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The lattices of Moore families and closure operators on a finite set: A survey. (English) Zbl 0971.06009

Janowitz, Melvin F. (ed.), OSDA98. Ordinal and symbolic data analysis. Univ. of Massachusetts, Amherst, MA, USA, September 28-30, 1998. Amsterdam: Elsevier, Electron. Notes Discrete Math. 2, no pag., electronic only (1999).
Summary: We present a survey of properties of the lattice of Moore families (families of subsets of a set \(S\) containing \(S\) and closed by set intersection) on a finite set \(S\), with proofs of the more significant results. In particular, we prove that this lattice is atomistic and lower bounded and that there exists a canonical basis allowing to represent any Moore family by “implicational” Moore families. The notion of Moore family has many cryptomorphic versions, especially the notions of closure operator and of (full) implicational system, which occur in many fields in pure or applied mathematics and computer science.
For the entire collection see [Zbl 0968.00024].

MSC:

06B05 Structure theory of lattices
06A15 Galois correspondences, closure operators (in relation to ordered sets)

Software:

SAS; SAS/STAT; C4.5
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References:

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