This talk is based on joint work in collaboration with V. Giustarini and F. Manfucci (IIIA-CSIC and Universitat de Barcelona) aimed at a better understanding of the structure of free Nelson algebras and their connection with free Heyting algebras. Nelson algebras, introduced by Rasiowa, provide the algebraic semantics for the constructive logic with strong negation developed by Nelson and Markov. They consist of bounded distributive lattices equipped with two additional operations, which can be interpreted as an implication and a strong negation. It is well known that they form a variety and can also be regarded as residuated lattices (see [3] for a recent survey collecting various facts about Nelson algebras). Free Nelson algebras play an important role in the study of the constructive logic with strong negation, as they are, up to isomorphism, the Lindenbaum–Tarski algebras of the logic.

The twist construction establishes a strong connection between Nelson algebras and Heyting algebras. In fact, the category of Nelson algebras is equivalent to a category whose objects are pairs consisting of a Heyting algebra and a Boolean filter [5]. We will see that this equivalence can be viewed as a restriction of an adjunction involving the category of Kleene lattices, and that this perspective allows us to describe the pair corresponding to a Nelson algebra free over a Kleene lattice using Heyting algebras free over bounded distributive lattices. Since free Kleene lattices are well understood, this provides an algebraic description of free Nelson algebras in terms of Heyting algebras free over bounded distributive lattices.

The connection between Nelson algebras and Heyting algebras via the twist construction also yields a duality for Nelson algebras reminiscent of Esakia duality for Heyting algebras [5]. We will exploit this duality to provide a tangible description of free algebras in various varieties of Nelson algebras. In particular, we will use the dual description of free Gödel algebras from [2] to describe the duals of free algebras in every variety of NM algebras (the algebraic counterparts of the nilpotent minimum logic, which can be thought of as particular Nelson algebras [1]), without any restriction on the cardinality of the set of free generators. A similar approach, together with the well known dual description of free pseudocomplemented distributive lattices, yields a dual description of free algebras in varieties of implication-free subreducts of Nelson algebras introduced in [6] and further studied in [4].

References:
[1] Busaniche, M., & Cignoli, R., Constructive logic with strong negation as a substructural logic, Journal of Logic and Computation, 20(4) (2010), 761–793.
[2] Carai, L., Free algebras and coproducts in varieties of Gödel algebras, 2024. Manuscript. Available at arXiv:2406.05480.
[3] Järvinen, J., Radeleczki, S., & Rivieccio, U., Nelson algebras, residuated lattices and rough sets: A survey, Journal of Applied Non-Classical Logics, 34(2–3) (2024), 368–428.
[4] Gomez, C., Marcos, M. A., & San Martin, H. J., On the relation of negations in Nelson algebras, Reports on Mathematical Logic, 56 (2021), 15–56.
[5] Sendlewski, A., Nelson algebras through Heyting ones: I, Studia Logica, 49 (1990), 105–126.
[6] Sendlewski, A., Topologicality of Kleene algebras with a weak pseudocomplementation over distributive p-algebras, Reports on Mathematical Logic, 25 (1991), 13-56.