Quasi-Nelson Algebras

We introduce a generalization of Nelson algebras having a not-necessarily involutive negation; we suggest to dub this class quasi-Nelson algebras in analogy with quasi-De Morgan lattices, these being a non-involutive generalization of De Morgan lattices. We show that, similarly to the involutive case (and perhaps surprisingly), our new class of algebras can be equivalently presented as (1) quasi-Nelson residuated lattices, i.e. models of the well-known Full Lambek calculus with exchange and weakening, extended with the Nelson axiom; (2) non-involutive twist-structures, i.e. special products of Heyting algebras, which generalize the well-known construction for representing algebraic models of Nelson's constructive logic with strong negation; (3) quasi-Nelson algebras, i.e. models of non-involutive Nelson logic viewed as a conservative expansion of the negation-free fragment of intuitionistic logic. The equivalence of the three presentations, and in particular the extension of the twist-structure representation to the non-involutive case, is the main technical result of the paper. We hope, however, that the main impact may be the possibility of opening new ways to (i) obtain deeper insights into the distinguishing feature of Nelson's logic (the Nelson axiom) and its algebraic counterpart; (ii) be able to investigate certain purely algebraic properties (such as 3-potency and (0,1)-congruence orderability) in a more general setting.