Four interval-valued intuitionistic fuzzy modal-level operators

Four new interval-valued intuitionistic fuzzy operators are introduced. It is shown for them that they exhibit behaviour similar both to the modal, as well as to the level operators defined over interval-valued intuitionistic fuzzy sets, and for this reason, they are called intervalvalued intuitionistic fuzzy modal-level operators. Their basic properties are discussed.


Introduction
The Interval-Valued Intuitionistic Fuzzy Sets (IVIFSs) were introduced exactly 30 years ago in [10] and over the years their theory has been enriched with a lot of operators that do not have analogues in the theories of standard fuzzy sets, intuitionistic fuzzy sets (IFSs), as well as the rest of the fuzzy sets extensions. In the present paper, we introduce three operators that exhibit behaviour similar to the modal, as well as to the level operators.

Preliminary definitions
Following [4,5], we give the definitions of the basic concepts and the basic operations, relations and operators over IFSs.
Let us have a fixed universe E and its subset A. The set where M A (x) ⊆ [0, 1] and N A (x) ⊆ [0, 1] are intervals and for all x ∈ E such that: is called an Interval-Valued Intuitionistic Fuzzy Set (IVIFS) and the intervals M A (x), N A (x) represent the interval of the degree of membership (validity, etc.) and the interval of the degree of non-membership (non-validity, etc.), respectively. Obviously, this definition is constructed analogously to the definition of an IFS (cf. [4,5]). IVIFSs have geometrical interpretations similar to-but more complex than-these of the IFSs (see Fig. 1). @ @ @ @ @ @ @ @ @ @ @ @ @ @ 0, 0 1, 0 For brevity, everywhere hereafter we write A instead of A * . Following [4], we introduce some relations and operations for two IVIFSs A and B: x ∈ E .

The new operators
Now, we introduce the first two new operators, defined over a given IVIFS A. They have the forms: where α, β ∈ [0, 1] and α + β ≤ 1.
We check that i.e., the first definition is correct. Analogously, we check that the second definition is also correct. After this, we see that for each IVIFS A and for every α, β, γ, δ, ∈ [0, 1], such that α + β ≤ 1 and γ + δ ≤ 1: and the inequalities are transformed to equalities if and only if α = δ = 1 and hence β = γ = 0.
Similarly to the rest modal operators, defined over IVIFSs, the two new operators can be also extended to the forms We can see again that both operators are correctly defined and for them the inequalities Obviously, for each IVIFS A: By this reason, hereafter, we will only work with the two extended operators.
Below, we will discuss the four interval-valued intuitionistic fuzzy operators from three different perspectives.

The perspective of the interval-valued intuitionistic fuzzy modal operators of first type
The simplest intuitionistic fuzzy modal operators are analogous of modal operators "necessity" and "possibility". In the framework of the IVIFSs theory these operators are extended and modified in a "step-by-step" manner. The first group of modal operators is the following (see [4,7]): for every α, β, γ, δ ∈ [0, 1].
The composition of two -, ♦-, D-, F -and G-operators can be represented by only one of them, while this is impossible for the rest of the operators, but now, we see that the following assertion is valid.
The most extended intuitionistic fuzzy modal operator from the first type has the form: Now, we see directly that where r 1 , r 2 , r 3 , r 4 ∈ [0, 1] are arbitrary numbers and ext 1 , ext 2 , ext 3 , ext 4 , ext 5 , ext 6 ∈ {inf, sup} are one of the two symbols, regardless of which. Therefore, the new operators have a similar X-representation as the rest of the extended modal-type operators.

The perspective of the interval-valued intuitionistic fuzzy modal operators of second type
The Interval-Valued Intuitionistic Fuzzy Modal Operators of Second Type (IVIFMO2) are introduced for the first time in [8,9]. They also have two forms: shorter (introduced in [8]) and extended (introduced in [9]). These IVIFO2s are represented by one -the most extended operator that has the form given below. Let again ext 1 , ext 2 ∈ {inf, sup}. We define The components of this operator must satisfy the following conditions in a general form: The two new (more extended) modal operators have the following representations: where ext 1 , ext 2 ∈ {inf, sup}.
The intervals of the degrees of membership and non-membership of the elements of a given universe to its subset can be directly changed by these operators.
From the above discussion we see that the four new operators are simultaneously modal as well as level operators. By this reason, we legitimately can call them modal-level operators.
Finally, following the above notation, we can denote:

Conclusion
In [1,2,3], to each of the 189 intuitionistic fuzzy implications introduced in [5], three different intuitionistic fuzzy conjunctions and disjunctions are juxtaposed. Therefore, they can be basis for introducing a lot of new O-(and therefore, P -, Q-, Hand J-) operators. Each of these new operators can be extended in the ways the standard modal and level operators are extended. Simultaneously, some of the operators, introduced for the IFS-case, have to be extended to IVIFSform (e.g., the G. Ç uvalcıoglu's operators, [12]). The applications of these operators will be interesting. For example, they can be used in different Data Mining processes as of decision making, of pattern recognition and a lot of others. One of the novel Data Mining tools is the IFS-based intercriteria analysis (see, e.g., [6,11,13]). It can be easily seen that the operators, discussed in the paper, can be used for modification of InterCriteria Analysis results and this will be an object of future research.