Intermediate Logics Admitting a Structural Hypersequent Calculus

Lauridsen, Frederik

doi:10.1007/s11225-018-9791-y

Cited by 5 publications

(2 citation statements)

References 39 publications

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“…Using syntactic methods, an analogous result for residuated lattices was given in [9]. These results overlap in the setting of Heyting algebras (see [22]), and can be used to provide examples of stable classes that are closed under MacNeille completions based on omitting finite projective distributive lattices. However, since the lattices D ⊕ 1 and 0 ⊕ D ⊕ 1 are not projective in distributive lattices, the results of Theorem 6.3(2) do not fall in the scope of the results of [9].…”

Section: Stable Universal Classes and Stable Intermediate Logicsmentioning

confidence: 86%

MacNeille transferability and stable classes of Heyting algebras

et al. 2018

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Abstract.A lattice P is transferable for a class of lattices K if whenever P can be embedded into the ideal lattice IK of some K ∈ K, then P can be embedded into K. There is a rich theory of transferability for lattices. Here we introduce the analogous notion of MacNeille transferability, replacing the ideal lattice IK with the MacNeille completion K. Basic properties of MacNeille transferability are developed. Particular attention is paid to MacNeille transferability in the class of Heyting algebras where it relates to stable classes of Heyting algebras, and hence to stable intermediate logics.Mathematics Subject Classification. 06D20, 06B23, 06E15.

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Section: Stable Universal Classes and Stable Intermediate Logicsmentioning

confidence: 86%

MacNeille transferability and stable classes of Heyting algebras

et al. 2018

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“…Proof. The first three statements can be found in [39] and [38, Chap. 2], the fourth in [9], and the last in [38,Chap.…”

Section: Boolean Products and Related Sheavesmentioning

confidence: 99%

Hyper-MacNeille Completions of Heyting algebras

Harding¹,

Lauridsen²

2019

Preprint

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A Heyting algebra is supplemented if each element a has a dual pseudo-complement a + , and a Heyting algebra is centrally supplement if it is supplemented and each supplement is central. We show that each Heyting algebra has a centrally supplemented extension in the same variety of Heyting algebras as the original. We use this tool to investigate a new type of completion of Heyting algebras arising in the context of algebraic proof theory, the so-called hyper-MacNeille completion. We show that the hyper-MacNeille completion of a Heyting algebra is the MacNeille completion of its centrally supplemented extension. This provides an algebraic description of the hyper-MacNeille completion of a Heyting algebra, allows development of further properties of the hyper-MacNeille completion, and provides new examples of varieties of Heyting algebras that are closed under hyper-MacNeille completions. In particular, connections between the centrally supplemented extension and Boolean products allow us to show that any finitely generated variety of Heyting algebras is closed under hyper-MacNeille completions.

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Jankov Formulas and Axiomatization Techniques for Intermediate Logics

Bezhanishvili

2022

Outstanding Contributions to Logic

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Intermediate Logics Admitting a Structural Hypersequent Calculus

Cited by 5 publications

References 39 publications

MacNeille transferability and stable classes of Heyting algebras

MacNeille transferability and stable classes of Heyting algebras

Hyper-MacNeille Completions of Heyting algebras

Jankov Formulas and Axiomatization Techniques for Intermediate Logics

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