Abstract
A lattice P is transferable for a class of lattices \(\mathcal {K}\) if whenever P can be embedded into the ideal lattice \(\mathfrak {I}K\) of some \(K\in \mathcal {K}\), then P can be embedded into K. There is a rich theory of transferability for lattices. Here we introduce the analogous notion of MacNeille transferability, replacing the ideal lattice \(\mathfrak {I}K\) with the MacNeille completion \(\overline{K}\). Basic properties of MacNeille transferability are developed. Particular attention is paid to MacNeille transferability in the class of Heyting algebras where it relates to stable classes of Heyting algebras, and hence to stable intermediate logics.
Article PDF
Similar content being viewed by others
References
Baker, K.A., Hales, A.W.: From a lattice to its ideal lattice. Algebra Univ. 4, 250–258 (1974)
Balbes, R., Dwinger, P.: Distributive Lattices. University of Missouri Press, Columbia (1974)
Balbes, R., Horn, A.: Projective distributive lattices. Pac. J. Math. 33, 273–279 (1970)
Bezhanishvili, G. (ed.): Leo Esakia on Duality in Modal and Intuitionistic Logics. Springer, Dordrecht (2014)
Bezhanishvili, G., Bezhanishvili, N.: Locally finite reducts of Heyting algebras and canonical formulas. Notre Dame J. Form. Log. 58, 21–45 (2017)
Bezhanishvili, G., Bezhanishvili, N., Ilin, J.: Cofinal stable logics. Studia Logica 104, 1287–1317 (2016)
Bezhanishvili, G., Gehrke, M.: Completeness of S4 with respect to the real line: revisited. Ann. Pure Appl. Log. 131, 287–301 (2005)
Chang, C.C., Keisler, H.J.: Model Theory, vol. 73, 3rd edn. North-Holland Publishing Co, Amsterdam (1990)
Ciabattoni, A., Galatos, N., Terui, K.: MacNeille completions of FL-algebras. Algebra Univ. 66, 405–420 (2011)
Crawley, P., Dilworth, R.P.: Algebraic Theory of Lattices. Prentice-Hall, Upper Saddle River (1973)
Esakia, L.: Heyting Algebras I: Duality Theory (Russian). Metsniereba, Tbilisi (1985)
Galvin, F., Jónsson, B.: Distributive sublattices of a free lattice. Can. J. Math. 13, 265–272 (1961)
Gaskill, H.S.: On the relation of a distributive lattice to its lattice of ideals. Bull. Austral. Math. Soc. 7, 377–385 (1972). (corrigendum, ibid. 8 (1973), 317–318 )
Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 238, 345–371 (2001)
Gehrke, M., Harding, J., Venema, Y.: MacNeille completions and canonical extensions. Trans. Am. Math. Soc. 358, 573–590 (2006)
Grätzer, G.: Universal algebra. In: Trends in Lattice Theory ((Sympos., U.S. Naval Academy, Annapolis, Md., 1966), pp. 173–210. Van Nostrand Reinhold, New York (1970)
Grätzer, G.: Lattice Theory: Foundation. Springer, Basel (2011)
Harding, J.: Any lattice can be regularly embedded into the MacNeille completion of a distributive lattice. Houston J. Math. 19, 39–44 (1993)
Harding, J., Bezhanishvili, G.: MacNeille completions of Heyting algebras. Houston J. Math. 30, 937–952 (2004)
Horn, A., Kimura, N.: The category of semilattices. Algebra Univ. 1, 26–38 (1971)
Kostinsky, A.: Projective lattices and bounded homomorphisms. Pac. J. Math. 40, 111–119 (1972)
Lauridsen, F.M.: Intermediate logics admitting a structural hypersequent calculus. Studia Logica. https://doi.org/10.1007/s11225-018-9791-y
Nelson, E.: The embedding of a distributive lattice into its ideal lattice is pure. Algebra Univ. 4, 135–140 (1974)
Rasiowa, H., Sikorski, R.: The Mathematics of Metamathematics. Monografie Matematyczne, vol. 41. Państwowe Wydawnictwo Naukowe, Warsaw (1963)
Wehrung, F.: Relative projectivity and transferability for partial lattices. Order 35, 111–132 (2018)
Author information
Authors and Affiliations
Corresponding author
Additional information
This project has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Skodowska-Curie Grant agreement No 689176.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Bezhanishvili, G., Harding, J., Ilin, J. et al. MacNeille transferability and stable classes of Heyting algebras. Algebra Univers. 79, 55 (2018). https://doi.org/10.1007/s00012-018-0534-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00012-018-0534-8