Compatibility of t-norms with the concept of ϵ-partition

In this paper we study the behaviour of a kind of partitions formed by fuzzy sets, the ϵ-partitions, with respect to three important operations: refinement, union and product of partitions. In the crisp set theory, the previous operations lead to new partitions: every refinement of a partition is also a partition; the union of partitions of disjoint sets is a partition of the union set; the product of two partitions of two sets is a partition of the intersection of the partitioned sets. It has been proven that ϵ-partitions extend the three previous properties when the intersection of fuzzy sets is defined by the minimum t-norm and the union by the maximum t-conorm. In this paper we consider any t-norm defining the intersection of fuzzy sets and we characterize those t-norms for which refinements, unions and products of ϵ-partitions are ϵ-partitions. We pay special attention to these characterizations in the case of continuous t-norms.