Two Dualities for Weakly Pseudo-complemented quasi-Kleene Algebras

Abstract

Quasi-Nelson algebras are a non-involutive generalisation of Nelson algebras that can be characterised in several ways, e.g. as (i) the variety of bounded commutative integral (not necessarily involutive) residuated lattices that satisfy the Nelson identity; (ii) the class of (0, 1)-congruence orderable commutative integral residuated lattices; (iii) the algebraic counterpart of quasi-Nelson logic, i.e. the (algebraisable) extension of the substructural logic FLew by the Nelson axiom. In the present paper we focus on the subreducts of quasi-Nelson algebras obtained by eliding the implication while keeping the two term-definable negations. These form a variety that (following A. Sendlewski, who studied the corresponding fragment of Nelson algebras) we dub weakly pseudo-complemented quasi-Kleene algebras. We develop a Priestley-style duality for these algebras (in two different guises) which is essentially an application of the general approach proposed in the paper A duality for two-sorted lattices by A. Jung and U. Rivieccio.