Abstract
A generalized implication on a distributive lattice \(\varvec{A}\) is a function between \(\varvec{A} \times \varvec{A}\) to ideals of \(\varvec{A}\) satisfying similar conditions to strict implication of weak Heyting algebras. Relative anihilators and quasi-modal operators are examples of generalized implication in distributive lattices. The aim of this paper is to study some classes of distributive lattices with a generalized implication. In particular, we prove that the class of Boolean algebras endowed with a quasi-modal operator is equivalent to the class of Boolean algebras with a generalized implication. This equivalence allow us to give another presentation of the class of quasi-monadic algebras and the class of compingent algebras defined by H. De Vries. We also introduce the notion of gi-sublattice and we characterize the simple and subdirectly irreducible distributive lattices with a generalized implication through topological duality.
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We thank the referee for their useful comments that have contributed to improving the presentation of the paper.
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This work was supported by Consejo Nacional de Investigaciones Científicas y Técnicas (PIP 11220200101301, CONICET-Argentina) and Agencia Nacional de Promoción Científica y Tecnológica (PICT2019-2019-00882, 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).
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Calomino, I., Castro, J., Celani, S. et al. A Study on Some Classes of Distributive Lattices with a Generalized Implication. Order (2023). https://doi.org/10.1007/s11083-023-09652-8
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DOI: https://doi.org/10.1007/s11083-023-09652-8