Partition distances

Alternative novel measures of the distance between any two partitions of a n-set are proposed and compared, together with a main existing one, namely partition-distance D(·, ·). The comparison achieves by checking their restriction to modular elements of the partition lattice, as well as in terms of suitable classifiers. Two of the new measures obtain through the size, a function mapping every partition into the number of atoms finer than that partition. One of these size-based distances extends to geometric lattices the traditional Hamming distance between subsets, when these latter are regarded as hypercube vertexes or binary n-vectors. After carefully framing the environment, a main comparison finally results from the following bounding problem: for every value k, with 0 < k < n, of partition-distance D(·, ·), determine the minimum and maximum of the indicator-Hamming distance δ(P,Q) proposed here over all pairs of partitions P,Q such that D(P,Q) = k.