Three-way fuzzy partitions defined by shadowed sets.

Pedrycz introduces the notion of a shadowed set as a three-way approximation of a fuzzy set. Given a pair of thresholds (α, β), a shadowed set approximates a fuzzy set by three regions: an α-core of the fuzzy set, the difference between a β-support and the α-core, and the complement of the β-support. A fuzzy partition is a family of nonempty fuzzy sets satisfying certain conditions. When applying ideas of shadowed sets to a family of nonempty fuzzy sets, the result is a three-way approximation of the family, that is, a family of shadowed sets. This enables us to study a new type of three-way fuzzy partitions. We introduce and examine three pairs of properties for defining a three-way fuzzy partition: disjointness property, overlap property, and coverage property of α-cores and β-supports. We propose a general definition of a three-way fuzzy partition by using two sets of properties. By considering all possible non-equivalent subsets of properties, we obtain 21 classes of three-way fuzzy partition. We study three classes of three-way fuzzy partition and establish their connections to partitions, fuzzy δ-ɛ-partitions, and interval-set clusters.