On interval-valued intuitionistic fuzzy modal operators

An survey of the existing interval-valued intuitionistic fuzzy modal operators is given. Eight new operators are introduced that extend the older ones. Some of their basic properties are discussed. Open problems are formulated.

The definition of the IVIFS can be however rewritten to become an analogue of the second definition of the IFS (see [5]) -namely, if M A and N A are interpreted as functions. Then, an IVIFS A (over a basic set E) is given by functions 1]) and N A : E → IN T ([0, 1]) and the above inequality.
We must note that there is no difference in principle between the two approaches. And what is more, the same exist also in the ordinary fuzzy sets theory. The author originally used the first one influenced by the Kaufmann's book [11]. Perhaps it was this approach that helped him develop the theory of operators over IFS in its present form. The same notation was used in [1987][1988] in the research on IVIFSs, too (see [3,10]).
Because below we will use only notation A * , for brevity, the asterisk will be omitted.
IVIFSs have geometrical interpretations similar to, but more complex than these of the IFSs (Fig. 1). It is suitable to define First, we define some relations over IVFSs. For every two IVIFSs A and B the following relations hold ("iff"' is a abbriviation of "if and only if"): Second, we describe the basic operations, defined for every two IVIFSs A and B. They are: Now, over IFSs 185 different implications are defined (see, e.g., [6,7]). On their basis, three types of 185 conjunctions and discunctions can be introduced (see, e.g., [1,2]. After publishing of [7], 5 new implications were introduced, but for them there are not constructed new conjunctions and discunctions. Now, the following Open Problems are interesting: 1. To construct analogous of all 190 implicationss for the IVIFS-case; 2. On their basis, to construct the respective triples of conjunctions and discunctions. Third, we give the list of the operators of modal type that are defined over an IVIFS A (see also [4,5]): and by this reason, below we will not discuss these three operators.
In [5], the operators F α.β , ..., J * α,β are extended to the following operators, where α, β, γ, δ ∈ [0, 1] such that α ≤ β and γ ≤ δ: Obviosly, the new operators are extensions of the previous ones. In [6,8], the idea for changing of real number parameters α, β in the operators defined over IFS A with whole IFS B is discussed. Here, we discuss wimilar idea for the case, when set A is an IVIFS.

Main results
Here, following [8], we describe step by step 8 intuitionistic fuzzy extended modal operators and discuss some of their propertiers. We give the proof of only the first two properties, formulated below and the rest properties are proved by the same manner.

Operator F B
Let B be an IVIFS. Then for each IVIFS A: Theorem 1. For every six IFSs A, B, C, D, P, Q: Proof: Let the IVIFSs A and B be given. Then, for the first equality we obtain that For the second equality we obtain (let us denote conditionaly this set as) Now, we must prove that set X is an IVIFS. First, we see diretly, that Second, analogously, we see that Third, we check validity of condition (*) for set X. sup Therefore, set X is an IVIFS. Finally, se wee that Therefore, X = F B (C).

Operator G B,C
Let B and C be IVIFSs. Then for each IVIFS A:

Operator H B,C
It is defined for every three IVIFSs A, B and C by: [inf N A (x) + inf N C (x) sup P A (x), sup N A (x) + sup N C (x) sup P A (x)] |x ∈ E}.