Heyting Algebras with Kalman-Galois Connections

Abstract

Inspired by Kalman’s work, a categorical equivalence has been proved between Heyting algebras and centered Nelson algebras. In this paper, we investigate Galois connections on Heyting algebras, specifically introducing a type of Galois connection known as Kalman-Galois connections. This new concept allows us to extend the previously established equivalence to Heyting algebras with Kalman-Galois connections (KG-algebras) and to a specific class of Nelson algebras endowed with a unary operator (GNc-algebras). We provide several examples to illustrate and motivate the study of these new classes of algebras. Additionally, by utilizing Hasimoto´s results on Heyting algebras with unary operators, we characterize the subdirectly irreducible KG-algebras, with a particular focus on the simple ones. Furthermore, for a given GNc-algebra, we prove that three distinct KG-algebras can be constructed from it, which are ultimately isomorphic. Finally, we generalize Monteiro´s construction of centered Nelson algebras within the framework of GNc-algebras and establish the relationship between this construction and Kalman’s construction for KG-algebras. We also specialize the study of Kalman-Galois connections to the variety of prelinear Heyting algebras.