Hilbert Algebras with Hilbert–Galois Connections

CELANI, SERGIO A.; MONTANGIE, DANIELA

STUDIA LOGICA

Springer Science and Business Media B.V.

Año: 2023 vol. 111 p. 113 - 138

0039-3215

In this paper we introduce Hilbert algebras with Hilbert–Galois connections (HilGC-algebras) and we study the Hilbert–Galois connections defined in Heyting algebras, called HGC-algebras. We assign a categorical duality between the category HilGC-algebras with Hilbert homomorphisms that commutes with Hilbert–Galois connections and Hilbert spaces with certain binary relations and whose morphisms are special functional relations. We also prove a categorical duality between the category of Heyting Galois algebras with Heyting homomorphisms that commutes with Hilbert–Galois connections and the category of spectral Heyting spaces endowed with a binary relation with certain special continuous maps.