Interval-valued intuitionistic fuzzy sets over universes with special forms

An interval-valued intuitionistic fuzzy set (IVIFS) whose universe is an intuitionistic fuzzy set (IFS) is introduced here and its basic properties are discussed. One step further, it is discussed the case when the universe represents an IVIFS itself. Again, some basic properties of the so constructed object are formulated and proved. Ideas for further research are presented.


Introduction
In [5], the idea for an Intuitionistic Fuzzy Set (IFS) whose universe is an IFS with respect to another universe, is discussed. Here we will develop this idea for the case of an Interval-Valued Intuitionistic Fuzzy Set (IVIFS).
Following [5,6], we mention that for a fixed universe E and a subset A of E, the set is called IFS and the functions µ A : E → [0, 1] and ν A : E → [0, 1] represent the degree of membership (validity, etc.) and non-membership (non-validity, etc.) of element x ∈ E to a fixed set A ⊆ E. Thus, we can also define function π A : E → [0, 1] by means of and it corresponds to the degree of indeterminacy (uncertainty, etc. ). An IVIFS A over E is an object of the form: are intervals and for all x ∈ E: (see [3,5,7]. This definition is analogous to the definition of an IFS. It can be however rewritten to become an analogue of the definition of IFS, namely, if M A and N A are interpreted as functions. Then, an IVIFS A (over a universe set E) is given by the above inequality and the functions For every two IVIFSs A and B, the following relation holds:

Main results
In [4] the idea for a new IFS-construction was proposed. It was extended in [5] but only for the IFS-case. Here, we will do this for the IVIFS case. Partially, the idea was generated with regards to the definition of a soft set and some of its modifications [1,2,[8][9][10][11][12][13][14]. But in the present research, we will keep only the IFS-and IVIFS-forms of the sets without adding any soft-set component.
Let E be a fixed universe and let A be an IVIFS over E. Let F be another universe and let the set E be an IFS over F having the form: Therefore, the element x ∈ E has the form: and there exists a bijection between the Eand F -elements of xand y-types, respectively.
Thus we can use the symbol "y" for both yand x-elements.
Let A // E stands for "A is an IVIFS over E" and A // (E/F ) stands for "A is an IVIFS over E, which is an IFS over F ".

IVIFS whose universe is an IFS
If the degrees of membership and non-membership of an element y to a set A in the frames of a universe E are M A (y) and N A (y), and the element y, M A (y), N A (y) has degrees of membership and non-membership to the set E within the universe F that are µ E (y) and ν E (y), then we can define the first batch of six forms of the transformation of the universe of the IVIFS A: We can call all these transformations, respectively, very optimistic, optimistic, average, pessimistic, very pessimistic, standard. Last name is given for the older transformation described for the IFS-case in [5].
Here, for the first time we formulate and prove the following Theorem 1. For each set A in the universe E, that is an IFS over the universe F , A // i 2.2 IVIFS whose universe is an IVIFS Now, we can define the second batch of six new forms of the transformation of the universe of the IVIFS A: We can formulate and prove a theorem similar to the above one: Theorem 2. For each set A in the universe E, that is an IFS over the universe F ,

Conclusion
Out next research will fork in two directions. On the one hand, we will study the possibility to represent a soft set by some of the above defined new types of sets. On the other hand, we will search for new ways of constructing of IVIFSs over universes with with IFS-or IVIFS-forms. For this aim, we plan to modify the extended interval-valued intuitionistic fuzzy modal operators, similarly to the modification, discussed in [6], in which the parameters α and β of the operators were changes with whole IFSs.