Recursive decomposition tree of a Moore co-family and closure algorithm. (English) Zbl 1311.06004

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 \(\mathbb M_n\). This paper follows the work initiated by P. Colomb et al. [Ann. Math. Artif. Intell. 67, No. 2, 109-122 (2013; Zbl 1311.06003)] in which we have given an inductive definition of the lattice of Moore co-families. In the present paper we use this definition to define a recursive decomposition tree of any Moore co-family and we design an original algorithm to compute the closure under union of any family. Then we compare performance of this algorithm to performance of Ganter’s algorithm and Norris’ algorithm.


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


Zbl 1311.06003
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