Fragments of Quasi-Nelson: The Algebraizable Core

Abstract This is the second of a series papers that investigate fragments quasi-Nelson logic (QNL) from an algebraic standpoint. QNL, recently introduced as common generalization intuitionistic and Nelson’s constructive with strong negation, axiomatic extension substructural $FL_{ew}$ (full Lambek calculus exchange weakening) by Nelson axiom. The counterpart QNL (quasi-Nelson algebras) class commutative integral residuated lattices (a.k.a. $FL_{ew}$-algebras) includes both Heyting algebras can be characterized algebraically in several alternative ways. present paper focuses on (a we dub implication algebras, QNI-algebras) implication–negation fragment corresponding to connectives witness algebraizability QNL. We recall main known results QNI-algebras establish number new ones. Among these, show form congruence-distributive variety (Cor. 3.15) enjoys equationally definable principal congruences congruence property (Prop. 3.16); also characterize subdirectly irreducible terms underlying poset structure (Thm. 4.23). Most these are obtained thanks twist representations for QNI-algebras, which generalize ones algebras; further introduce Hilbert-style algebraizable has its equivalent semantics.