On the most extended interval-valued intuitionistic fuzzy modal operators from both types

: The most extended (by the moment) interval-valued intuitionistic fuzzy modal operators from both types are introduced. A theorem for equivalence of two of them is proved.


Introduction
The Interval Valued Intuitionistic Fuzzy Sets (IVIFSs) are extensions of the Intuitionistic Fuzzy Sets (IFSs) (see [1,2,5] and their theory was enriched with a lot of operators that do not have analogues in the standard fuzzy sets theory and in the rest of the fuzzy sets extensions. In the present paper, we introduce two groups of operators that are extensions of the existing by the moment interval-valued intuitionistic fuzzy modal operators from the two types.
2 Extensions of the interval-valued intuitionistic fuzzy modal operators from the first type The first of the extensions, introduced in the last years, was given in [3]. We use it as a basis of our research, correcting some misprints in it. Here, we introduce new operators, one of which is the operator from [3]. First, all forms of the operators are given and the conditions for the validity of each one of them is discussed and after this, we will reduce the more detailed research to its simplest case.
Let ext 1 , ext 2 , ext 3 , ext 4 , ext 5 , ext 6 , ext 7 , ext 8 ∈ {inf, sup}. Let The definition will be correct, if Not the most complex (while not the simplest either) form of the operator is: For brevity, in the upper index of we will write and instead of inf and sup, respectively. From the above records of the -operator it is clear that in the first case it will have 256 different forms and in the second case -16 different forms.
The simplest form of the -operator is So, here will take into consideration the simpler situation, that, obviously, has only 4 cases that we will study sequentially.
Let everywhere for the 4 cases, and to see that the operator is correct, we must find the conditions under which the inequalities (1) -(3) are valid. Below, we shall study the mentioned above four cases, each of which with four sub-cases. 1.1. We see that Therefore, the condition is 2
The second -operator is

2.1
We see that this case coincides with case 1.1.

This case follows directly from
Therefore, the conditions are 2 + 2 ≤ 1 and, respectively, for the second inequality 2 + 2 ≤ 1.

We obtain
Therefore, the condition is min i.e., now the condition is max Therefore, for the second case, the inequalities (1) -(3) have the concrete forms The third -operator is 3.1. We obtain directly that 3.2. This case coincides with 1.2. 3.3. We have: Therefore, for the third case, the inequalities (1) -(3) have the concrete forms The fourth -operator is This case coincides with 2.4. Therefore, for the fourth case, the inequalities (1) -(3) have the forms of (10) -(12), as for the third case.
For all cases, one basic condition must be valid From all these checks it follow the validity of the following theorem.
Theorem 1. For each IVIFS and for each one of the four -operators, is an IVIFS.

Extensions of the interval-valued intuitionistic fuzzy modal operators from the second type
Now, having in mind the four forms of the -operator, here, for a first time we will introduce an extension of the operator ∘ (︂ )︂ .
Now, there are four cases that we must study sequentially.