Abstract
The MacNeille completion of a poset (P, ≤ ) is the smallest (up to isomorphism) complete poset containing (P, ≤ ) that preserves existing joins and existing meets. It is wellknown that the MacNeille completion of a Boolean algebra is a Boolean algebra. It is also wellknown that the MacNeille completion of a distributive lattice is not always a distributive lattice (see [Fu44]). The MacNeille completion even seems to destroy many properties of the initial lattice (see [Ha93]). Weakly dicomplemented lattices are bounded lattices equipped with two unary operations satisfying the equations (1) to (3’) of Theorem 3. They generalise Boolean algebras (see [Kw04]). The main result of this contribution states that under chain conditions the MacNeille completion of a weakly dicomplemented lattice is a weakly dicomplemented lattice. The needed definitions are given in subsections 1.2 and 1.3.
2000 Mathematics Subject Classification: 06B23.
Keywords
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
Erne, M.: Distributivgesetze und Dedekind’sche Schnitte. Abh. Braunschweig Wiss. Ges. 33, 117–145 (1982)
Ganter, B., Wille, R.: Formal Concept Analysis. Mathematical Foundations. Springer, Heidelberg (1999)
Kwuida, L.: Dicomplemented Lattices. A Contextual Generalization of Boolean algebras. Shaker Verlag, Aachen (2004)
Funayama, N.: On the completion by cuts of distributive lattices. Proc. Imp. Acad. Tokyo 20, 1–2 (1944)
Harding, J.: Any lattice can be regularly embedded into the MacNeille completion of a distributive lattice. Houston J. Math. 19(1), 39–44 (1993)
Wille, R.: Boolean Concept Logic. In: Ganter, B., Mineau, G.W. (eds.) ICCS 2000. LNCS, vol. 1867, pp. 317–331. Springer, Heidelberg (2000)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Kwuida, L., Seselja, B., Tepavčević, A. (2007). On the MacNeille Completion of Weakly Dicomplemented Lattices. In: Kuznetsov, S.O., Schmidt, S. (eds) Formal Concept Analysis. ICFCA 2007. Lecture Notes in Computer Science(), vol 4390. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70901-5_17
Download citation
DOI: https://doi.org/10.1007/978-3-540-70901-5_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70828-5
Online ISBN: 978-3-540-70901-5
eBook Packages: Computer ScienceComputer Science (R0)