|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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