Intuitionistic fuzzy level operators related to the degree of uncertainty

: Two new intuitionistic fuzzy operators of level type are introduced and some of their properties are studied. They are related to the degree of uncertainty of a given intuitionistic fuzzy set. Geometrical interpretations onto the intuitionistic fuzzy interpretation triangle are provided

Their properties and geometrical interpretations were studied by Todorova and Vassilev in [6].
Another level operator, N γ , was recently defined and investigated by Atanassova, [3], later extended to the level operator N γ 2 γ 1 for the case of interval-valued fuzzy sets, [4].All IFLOs change (to some extent) the values of the membership and non-membership functions of the IFSs.
In the present paper, for the first time, we will discuss IFLOs that change the degree of uncertainty, that is defined (see [1]) as:

As by definition
Various aspects of uncertainty (also called hesitation) have been studied by Vassilev alone or with co-authors.Noteworthy are the research related to isohesitant intuitionistic fuzzy sets, i.e., ones that maintain the same hesitancy distribution, [8], and the research related to intuitionistic fuzzy subsets with diminishing hesitancy values, [7].

Intuitionistic fuzzy level operator R α
Let (see [1]) For it, we see that π U * (x) = 1 for each x ∈ E. Now, using the two Intuitionistic Fuzzy Topological Operators (IFTO) we can see that for each x ∈ E in the first case: in the second case: and in general case: In general, in each one of these three cases, π A (x) can be a member of a larger interval, that included the above mentioned intervals.
Let the IFSs used below be different than U * .For each IFS A and for each α ∈ [0, 1] we define: Theorem 1.The set R α is an IFS.
Proof.First, we see that for each α ∈ [0, 1], because π A (x) > 0: The check of the non-membership degree is similar.Second, for operator R α we see that Therefore, the definition of the new operator is correct and it is an IFS.This completes the proof.
Let R α (A)(x) denote the result of applying of R α over element ⟨x, µ A (x), ν A (x)⟩.Obviously, when α = π A (x): i.e., in a result of applying R α over element x ∈ E does not change its degrees.
Theorem 2. For each IFS A ̸ = U * and for each α ∈ [0, 1]: Proof.Let α ∈ [0, 1] and the IFS A be given.The check of the equality is the following: This completes the proof.
Theorem 3.For each IFS A ̸ = U * and for every α, β ∈ [0, 1]: Proof.Let α, β ∈ [0, 1] and the IFS A be given.The check of the equality is the following: This completes the proof.
The simplest IF modal operators are (see [1]): For them, the following assertions are valid.
Theorem 5.For each IFS A ̸ = U * and for each α ∈ [0, 1]: Proof.For (a), we calculate: Now, we must prove that the membership degree of C(R α (A)) is higher than this of R α (CA) and that the non-membership degree of R α (CA) is higher than this of C(R α (A)).
First, we must mention that for every three real numbers a, b, c > 0, from a ≥ b it follows that because ab Then, having in mind that n ≤ q and (2), we obtain and let Then, having in mind that m ≥ s and (2), we obtain Therefore, (a) is valid.The check of the validity of (b) is similar.This completes the proof.Theorem 8.For each IFS A ̸ = U * and for every α, β, γ ∈ [0, 1]: Theorems 4 and 5 are modified to the following forms.

Conclusion
The two operators, discussed in the present paper, modify the degree of uncertainty of the elements of a given IFS.So, in the IFS theory already there are tools for modifying of each one of the degrees of the IFS elements.
The new operators can be used, e.g., in procedures of decision making in which it is necessary to change degree of uncertainty of some expert's evaluations, when they are wrong.It is important, because in [1] there are procedures for changing only of the two standard degrees (of membership and of non-membership).
Another application of the new operators will be in intercriteria analysis (see, e.g., [2,5], when the degree of uncertainty must be corrected because existing of lacks of evaluations of objects by given criteria.This application will be discussed in near future. µ A (y) + ν A (y) .