On algebraic properties of fundamental group of intuitionistic fuzzy topological spaces (IFTSs)

In this paper, the notion of some algebraic properties of fundamental group of intuitionistic fuzzy topological spaces (IFTSs) are introduced. We give a necessary and sufficient condition for a fundamental group of IFTSs to be abelian, a necessary and sufficient conditions for a subset of fundamental group of IFTSs to be subgroup, a necessary and sufficient condition for a subgroup of fundamental group of IFTSs to be normal and a necessary and sufficient condition for an element to be in a center of fundamental group of IFTSs. We also describe the set of centralizers of an element in a fundamental group of IFTSs and the quotient fundamental group of IFTSs.


Introduction
T he concept of intuitionistic fuzzy set was introduced as a tool for dealing with uncertainties. It was first defined by Atanassove [1] as generalization of fuzzy set introduced by Zadeh [2]. After the work of Atanassove [1], many researcher worked in this direction, for example, Az-zo'bi et al. [3] defined and studied the fundamental group of intuitionistic fuzzy topological spaces which depends on the concepts of intuitionistic fuzzy sets and intuitionistic fuzzy topology. This concept was completely described as a generalization of the fundamental group of fuzzy topological spaces. Osmanglu and Tokat [4] discussed the intuitionistic fuzzy soft topology and Babitha and John [5] generalized intuitionistic fuzzy soft sets and set theoretical operations with illustrating examples.
Soroja and Kalaichelvi [6] construct a topology on an intuitionistic fuzzy neutrosophic soft sets. The concepts of intuitionistic fuzzy neutrosophic soft closure, intuitionistic fuzzy neutrosophic soft interior, intuitionistic fuzzy neutrosophic soft exterior, intuitionistic fuzzy neutrosophic soft boundary were introduced and some of its properties were also studied. Deschrijver et al. [7] introduced the notion of intuitionistic fuzzy t-norm and t-conorm, and investigate under which conditions a similar representation theorem can be obtained. They also argued in [8] that some of the existing definitions that appear in intuitionistic fuzzy literature (i. e., intuitionistic fuzzy connectives: negation, conjunction, disjunction and implication) were not sufficiently general for all pratical purposes, and suggest to replace them with new ones.
Liu and Wang [9] introduced new methods for solving multi-criteria decision-making problem in an intuitionistic fuzzy environment. Ersoy et al. [10] introduced the concept of intuitionistic fuzzy soft rings, and some basic properties of intuitionistic fuzzy soft rings were also given.
Jiang [11] studied a new entropy and its properties based on the improved axiomatic definition of intuitionistic fuzzy entropy and the nonlinear triangular intuitionistic fuzzy number and its application in linear integral equation was proposed by Mondal et al. [12]. And also in the pipeline of the literature Li and Jin [13] proposed a scalar expected value of intuitionistic fuzzy random individuals and its application to risk evaluation in insurance companies.
As described by Az-zo'bi et al. [3], the operation on the fundamental group of intuitionistic fuzzy topological spaces is not trivial, hence the study of its algebraic properties is very essential, which is the main task of this paper.

Definition 7.
[17] Let 0 ∼ = { x, 0, 1 : x ∈ X} and 1 ∼ = { x, 1, 0 : x ∈ X}. An intuitionistic fuzzy topology (IFT) on a nonempty set X is a family τ of IFSs in X satisfying the following axioms: The pair (X, τ) is called an intuitionistic fuzzy topological space (IFTS) and IFS in τ is known as intuitionistic fuzzy open set (IFOS) in X. The complementĀ of an IFOS A in an IFTS (X, τ) is called an intuitionistic fuzzy closed set (IFCS) in X. Definition 8. [17] Let (X, τ) and (Y, σ) be two IFTSs and let f : X → Y be a function. Then f is said to be fuzzy continuous if and only if the pre-image of each IFS in σ is IFS in τ. Definition 9. [17] Let (X, τ) and (Y, σ) be two IFTSs and let f : X → Y be a function. Then f is said to be fuzzy open if and only if the image of each IFS in τ is IFS in σ.

Definition 10.
[17] The f : (X, τ) → (Y, σ) is said to be fuzzy continuous if and only if the pre-image of each IFCS in σ is IFCS in τ. Definition 11. [17] Let (X, τ) be an IFTS. Then (a) X is said to be fuzzy C 5 -disconnected if there exists an intuitionistic fuzzy open and fuzzy closed set G such that G = 1 ∼ and G = 0 ∼ . (b) X is said to be fuzzy C 5 -connected if is not fuzzy C 5 -disconnected. Theorem 1. [17] Let f : (X, τ) → (Y, φ) be a fuzzy continuous surjection. If (X, τ) is fuzzy C 5 -connected then so (Y, φ). Definition 12. [17] Let (X, T) be ordinary topological. The collectioñ τ = {G : G is an intuitionistic fuzzy set on X and G • ∈ T} is an intuitionistic fuzzy topology on X induced by T. The pair (X,τ) is called the intuitionistic fuzzy topological space induced by (X, T). Thus if ε I is the Euclidean subspace topology on I then (I, ε I ) denotes the IFTS induced by the (usual) topological space (I, ε I ). Definition 13. [3] Let (X, τ) be an IFTS. If α : (I, ε I ) → (X, τ) is a fuzzy continuous E is a fuzzy C 5 -connected in (I, ε I ) with µ E (0), µ E (1) > 0 and ν E (0), ν E (1) < 1 then the IFS α(E) is called an intuitionistic path in (X, τ). The intuitionistic fuzzy points c 1 (λ, δ) and c 2 (γ, θ) in (X, τ) are called the initial point and the terminal point of the intuitionistic fuzzy path α(E) in respectively.
If the initial point equals to the terminal point, then we call it an intuitionistic path an intuitionistic fuzzy loop. The collection of all intuitionistic fuzzy loops in (X, τ) by Ω((X, τ), c(λ, δ)) The IFP c(λ, δ) is called an intuitionistic fuzzy base point of (X, τ) and ((X, τ), c(λ, δ)) is called pointed intuitionistic fuzzy space.
is a fuzzy continuous function.
Then the intuitionistic fuzzy set N defined by
Proof. The proof follows directly from the following lemmas.

Necessary and sufficient conditions for a fundamental group of IFTSs to be abelian
In this section, we characterized the necessary and sufficient conditions for a fundamental group of IFTS to be abelian. As it is generally described in [18] that a group G is said to be abelian if it is commutative. Further applications includes; the unitary character group of abelian unipotent groups [19] and a theorem on the action of abelian unitary groups in [20]. Now, follow from the above descriptions, it is discovered that the study of abelian fundamental group of IFTSs is very essential. Therefore we have the following definition and theorem: for some connected IFSs F and H in (I,ε I ).
for some connected IFSs L, M, N and P in (I,ε I ).

Necessary and sufficient conditions for a subset of fundamental group of IFTSs to be a subgroup
In this section, we characterized the necessary and sufficient conditions for a subset of fundamental group to be its subgroups. A necessary and sufficient conditions for a subset of a group to be a subgroup have been explained in [21], that is given a group G with a particular binary operation, if a non-empty subset H of G is closed and every elements in it has inverse under that particular operation, then we say H is a subgroup of G. Further applications includes; the maximal subgroups of symplectic groups stabilizing spreads [22]; 3-permutable subgroups and p-nilpotency of finite group II [23]; fuzzy subgroups of nilpotent groups [24]; subgroup of nilpotent group [25]; two theorems about nilpotent subgroup [26]. Now, follow from the above descriptions, it is discovered that the study of subgroups of fundamental group is very essential. Therefore we have the following definition: Definition 17. Let π 1 ((X, τ), c(λ, δ)) be fundamental group of IFTS X and U be nonempty subspace of X. If π 1 ((U, τ), c(λ, δ)) is a group under the operation · of π 1 ((X, τ), c(λ, δ)), then π 1 ((U, τ), c(λ, δ)) is a subgroup of π 1 ((X, τ), c(λ, δ)). Theorem 6. Let π 1 ((X, τ), c(λ, δ)) be fundamental group of IFTS X and U be nonempty subspace of X. Then π 1 ((U, τ), c(λ, δ)) is a subgroup of π 1 ((X, τ), c(λ, δ)) if and only if the following conditions hold and for some connected IFSs A, E, L and N ∈ (I,ε I ). c(λ, δ)) and for some connected IFSs A, B and C ∈ (I,ε I ).
For (i): It is suffices to show that the following conditions are satisfied Consider the function α depend on (I,ε I ) to (X, τ) such that α((I,ε I )) ⊆ (U, τ) ⊆ (X, τ) where U is a subspace of X. Thus α(A) depend on (I,ε I ) to (U, τ).
(⇐) Suppose the three conditions above satisfied.

Necessary and sufficient condition for an element to be in the center of a fundamental group of IFTSs
In this section, we give a necessary and sufficient condition for an element to be in a center of a fundamental group of IFTS. A necessary and sufficient condition for an element to be in a center of a group as been described in [25] that given a group G, the center of G is define as the set of element x ∈ G that commutes with every element of G i. e., Z(G) = {x ∈ G : xy = yx ∀y ∈ G}. Now, follow from the above descriptions we have the following definition: Definition 20. The center of the fundamental group of IFTS X is define as the set of all homotopy classes of loop [α(A)] ∈ π 1 ((X, τ), c(λ, δ)) that commutes with every homotopy classes of loop [β(B)] ∈ π 1 ((X, τ), c(λ, δ)) i. e., Z(π 1 ((X, τ), c(λ, δ))) = {[α(A)] ∈ π 1 ((X, τ), c(λ, δ)) : Theorem 8. Let π 1 ((X, τ), c(λ, δ)) be a fundamental group of IFTS X. Then [α(A)] ∈ Z(π 1 ((X, τ), c(λ, δ))) if and and [o(E)] ∈ π 1 ((X, τ), c(λ, δ)) and for some connected IFSs L, M, N and P in (I,ε I ).

Quotient Fundamental Group of IFTSs
In this section, we deduced the quotient group under fundamental group of IFTS from general description of quotient group. As it is generally described in [25] that given a group G and a normal subgroup N, a set G/N define by G/N = {gN : g ∈ G} is called a quotient group.

Description of centralizer of the fundamental group of IFTSs
In this section, we deduced the centralizer of an element in a fundamental group of IFTS from general description of centralizer of an element in a group. As it is generally described that given a group G, the centralizer of an element in x ∈ G is define as a set of elements G that commutes with every element of x i. e., C(x) = {y ∈ G : xy = yx∀x ∈ G}.
Further applications includes; centralizers of nilpotent group elements in semisimple algebraic groups [27]; the isomorphism type of the centralizer of an element in a lie group [28]; centralizers of semisimple elements in the finite classical groups [29]. Now, we have the following definition: ∈ π 1 ((X, τ), c(λ, δ)) and for some connected IFSs F, G, H and J in (I,ε I ).