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

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.