Abstract
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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Acknowledgements
The authors are thankful to the anonymous referees for several remarks which helped correcting and improving our results.
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Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This research was supported by: Consejo Nacional de Investigaciones Científicas y Técnicas (PIP 11220200101301CO) and Agencia Nacional de Promoción Científica y Tecnológica (PICT2019-2019-00882, ANPCyT-Argentina); MOSAIC Project 101007627 (European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie); the I+D+i research project (grant number PID2019-110843GA-I00) La geometría de las lógicas no-clásicas, funded by the Ministry of Science and Innovation of Spain.
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Celani, S.A., Rivieccio, U. Intuitionistic Modal Algebras. Stud Logica 112, 611–660 (2024). https://doi.org/10.1007/s11225-023-10065-2
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DOI: https://doi.org/10.1007/s11225-023-10065-2