Topological Duality and Algebraic Completions

Gehrke, Mai

doi:10.1007/978-3-030-76920-8_4

Cited by 2 publications

(4 citation statements)

References 46 publications

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“…To define canonical extensions for Σ-algebras, we combine techiques for residuated lattices and distributive modal algebras developed in [7] and [9]. Let R = (R, ∨, ∧, •, \, /, (!…”

Section: Canonical Extensionsmentioning

confidence: 99%

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Semantic Analysis of Subexponential Modalities in Distributive Non-commutative Linear Logic

Rogozin

2023

Electron. Proc. Theor. Comput. Sci.

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In this paper, we consider the full Lambek calculus enriched with subexponential modalities in a distributive setting. We show that the distributive Lambek calculus with subexponentials is complete with respect to its Kripke frames via canonical extensions. In this approach, we consider subexponentials as S4-like modalities and each modality is interpreted with a reflexive and transitive relation similarly to usual Kripke semantics.

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“…To define canonical extensions for Σ-algebras, we combine techiques for residuated lattices and distributive modal algebras developed in [7] and [9]. Let R = (R, ∨, ∧, •, \, /, (!…”

Section: Canonical Extensionsmentioning

confidence: 99%

“…The fact that the canonical extension of the lattice reduct is a perfect distributive lattice is by Jonsson, see [6,Theorem 2.3]. The canonical extension of the residuated lattice reduct of R is also a perfect residuated lattice, see [7,Proposition 5]. The canonical extension of the !…”

Section: Canonical Extensionsmentioning

confidence: 99%

Semantic Analysis of Subexponential Modalities in Distributive Non-commutative Linear Logic

Rogozin

2023

Electron. Proc. Theor. Comput. Sci.

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“…• Its lattice reduct is perfect distributive Here we formulate canonical extensions for bounded distributive lattices with a residuated family in the fashion of [18], where canonical extensions for Heyting algebras were studied. Here we take a generalised version of that contruction formulated for residuated lattices, see [17].…”

Section: The Distributive Lambek Calculus With Modal Operatorsmentioning

confidence: 99%

“…Instead of proof that repeats this one [17], we just define • σ , \ π , and / π explicitly. Here we note that the canonical extension of a lattice reduct is a perfect distributive lattice [20].…”

Section: The Distributive Lambek Calculus With Modal Operatorsmentioning

confidence: 99%

The Distributive Full Lambek Calculus with Modal Operators

Rogozin¹

2020

Preprint

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In this paper, we study logics of bounded distributive residuated lattices with modal operators considering ✷ and ✸ in a noncommutative setting. We introduce relational semantics for such noncommutative modal logics. We prove that any canonical logic is Kripke complete via discrete duality and canonical extensions. That is, we show that a modal extension of the distributive full Lambek calculus is the logic of its frames if its variety of algebras is closed under canonical extensions. After that, we establish a Priestley-style duality between residuated distributive modal algebras and certain topological Kripke structures based on Priestley spaces.

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Topological Duality and Algebraic Completions

Cited by 2 publications

References 46 publications

Semantic Analysis of Subexponential Modalities in Distributive Non-commutative Linear Logic

Semantic Analysis of Subexponential Modalities in Distributive Non-commutative Linear Logic

The Distributive Full Lambek Calculus with Modal Operators

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