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Recursive decomposition and bounds of the lattice of Moore co-families. (English) Zbl 1311.06003

Summary: A collection of sets on a ground set \(U_n\) (\(U_n=\{1,2,\ldots,n\}\)) closed under intersection and containing \(U_n\) is known as a Moore family. The set of Moore families for a fixed \(n\) is in bijection with the set of Moore co-families (union-closed families containing the empty set) denoted \(\mathbb M_n\). In this paper, we propose a recursive definition of the set of Moore co-families on \(U_n\). Then we apply this decomposition result to compute a lower bound on \(|\mathbb M_n|\) as a function of \(|\mathbb M_{n-1}|\), the Dedekind numbers and the binomial coefficients. These results follow the work carried out by P. Colomb et al. [Lect. Notes Comput. Sci. 5986, 72-87 (2010; Zbl 1274.05013)] to enumerate the number of Moore families on \(U_7\).

MSC:

06A15 Galois correspondences, closure operators (in relation to ordered sets)
05A15 Exact enumeration problems, generating functions
06B05 Structure theory of lattices

Citations:

Zbl 1274.05013
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References:

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