On verifying and engineering the wellgradedness of a union-closed family

Current techniques for generating a knowledge space, such as QUERY, guarantee that the resulting structure is closed under union, but not that it satisfies wellgradedness, which is one of the defining conditions for a learning space. We give necessary and sufficient conditions on the base of a union-closed set family that ensures that the family is well-graded. We consider two cases, depending on whether or not the family contains the empty set. We also provide algorithms for efficiently testing these conditions, and for augmenting a set family in a minimal way, to one that satisfies these conditions.