The class of Kleene algebras satisfying an interpolation property and Nelson algebras
The paper intends to clarify the role of the interpolation property in the theory of Kleene algebras and in the representation theory of Nelson algebras. Firstly it is shown that the Kalman construction, which associates each bounded distributive lattice \({\mathcal L}\) to a Kleene algebra K(\({\mathcal L})\), yields a functor K and that K has a left adjoint L. The adjunction is an equivalence, if \({\mathcal L}\) is restricted to the category of the centered Kleene algebras satisfying the interpolation property. By restricting K to subcategories, various adjunctions are obtained in the paper: adjunctions from the category of Heyting algebras to the one of Nelson algebras, from the one of Boolean algebras to the one of three valued Lukasiewicz algebras, and from the one of Stone algebras to the one of regular \(\alpha\)-De Morgan algebras. The characterizations of injective Nelson algebras, of subdirectly irreducible ones, and of simple ones are given by using the adjunctions. The congruence lattices of Nelson algebras and of \(\alpha\)-De Morgan algebras are also studied.
scientific article; zbMATH DE number 4006314