On the redundancy of fuzzy partitions

We introduce a concept of redundancy for a family of fuzzy subsets, based on a left continuous t-norm and its corresponding residual operator, and study its basic properties, including a characterization of families with redundancy zero in some important special cases. Examples illustrate that, in the case of t-norms with zero divisors, several definitions for fuzzy partitions from the literature are insensitive to the phenomenon of redundancy. Finally, a geometric interpretation of the redundancy based on the Lukasiewicz t-norm and an upper bound for its value are given.