Structural properties and algorithms on the
                   lattice of Moore co-families


         Laurent Beaudou1 and Pierre Colomb1 and Olivier Raynaud1
 Université Blaise Pascal, Campus Universitaire des Cézeaux, 63173 Aubière, France



       Abstract. A collection of sets on a ground set Un (Un denotes the set
       {1, 2, ..., n}) closed under intersection and containing Un is known as a
       Moore family. The set of Moore families for a xed n is in bijection
       with the set of Moore co-families (union-closed families containing the
       empty set) denoted Mn . In this paper, we show that the set Mn can be
       endowed with the quotient partition associated with some operator h.
       This operator h is the main concept underlying a recursive description
       of Mn . By this way each class of the partition contains all the families
       which have the same image by h. Then we prove some structural results
       linking any Moore co-family to its image by h. From these results we
       derive an algorithm which computes eciently the image by h of any
       given Moore co-family.

       Key words: Moore co-families, Formal Concept Analysis, lattices



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