Closure systems, implicational systems, overhanging relations and the case of hierarchical classification

Abstract Moore families and closure operators, especially those appearing in hierarchical classification, are considered here from the point of view of their related implicational systems and overhanging relations. The last ones, newly introduced here, generalize the “nesting relations” defined by Adams [J. Classification 3 (1986) 299] in the case of classification trees. Here we characterize overhanging relations by three axioms, and prove that they are in a one-to-one correspondence with Moore families. We study which properties of implicational systems and overhanging relations are general, and which are specific to hierarchical structures. We also characterize canonical implication bases of hierarchies and obtain a similar result for overhangings.

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