Intuitionistic Modal Algebras

Studia Logica:1-50 (forthcoming)
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Recent research on algebraic models of _quasi-Nelson logic_ has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special modal operator, known in the literature as a _nucleus_. Among these various algebraic structures, for which we employ the umbrella term _intuitionistic modal algebras_, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for their primitive operations arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of _weak implicative semilattices_, whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisfies the postulates of Celani and Jansana’s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus.

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Author Profiles

Umberto Rivieccio
Universidad Nacional de Educación a Distancia

References found in this work

Constructible falsity.David Nelson - 1949 - Journal of Symbolic Logic 14 (1):16-26.
Fragments of quasi-Nelson: residuation.U. Rivieccio - 2023 - Journal of Applied Non-Classical Logics 33 (1):52-119.
Fragments of Quasi-Nelson: The Algebraizable Core.Umberto Rivieccio - 2022 - Logic Journal of the IGPL 30 (5):807-839.
Nelson algebras through Heyting ones: I.Andrzej Sendlewski - 1990 - Studia Logica 49 (1):105-126.

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