Abstract: In this paper we survey aspects of the theory of quantales, starting from its connection to locales and C*-algebras and finishing with recent developments involving the notions simplicity and spatiality.

Cited by 20 publications

References 14 publications

An algebraic generalization of Kripke structures

An algebraic generalization of Kripke structures

Topographic spaces over ordered monoids

Operational Galois Adjunctions

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