Authors: Kwuida, Léonard
Šešelja, Branimir
Tepavčević, Andreja 
Title: On the MacNeille completion of weakly dicomplemented lattices
Journal: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume: 4390 LNAI
First page: 271
Last page: 280
Conference: 5th International Conference on Formal Concept Analysis, ICFCA 2007; Clermont-Ferrand; France; 12 February 2007 through 16 February 2007
Issue Date: 1-Jan-2007
Rank: M23
ISBN: 978-3-540-70901-5
ISSN: 0302-9743
DOI: 10.1007/978-3-540-70901-5_17
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.
Keywords: Formal concept analysis | MacNeille completion | Weakly dicomplemneted lattices
Publisher: Springer Link

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