From fuzzy sets to the decompositions of non-rigid sets

On the uniqueness of L -fuzzy sets in the representation of families of sets

“…Among all the topics on L-fuzzy mathematics, the representation of a poset by an L-fuzzy set is very interesting, which in the case of a fuzzy set had been studied by Šešelja and Tepavčević [14,15,17], and by Jaballah and Saidi [8] who investigated the characterization of all fuzzy sets of X that can be identified with a given arbitrary family C of subsets of X together with a given arbitrary subset of [0,1], in particular, they gave necessary and sufficient conditions for both existence and uniqueness of such fuzzy sets. Later, Saidi and Jaballah investigated the problem of uniqueness under different considerations (see e.g., [11][12][13]). Also, Gorjanac-Ranitović and Tepavčević [5] formulated a necessary and sufficient condition, under which for a given family of subsets F of a set X and a fixed complete lattice L there is an L-fuzzy set µ such that the collection of cuts of µ coincides with F .…”

Generalized representation theorem and its application to the construction of fuzzy sets: Existence and uniqueness

“…Jaballah and Saidi [7,[15][16][17][18] have studied the problem, which says whether we can construct a fuzzy subsetà of U satisfying R(ξÃ) = S and M S = A S , where R(ξÃ) denotes the rage of ξÃ. However, the 0-level set was also defined as the whole universal set U in Jaballah and Saidi [7,[15][16][17][18]. As we mentioned above, the 0-level set will not be considered as the whole universal set in this paper, the set M 0 will not be assumed in the family M = {M α : α ∈ (0, 1]}.…”

“…Therefore, we shall consider the subset S of (0, 1] in this paper, and try to construct a fuzzy subsetà of U such that R(ξÃ) \ {0} = S. On the other hand, whether the supremum α * = sup S belongs to S or not is an important issue. We are going to construct a fuzzy subsetà of U such that R(ξÃ) \ {0} = S ∪ {α * }, which was not discussed in Jaballah and Saidi [7,[15][16][17][18]. The existence of construction was obtained in Jaballah and Saidi [7, Theorem 1], which was not related the representation theorem as shown in Ralescu [14,Theorem 2].…”