MacNeille transferability and stable classes of Heyting algebras

Bezhanishvili, Guram; Harding, John; Ilin, Julia; Lauridsen, Frederik

doi:10.1007/s00012-018-0534-8

Cited by 3 publications

(1 citation statement)

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“…7.3] without recourse to syntactic analysis. We note that for any such variety the class of finitely subdirectly irreducible algebras is always closed under MacNeille completions [9], see also [3] for an alternative argument. A positive answer to the following question would solve this problem.…”

Section: Discussionmentioning

confidence: 97%

Hyper-MacNeille Completions of Heyting algebras

Harding¹,

Lauridsen²

2019

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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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Section: Discussionmentioning

confidence: 97%