Algebraization in quasi-Nelson logics

Abstract

Quasi-Nelson logic is a recently introduced generalization of Nelson’s constructive logic with strong negation to a non-involutive setting. The present work proposes to study the logic of some fragments of quasi-Nelson logic, namely: pocrims (ℒQNP) and semihoops (ℒQNS); in addition to the logic of quasi-N4-lattices (ℒQN4). This is done by means of an axiomatization via a finite Hilbert-style calculus. The principal question which we will address is whether the algebraic semantics of a given fragment of quasi-Nelson logic (or class of quasi-N4-lattices) can be axiomatized by means of equations or quasi-equations. The mathematical tool used in this investigation will be the twist-algebra representation. Coming to the question of algebraizability, we recall that quasi-Nelson logic (as extensions of ℱℒew) is algebraizable in the sense of Blok and Pigozzi. Furthermore, we showed the algebraizability of ℒQNP, ℒQNS and ℒQN4, which is BP-algebraizable with the set of defining equations E(x) := {x = x → x} and the set of equivalence formulas ∆(x, y) := {x → y, y → x, ∼ x → ∼ y, ∼ y → ∼ x}.