Fragments of Quasi-Nelson: Two Negations.

The variety of quasi-Nelson algebras has been recently singled out and characterised in several equivalent ways: among others, as (1) the class of bounded commutative integral (but not necessarily involutive) residuated lattices satisfying the Nelson identity, as well as (2) the class of (0, 1)-congruence orderable commutative integral residuated lattices. Logically, quasi-Nelson algebras are the algebraic counterpart of quasi-Nelson logic, which is the (algebraisable) extension of the substructural logic FLew (Full Lambek calculus with Exchange and Weakening) by the Nelson axiom. Quasi-Nelson logic may also be viewed as a common generalisation of both Nelson’s constructive logic with strong negation and intuitionistic logic. The present paper focusses on the subreducts of quasi-Nelson algebras obtained by eliding the implication while keeping the two term-definable negations. It is shown that, similarly to the involutive case (treated by A. Sendlewski in 1991), this class of algebras is a variety that can be characterised by means of twist-structures over pseudo-complemented distributive lattices. In this way we extend to a non-involutive setting the well-known connection between Nelson and Heyting algebras, as well as Sendlewski’s result relating Kleene algebras with a weak pseudo-complementation and pseudocomplemented distributive lattices.

Authors

• Umberto Rivieccio

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