A Categorical Equivalence for Stonean Residuated Lattices
We follow the ideas given by Chen and Grätzer to represent Stone algebras and adapt them for the case of Stonean residuated lattices. Given a Stonean residuated lattice, we consider the triple formed by its Boolean skeleton, its algebra of dense elements and a connecting map. We define a category whose objects are these triples and suitably defined morphisms, and prove that we have a categorical equivalence between this category and that of Stonean residuated lattices. We compare our results with other works and show some applications of the equivalence.
KeywordsStonean residuated lattices Boolean algebras Triples
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- 1.Aguzzoli, S., T. Flaminio, and S. Ugolini, Equivalences between the subcategories of MTL-algebras via Boolean algebras and prelinear semihoops, Journal of Logic and Computation, 2017. https://doi.org/10.1093/logcom/exx014.
- 2.Bredon, G.E., Sheaf Theory, Second Edition, Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, Heidelberg, Berlin, 1997.Google Scholar
- 3.Burris, S., and H.P. Sankappanavar, A Course in Universal Algebra Graduate Texts in Mathematics, vol. 78. Springer-Verlag, New York, Heidelberg, Berlin, 1981.Google Scholar
- 7.Cignoli, R., and A. Torrens, An algebraic analysis of product logic, Multiple-Valued Logic 5:45–65, 2000.Google Scholar
- 14.Galatos, N., P. Jipsen, T. Kowalski, and H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Elsevier, New York, 2007.Google Scholar
- 15.Grätzer, G., Universal Algebra, Van Nostrand, Princeton, 1968.Google Scholar
- 16.Grätzer, G., General Lattice Theory, Academic Press, New York San Francisco, 1973.Google Scholar
- 17.Jacobson, N., Basic algebra. II, Second Edition, W. H. Freeman and Company, New York, 1989.Google Scholar
- 18.Katriňák, T., A new proof of the construction theorem for Stone algebras, Proceedings of the American Mathematical Society 40:75–78, 1973.Google Scholar
- 21.Kowalski, T., and H. Ono, Residuated lattices: An algebraic glimpse at logics without contraction. Preliminary report.Google Scholar
- 22.Mac Lane, S., Categories for the Working Mathematician. 2nd edition, Graduate Texts in Mathematics, Volume 5, Springer, Berlin, 1998.Google Scholar
- 25.McKenzie, R., G.F. McNulty, and W.E. Taylor, Algebras, Lattices, Varieties. Volume I, Wadsworth and Brooks/Cole, Monterey, 1987.Google Scholar
- 27.Schmidt, J., Quasi-decompositions, exact sequences, and triple sums of semigroups I–II, vol. 17 of Colloquia Mathematica Societatis, János Bolyai, Constributions to Universal Algebra, Szeged, 1975.Google Scholar