Given a set Un = {1,..,n}, a collection M of subsets of Un that is closed under intersection and contains 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 itself Mn. This paper follows the work initiated in [7] and [8] about the recursive decomposition of the lattice of Moore co-families. We first show that each Moore co-family can be represented by a decomposition tree and we use this principle to design an original algorithm to close under union any given family. Then we discuss about the time complexity and give some experimental results.