Existence and uniqueness of fuzzy ideals

Uniqueness results in the representation of families of sets by fuzzy sets

“…The following theorem shows that Theorem 1 of [24] still holds true in the context of fuzzy sets instead of fuzzy ideals.…”

“…Also, given any set A, we denote its cardinality by |A|. We start this section by giving necessary and sufficient conditions for uniqueness, which show that Theorem 2 of [24] still holds true in the context of fuzzy sets instead of fuzzy ideals. …”

“…In [24], the authors studied the problem of characterization of the set of all fuzzy ideals corresponding to a given collection of ideals C of a ring R and a given subset S of the interval [0, 1]. Among other things, they obtained two characterization theorems, one on existence and one on uniqueness.…”

“…Among other things, they obtained two characterization theorems, one on existence and one on uniqueness. We start this paper by considering the same problem dealt with in [24] but in the context of fuzzy sets instead of fuzzy ideals: Problem 1. Let R be a nonempty set, S be a subset of the interval [0, 1], and C be a family of nonempty subsets of R.…”

“…It turns out that, after minor modification of their proofs, the two characterization theorems of [24] on existence and uniqueness still hold true when considering fuzzy sets instead of fuzzy ideals, see Theorems 1 and 2 below.…”