Abstract
Any Galois connection (between two sets) has associated with it two closure operators, and hence two complete lattices (dually isomorphic to each other). Set-theoretical Galois connections are induced by relations, and it is interesting to know which subrelations of an inducing relation will in turn induce complete sublattices of the two lattices. Complete sublattices of the Galois-lattices may also be obtained using conjugate pairs of closure operators. It is also well known that the set of fixed points of a closure (or a kernel) operator defined on a complete lattice forms a complete lattice, and in this case too we can look for properties which ensure that this lattice is a complete sublattice. This paper surveys the results known about such subrelations, conjugate pairs of closure operators, and closure and kernel operators. We also give an application of the closure and kernel operator method, which generalizes several well-known examples from universal algebra using the Galois connection (Id,Mod).
Research of the second author supported by NSERC of Canada.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arworn, S. Groupoids of Hypersubstitutions and G-solid varieties, Shaker-Verlag, Aachen 2000.
Arworn, S, K. Denecke and R. Pöschel, Closure Operators on Complete Lattices, to appear in Proc. International Conference on Ordered Algebraic Structures, Nanjing, 1998.
Denecke, K., Wismath, S. L., Universal Algebra and Applicatuions in Theoretical Computer Science, Chapman and Hall/CRC, Boca Raton 2002.
Denecke, K., Clones closed with respect to closure operators, Multi. Val. Logic, Vol. 4 (1999), 229 - 247.
Denecke, K. and S. L. Wismath, Complexity of Terms and the Galois-Connection Id-Mod, this volume pp. 91–107.
Dikranjan, D. and E. Giuli, Closure Operators I, Topology Appl. 27 (1987), 129 - 143.
Ganter, B. and R. Wille, Formale Begriffsanalyse, Springer 1996.
Graczyńska, E., On normal and regular identities and hyperidentities, Proc. Universal Algebra and Applied Algebra (Turawa 1988), World Scientific Publishing, Teaneck, N.J. 1989, 107–135.
Mel’nik, I. I., Nilpotent shifts of varieties (in Russian), Mat. Zametki 14, No. 5, 1973. English translation: Math. Notes 14, 1973, 962–966.
PÅ‚onka, J., On Varieties of Algebras Defined by Identities of Some Special Forms, Houston Journal of Mathematics, Vol. 14, no. 2, 1988, 253 - 263.
Reichel, M., Bi-Homomorphismen und Hyperidentitäten, Dissertation, Universität Potsdam 1994.
Tarski, A., A lattice theoretical fix point theorem and its application, Pacific. J. Math. 5 (1955), 285 - 310.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Denecke, K., Wismath, S.L. (2004). Galois Connections and Complete Sublattices. In: Denecke, K., Erné, M., Wismath, S.L. (eds) Galois Connections and Applications. Mathematics and Its Applications, vol 565. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-1898-5_4
Download citation
DOI: https://doi.org/10.1007/978-1-4020-1898-5_4
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-6540-7
Online ISBN: 978-1-4020-1898-5
eBook Packages: Springer Book Archive