Recursive decomposition and bounds of the lattice of Moore co-families

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.