A note on a natural equivalence relation on fuzzy power set

Some notes on equivalent fuzzy sets and fuzzy subgroups

“…By their definition, the two fuzzy sets in our Example 2.2 are not equivalent. On the other hand, also in Example 2.2 the family of cuts of the empty set is {, X}, so is equivalent to by the Definition of equivalence of fuzzy sets in [1]. But is equivalent to by our Definition 2.5 since their support sets are not equal to each other, so the definition of equivalence of fuzzy sets in [1] is not the same as our Definition 2.5.…”

“…In Example 2.2, the collection of level subsets of is {, X}, while the collection of level subsets of is {X}, but it is easy to examine ≈ holds. In [1] they define another kind of equivalence of two fuzzy sets, i.e., two fuzzy sets is equivalent if and only if they have equal families of cuts. Their method can be viewed as the algebra method since their equivalence of fuzzy sets is defined by using of isomorphism between two lattices.…”

Non-standard cut classification of fuzzy sets

“…Systematic investigation of the analogue equality for ordinary cut sets (in finite case) can be found in Murali and Makamba [10] and in some previous investigation (as cited in [10]) in Makamba [9] and Alkhamees [1]. The problem has been completely solved in Šešelja and Tepavčević [19] (in case of lattice-valued fuzzy sets) and by the same authors in [20] in most general settings (with a poset as a co-domain). Here we give necessary and sufficient conditions under which two lattice-valued fuzzy sets with the same domain and co-domain have equal collections of these non-standard cut sets.…”

Soft Intervals and Soft Ordered Topology

“…[7] gave a matrix representation of soft sets and using this representations made soft set have a rich computer application potential. Recently, the properties and applications on the soft set theory [1], [12], [13], [14], [15], [23], [21] and the fuzzy soft set theory [16], [4], [26], [9], [25], [5] have been studied increasingly. Babitha and Sunil [2] defined soft set relation and [3] ordering on soft sets.…”