Some binary relations and operations on the set of fuzzy partitions (Q1311844)

A fuzzy partition of \(n\) objects into \(k\) clusters can be described by a \(k \times n\) matrix where the \(i\)-th row is a fuzzy set on the universe of objects (i.e. the \(i\)-th row contains the \(n\) membership values of the objects reflecting the degree to which they belong to the \(i\)-th cluster) and where the \(j\)-th column (the ``partition vector'') reflects for the \(j\)-th object the partition of the total membership 1 on the \(k\) clusters. The author considers the aggregation of two different fuzzy partitions. The usual union or intersection between fuzzy sets do not form a fuzzy partition. Therefore, the author defines a special union \(\vee\) and intersection \(\land\) of fuzzy partitions which preserve the partition property and proves that fuzzy partitions furnished with \(\vee\) and \(\wedge\) form a distributive lattice and that \(\vee\) generates a partial ordering in the usual way: \(U\leq V\) iff \(U \vee V=V\).