(In)consistency Operators on Quasi-Nelson Algebras

We propose a preliminary study of (in)consistency operators on quasi-Nelson algebras, a variety that generalizes both Nelson and Heyting algebras; our aim is to pave the way for introducing logics of formal inconsistency (LFIs) in a non-necessarily involutive setting. We show how several results that were obtained for LFIs based on distributive involutive residuated lattices can be extended to quasi-Nelson algebras and their logic. We prove that the classes of algebras thus obtained are equationally axiomatizable, and provide a twist representation for them. Having obtained some insight on filters and congruences, we characterize the directly indecomposable members of these varieties, showing in particular that two of them are semisimple. Further logical developments and extensions of the present approach are also discussed.