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 | 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. |
Keywords: | Formal concept analysis | MacNeille completion | Weakly dicomplemneted lattices | Publisher: | Springer Link |
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