The number of Moore families on image

This paper studies the generating problem for Moore families on an n-set (i.e. families closed under intersection containing the n-set) or closure operators. We show a bijection between Moore families and ideal color sets of the colored poset based on image, where image is the sum of n Boolean lattices with image atoms. By applying an algorithm to generate ideal color sets, we can determine that the number of Moore families on 6 elements is exactly 75 973 751 474.