Hyper-MacNeille Completions of Heyting Algebras

Harding, John; Lauridsen, Frederik

doi:10.1007/s11225-021-09941-6

Cited by 2 publications

(3 citation statements)

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“…If in addition X is a BH-space, then the interior of an upset is an upset (see e.g. [11,Lem. 3.6]), and hence joins and meets are computed as follows (see, e.g., [11,Thm.…”

Section: 3mentioning

confidence: 99%

“…More specifically, in (10) we use that finite joins distribute over infinite meets; this is not true in a complete Heyting algebra that is not bi-Heyting. In (11) we use that the interior of a closed upset is a clopen upset; this is not true in an extremally order-disconnected Esakia space that is not a BH-space. Thus, further insight is needed to attack Kuznetsov's original problem.…”

Section: Incompletenessmentioning

confidence: 99%

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A negative solution of Kuznetsov's problem for varieties of bi-Heyting algebras

Bezhanishvili¹,

Gabelaia²,

Jibladze³

2021

Preprint

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We show that there exist (continuum many) varieties of bi-Heyting algebras that are not generated by their complete members. It follows that there exist (continuum many) extensions of the Heyting-Brouwer logic HB that are topologically incomplete. This result provides further insight into the long-standing open problem of Kuznetsov by yielding a negative solution of the reformulation of the problem from extensions of IPC to extensions of HB.

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“…If in addition X is a BH-space, then the interior of an upset is an upset (see e.g. [11,Lem. 3.6]), and hence joins and meets are computed as follows (see, e.g., [11,Thm.…”

Section: 3mentioning

confidence: 99%

Section: Incompletenessmentioning

confidence: 99%

A negative solution of Kuznetsov's problem for varieties of bi-Heyting algebras

Bezhanishvili¹,

Gabelaia²,

Jibladze³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Let A be a centrally supplemented Heyting algebra. It is well known that the Mac-Neille completion of a Heyting algebra is also a Heyting algebra, see, e.g., [36,Thm. 2.3].…”

Section: Supplemented Heyting Algebrasmentioning

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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Hyper-MacNeille Completions of Heyting Algebras

Cited by 2 publications

References 43 publications

A negative solution of Kuznetsov's problem for varieties of bi-Heyting algebras

A negative solution of Kuznetsov's problem for varieties of bi-Heyting algebras

Hyper-MacNeille Completions of Heyting algebras

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