Recursive decomposition tree of a Moore co-family and closure algorithm

Abstract

A collection of sets on a ground set Un (Un = {1,2,...,n}) closed under intersection and containing Un 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 in Colomb et al. (Ann Math Artif Intell 67(2):109–122, 2013) 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.