Fuzzy δ-ɛ-partitions

• Authors: Susana Montes;Inés Couso;Pedro Gil
• Venue: Information Sciences: an International Journal
• Year: 2003

In this paper, the concept of a fuzzy δ-ε-partitions of any fuzzy subset of the universe Ω is introduced as a generalization of that of a classical partition. The approach is based on four observations: the first one is that for any two members of a classical partition, their intersection is the empty set if they are not equal; the second one is that the union of all members of a classical partition is equal to the partitioned set; the third one is that this definition has to be applicable for any extension of the union and intersection for fuzzy subsets; and the fourth one is that this definition has to allow us to establish a one-to-one correspondence between fuzzy partitions and some kind of fuzzy equivalence relations. Once this new concept is introduced, by considering that the constrains for union and intersection are too limiting, and that they have to be slightly modified, some equivalence definitions of δ-ε-partitions are proposed. In particular, two very important cases are studied, they are the case when δ is equal to ε, because this is very useful in practical applications, and the case when δ is equal to 1 - ε, that is, when the degree of fuzziness for the union and the intersection are equal. This case is very useful in theoretical applications and it will be related to the concept of ε-equal fuzzy subsets. Apart from the fact that they are a generalization of that of a classical partition, the new concept of δ-ε-partition is also a generalization of that of the best known definitions of fuzzy partitions proposed in the literature.