Abstract
In this paper we study the variety \(\textsf{WL}\) of bounded distributive lattices endowed with an implication, called weak Lewis distributive lattices. This variety corresponds to the algebraic semantics of the \(\{\vee ,\wedge ,\Rightarrow ,\bot ,\top \}\)-fragment of the arithmetical base preservativity logic \(\mathsf {iP^{-}}\). The variety \(\textsf{WL}\) properly contains the variety of bounded distributive lattices with strict implication, also known as weak Heyting algebras. We introduce the notion of WL-frame and we prove a representation theorem for WL-lattices by means of WL-frames. We extended this representation to a topological duality by means of Priestley spaces endowed with a special neighbourhood relation between points and closed upsets of the space. These results are applied in order to give a representation and a topological duality for the variety of weak Heyting–Lewis algebras, i.e., for the algebraic semantics of the arithmetical base preservativity logic \(\textsf{iP}^{-}\).
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Acknowledgements
This work was supported by Consejo Nacional de Investigaciones Científicas y Técnicas (PIP 11220170100195CO and PIP 11220200 100912CO, CONICET-Argentina), Universidad Nacional de La Plata (11X /921) and Agencia Nacional de Promoción Científica y Tecnológica (PICT20 19-2019-00882, ANPCyT-Argentina). This project has also received funding from MOSAIC Project 101007627 (European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie). We are also indebted to the anonymous referees for several improvements over the original manuscript.
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Presented by Francesco Paoli; Received July 2, 2023.
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Calomino, I., Celani, S.A. & Martín, H.J.S. On Weak Lewis Distributive Lattices. Stud Logica (2024). https://doi.org/10.1007/s11225-024-10112-6
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DOI: https://doi.org/10.1007/s11225-024-10112-6
Keywords
- Subintuitionistic logic
- Preservativity logics
- Weak Heyting algebras
- Heyting algebras
- Neighbourhood frames
- Duality theory