On the mutual definability of fuzzy equivalence relations and fuzzy partitions

This paper deals with the concepts of fuzzy equivalence relations and fuzzy partitions. We define these concepts in such ways that it is possible to construct fuzzy partitions from fuzzy equivalence relations and vice versa. Our definitions have the property that a fuzzy partition 𝒫 constructed from a fuzzy equivalence relation which itself was constructed from a fuzzy partition 𝒫´ is equal to 𝒫´. The same property can be proved if the process of construction starts and ends with an equivalence relation. These two properties are called involution properties and are the essential parts of our considerations. We underline that these involution properties are very important though frequently forgotten in publications and textbooks. In this paper we consider two different definitions for transitivity and present the definitions of the corresponding fuzzy partitions