Abstract
The variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., \(\ell \)-groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated \(\ell \)-groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated \(\ell \)-groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated \(\ell \)-groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.
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Acknowledgements
We thank Ivan Chajda, Roberto Giuntini, Tomasz Kowalski, Antonino Salibra and Gavin St John for their precious suggestions. All authors gratefully acknowledge the support of Fondazione di Sardegna within the project “Resource sensitive reasoning and logic”, Cagliari, CUP: F72F20000410007, and the Regione Autonoma della Sardegna within the project “Per un’estensione semantica della Logica Computazionale Quantistica-Impatto teorico e ricadute implementative”, RAS: SR40341. F. Paoli acknowledges the support of MIUR within the project PRIN 2017: “Theory and applications of resource sensitive logics”, CUP: 20173WKCM5. D. Fazio and A. Ledda acknowledge the support of MIUR within the project PRIN 2017: “Logic and cognition. Theory, experiments, and applications”, CUP: 2013YP4N3.
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Fazio, D., Ledda, A. & Paoli, F. Residuated Structures and Orthomodular Lattices. Stud Logica 109, 1201–1239 (2021). https://doi.org/10.1007/s11225-021-09946-1
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DOI: https://doi.org/10.1007/s11225-021-09946-1
Keywords
- Residuated groupoids
- Residuated lattices
- Left-residuated groupoids
- Orthomodular lattices
- Quantum structures
- Completions
- Dedekind–MacNeille completions