Abstract
A collection of sets on a ground set U n (U n = {1,2,...,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 in [1] to enumerate the number of Moore families on U 7.
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Colomb, P., Irlande, A., Raynaud, O. et al. Recursive decomposition and bounds of the lattice of Moore co-families. Ann Math Artif Intell 67, 109–122 (2013). https://doi.org/10.1007/s10472-013-9345-y
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DOI: https://doi.org/10.1007/s10472-013-9345-y