On the intuitionistic fuzzy modal feeble topological structures

: In the paper, for the ﬁrst time, ideas for intuitionistic fuzzy modal feeble topological structures (of two types) are introduced, and some of their properties are discussed. These topologies are based on the intuitionistic fuzzy operation @ , intuitionistic fuzzy operator W , on the two intuitionistic fuzzy modal operators , ♦ and of simplest intuitionistic fuzzy extended modal operator D α .


Introduction
In the present paper, we combine ideas and definitions from the areas of (general) topology (see, e.g., [8,11]), of (standard) modal logic (see, e.g., [7,9,10]) and of intuitionistic fuzziness (see, e.g., [3][4][5]), and we introduce the concepts of Intuitionistic Fuzzy Feeble Topological Structures of First and Second Type (IFFTS1 and IFFTS2) and Intuitionistic Fuzzy Modal Feeble Topological Structures of First and Second Types (IFMFTS1 and IFMFTS2). It is important to mention that introducing the term "feeble" for some types of topological structures, we aim to make a distinction with term "weak" which is used in topology.
The paper is organized as follows. Initially, short remarks over Intuitionistic Fuzzy Sets (IFSs) are given in Section 2. In the next sections, new topology related objects are introduced and some of their basic properties are discussed. In the Conclusion, new directions of the development of the present ideas are formulated.

Short remarks on intuitionistic fuzzy sets
Let a set E be fixed. An IFS A in E is an object of the following form: where functions µ A : E → [0, 1] and ν A : E → [0, 1] define the degree of membership and the degree of non-membership of the element x ∈ E, respectively, and for every x ∈ E: Let for every x ∈ E, π A (x) = 1 − µ A (x) − ν A (x). Therefore, the function π determines the degree of uncertainty of the element x.
Obviously, for every ordinary fuzzy set π A (x) = 0 for each x ∈ E and these sets have the form: Let everywhere below, the universe E be given. One of the geometrical interpretations of the IFSs uses the intuitionistic fuzzy interpretational triangle on Figure 1. For every two IFSs A and B many relations and operations have been defined (see, e.g. [1,3,4]). The most important of them are the following: Here, we give definitions of only the first two modal operators (see, e.g., [3,4]) that are intuitionistic fuzzy interpretations of the classical modal logic operators (see, e.g., [7,9,10]): Let for each set X P(X) = {Y |Y ⊆ X}.
Let for each set E, F S(E) and IF S(E) be the sets of all FSs and IFSs, respectively, with universe E. Then we see that Therefore, P(E * ) coincides with IF S(E). On the other hand side, F S(E) coincides with the set 3 Definition of a First type of Intuitionistic fuzzy feeble topological structure with a self-dual operation and operator As a basis of the present research we use the definition from [11], where the conditions that characterize the topological structure X, O, • are where X is a set of subsets, A, B ∈ X, O : X → X and • : X × X → X.
Below, we will call the object X, O, • a Feeble Topological Structure (FTS), when some of the four conditions is not valid or if it is valid only partially. The reason for this is that, below we will use a special operation • and in this and the next two sections, a special operator O that are self-dual and this fact will be included in the names of the introduced objects. In sections 6, 7 and 8 we will use the same operation • and other (standard) IF topological operators.
By analogy with [11], and extending the definitions from there, we will call a first type of an Intuitionistic Fuzzy FTS with a Self-Dual Operation and operator (IFFTSDO1) the object is a (self-dual) operation, and for every two IFSs A, B ∈ P(E * ) : As it is seen, in the new definition condition C2 is omitted and this is exactly the reason for the term "feeble" in the name of the new object. In the definition we add three new conditions (D1, D2 and D3) that are related to the self-duality of the operator and operation.
In [6], Adrian Ban and the author introduced "weight-center operator" over a given IFS A by: where card(E) is the number of the elements of a finite set E (see also [4]). For the continuous case, the "summation" may be replaced by integrals over E.
Theorem 1. P(E * ), W, @ is an IFFTSDO1 with the (self-dual) operator "W" and the (self-dual) operation "@". Proof. Let the IFSs A, B ∈ P(E * ) be given. Then we check sequentially that Therefore, Theorem 1 is valid.
4 Definition of a First type of Intuitionistic fuzzy modal feeble topological structure with a self-dual operation and operator, and a standard modal operator We will call First Type of an Intuitionistic Fuzzy Modal Feeble Topological Structure with a Self-Dual Operation and operator (IFMFTSDO1) the object modal operator, and for every two IFSs A, B ∈ P(E * ) : Theorem 2. P(E * ), W, @, is an IFMFTSDO1 with the (self-dual) operator W and the (self-dual) operation @, and with the (standard) modal operator . Proof. The checks of the first six conditions from the definition of an IFMFTSDO1 coincide with the checks of the conditions from the definition for IFTFSDO1 (see Theorem 1). So, we will check sequentially the remaining four conditions.
Let the IFSs A, B ∈ P(E * ) be given. Then we check that: Therefore, Theorem 2 is valid.
Proof. The first six conditions from the definition of an IFMFTSDO1 again coincide with the conditions from the definition of an IFTFSDO1. So, we will again check sequentially the remaining four conditions. Let the IFSs A, B ∈ P(E * ) be given. Then we check that: This completes the proof.
5 Definition of a First type of Intuitionistic fuzzy extended modal feeble topological structure with a self-dual operation and operator, and an extended self-dual modal operator The first extended modal operator (see [2][3][4]) has the form: It is extended modal operator, because for each IFS A: Operator D α is self-dual, because for each IFS A:  Proof. The first six conditions from the definition of an IFMFTSDO1 again coincide with the conditions from the definition of an IFTFSDO1. So, we will again check sequentially the remaining four conditions. Let the IFSs A, B ∈ P(E * ) be given. Then we check sequentially that: As we mentioned above, Condition D8 is valid because of the self-duality of the extended modal operator D α .
This completes the proof. .

Definition of a Second type of Intuitionistic fuzzy feeble topological structure with a self-dual operation
Below, we will call the object X, O, • a second type of a Feeble Topological Structure with a self-dual operation (FTSSDO2), when condition C1 is changed with the feeble condition: keeping conditions C2, C3, C4, D2 and D3. We must mention that now condition D1 here and thereafter is omitted because below we will use an operator that is not self-dual, while condition D2 must remain because the operation @ is a self-dual one. When this operator and the (self-dual) operation are intuitionistic fuzzy ones, then the structure is an Intuitionistic Fuzzy FTSSDO2 (IFFTSDO2). Now, the used operator "O 1 " has a dual one -"O 2 " so that for each IFS A: The first operator will be an analogous of the standard topological operator "closure" (cl) and the second -of the standard topological operator "interior" (in). By this reason, we will denote the respective structures by cland in-ones.
Theorem 5. P(E * ), C, @ is a cl-IFFTSDO2 with an (self-dual) operation "@". Proof. Let the IFSs A, B ∈ P(E * ) be given. Then we check sequentially that conditions C1', C2, C3, C4 are valid: i.e., condition C1' is valid, but obviously, condition C1 will not be always valid: Condition D2 is checked in Section 3 (Theorem 1), while the validity of condition D3 follows from: This completes the proof.
In the same manner, we can prove the next theorem.
Theorem 6. P(E * ), I, @ is an in-IFFTSDO2 with an (self-dual) operation "@". 7 Definition of a Second type of Intuitionistic fuzzy modal feeble topological structure with a self-dual operation We will call Second type of Intuitionistic Fuzzy Modal Feeble Topological Structure with a Self-Dual Operation and operator (IFMFTSDO2) the object is a (standard) modal operator, and for every two IFSs A, B ∈ P(E * ) : the conditions C1', C2, C3, C4, D2, . . . , D7 are satisfied.
By analogy, we can prove the validity of the following three assertions.
Theorem 9. P(E * ), I, @, is an in-IFMFTSDO2 with the operator I, the (self-dual) operation @, and with the (standard) modal operator .
9 Conclusion and ideas for future research The described above idea opens some directions for future research. Below, we discuss some of them. First, we plan to introduce an intuitionistic fuzzy topological structure and an intuitionistic fuzzy modal topological structure, in which all standard conditions for topological structures are valid and operation is not self-dual (that is the reason for the above discussed change of the conditions for topological structures).
Second, we can extend the last type of an intuitionistic fuzzy topological structure with some of the extended modal operators that are not already self-dual ones.