We introduce the category of Heyting frames, those coherent frames L in which the compact elements form a Heyting subalgebra of L, and show that it is equivalent to the category of Heyting algebras and dually equivalent to the category of Esakia spaces. This provides a frame-theoretic perspective on Esakia duality for Heyting algebras. We also generalize these results to the setting of Brouwerian algebras and Brouwerian semilattices by introducing the corresponding categories of Brouwerian frames and extending the above equivalences and dual equivalences. This provides a frame-theoretic perspective on generalized Esakia duality for Brouwerian algebras and Brouwerian semilattices.

A frame-theoretic perspective on Esakia duality / G. Bezhanishvili, L. Carai, P.J. Morandi. - In: ALGEBRA UNIVERSALIS. - ISSN 0002-5240. - 84:4(2023), pp. 30.1-30.24. [10.1007/s00012-023-00827-3]

A frame-theoretic perspective on Esakia duality

L. Carai
Secondo
;
2023

Abstract

We introduce the category of Heyting frames, those coherent frames L in which the compact elements form a Heyting subalgebra of L, and show that it is equivalent to the category of Heyting algebras and dually equivalent to the category of Esakia spaces. This provides a frame-theoretic perspective on Esakia duality for Heyting algebras. We also generalize these results to the setting of Brouwerian algebras and Brouwerian semilattices by introducing the corresponding categories of Brouwerian frames and extending the above equivalences and dual equivalences. This provides a frame-theoretic perspective on generalized Esakia duality for Brouwerian algebras and Brouwerian semilattices.
Heyting algebra; Esakia duality; Coherent frame; Algebraic frame; Brouwerian algebra; Brouwerian semilattice
Settore MAT/01 - Logica Matematica
Settore MAT/02 - Algebra
2023
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/1011749
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