International Journal of Analysis and Applications
ISSN 2291-8639
Volume 12, Number 2 (2016), 163-179
http://www.etamaths.com

INTUITIONISTIC FUZZY TOPOLOGICAL POLYGROUPS

N. ABBASIZADEH AND B. DAVVAZ∗

Abstract. The notion of intuitionistic fuzzy set was introduced by Atanassov as a generalization
of the notion of fuzzy set. Intuitionistic fuzzy topological spaces were introduced by Coker. This

paper provides a new connection between algebraic hyperstructures and intuitionistic fuzzy sets. In

this paper, we introduce and study the concept of intuitionistic fuzzy subpolygroup and intuitionistic
fuzzy topological polygroup. We also investigate some interesting properties of an intuitionistic fuzzy

subpolygroup and intuitionistic fuzzy normal subpolygroup.

1. Introduction

The hyperstructure theory was born in 1934 when Marty introduced the notion of hypergroup [24].
The concept of intuitionistic fuzzy sets was introduced by Atanassov [5]. Coker [7] has introduced the
notions of intuitionistic fuzzy topological spaces. Biswas [6] introduced the concept of intuitionistic
fuzzy subgroup and some other concepts. The concepts of quasi-coincidence for intuitionistic fuzzy
point was introduced and developed by Gallego Lupianez [14]. On the other hand, in the last few
decades, many connections between hyperstructures and intuitionistic fuzzy sets has been established
and investigated. In [18], Heidari et, al introduced the notion of topological polygroups. Then in
[1, 2, 3] Abbasizadeh et, al investigated to notion of fuzzy topological polygroups.

We recall some basic definitions and results to be used in the sequel.
Let H be a non-empty set. Then a mapping ◦ : H ×H −→ P∗(H) is called a hyperoperation, where
P∗(H) is the family of non-empty subsets of H. The couple (H,◦) is called a hypergroupoid. In the
above definition, if A and B are two non-empty subsets of H and x ∈ H, then we define

A◦B =
⋃
a∈A
b∈B

a◦ b, x◦A = {x}◦A and A◦x = A◦{x}.

A hypergroupoid (H,◦) is called a semihypergroup if for every x,y,z ∈ H, we have x◦(y◦z)=(x◦y)◦z
and is called a quasihypergroup if for every x ∈ H, we have x ◦ H = H = H ◦ x. This condition is
called the reproduction axiom. The couple (H,◦) is called a hypergroup if it is a semihypergroup and
a quasihypergroup [9, 24].

For all n > 1, we define the relation βn on a semihypergroup H, as follows:

a βn b ⇔∃ (x1, . . . ,xn) ∈ Hn : {a,b}⊆
n∏
i=1

xi,

and β =
⋃
βn, where β1 = {(x,x) | x ∈ H} is the diagonal relation on H. This relation was introduced

by Koskas [21] and studied mainly by Corsini, Davvaz, Freni, Leoreanu, Vougiouklis and many others.
Suppose that β∗ is the smallest equivalence relation on a hypergroup (semihypergroup) H such that
the qoutient H/β∗ is a group (semigroup). If H is a hypergroup, then β = β∗ [13]. The relation β∗ is
called the fundamental relation on H and H/β∗ is called the fundamental groups.

A special subclass of hypergroups is the class of polygroups. We recall the following definition from
[8]. A polygroup is a system P = 〈P,◦,e,−1 〉, where ◦ : P × P −→ P∗(P), e ∈ P , −1 is a unitary
operation P and the following axioms hold for all x,y,z ∈ P:

(1) (x◦y) ◦z =x◦ (y ◦z),
(2) e◦x = x =x◦e,

2010 Mathematics Subject Classification. 20N20, 20N25, 03E72.
Key words and phrases. intuitionistic fuzzy set; polygroup; topological polygroup.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

163



164 ABBASIZADEH AND DAVVAZ

(3) x ∈ y ◦z implies y ∈ x◦z−1 and z ∈ y−1 ◦x.
The following elementary facts about polygroups follow easily from the axioms:
e ∈ x◦x−1 ∩x−1 ◦x, e−1 = e, (x−1)−1 = x, and (x◦y)−1=y−1 ◦x−1.

A non-empty subset K of a polygroup P is a subpolygroup of P if and only if a,b ∈ K implies
a ◦ b ⊆ K and a ∈ K implies a−1 ∈ K. The subpolygroup N of P is normal in P if and only if
a−1 ◦N ◦a ⊆ N for all a ∈ P . For a subpolygroup K of P and x ∈ P , denote the right coset of K by
K ◦x and let P/K be the set of all right cosets of K in P . If N is a normal subpolygroup of P, then
(P/N,�,N,−1 ) is a polygroup, where N ◦x�N ◦y={N ◦z|z ∈ N ◦x◦y} and (N ◦x)−1 = N ◦x−1.
For more details about polygroups we refer to [11, 12, 15].

Let P = 〈P,◦,e,−1 〉 be a polygroup and (P,T ) be a topological space. Then, the system P =
〈P,◦,e,−1 ,T 〉 is called a topological polygroup if the mapping ◦ : P ×P −→ ℘∗(P) and −1 : P −→ P
are continuous (see [18]).
Let P = 〈P,◦,e,−1 〉 be a polygroup and (P,T ) be a fuzzy topological space. A triad (P,◦,T ) is called
a fuzzy topological polygroup or FTP for short, if (see [1, 2, 3]):

(i) For all x,y ∈ P and any fuzzy open Q-neighborhood W of any fuzzy point zλ of x◦y, there
are fuzzy open Q-neighborhood U of xλ and V of yλ such that:
U •V ≤ W .

(ii) For all x ∈ P and any fuzzy open Q-neighborhood V of x−1λ , there exists a fuzzy open Q-
neighborhood U of xλ such that:
U−1 ≤ V .

2. Preliminaries

For the sake of convenience and completeness of our study, in this section some basic definition and
results of [4, 5, 7, 16, 17, 22, 23], which will be needed in the sequel are recalled here.

Let X be a non-empty set and I be the closed interval [0, 1]. A complex mapping A = (µA,νA) :
X −→ I × I is called an intuitionistic fuzzy set (in short, IFS) on X if µA(x) + νA(x) ≤ 1 for each
x ∈ X, where the mapping µA : X −→ I and νA : X −→ I denote the degree of membership (namely
µA(x)) and the degree of nonmembership (namely νA(x)) of each x ∈ X, respectively. In particular,
0∼ and 1∼ denote the intuitionistic fuzzy empty set and the intuitionistic fuzzy whole set in X defined
by 0∼(x) = (0, 1) and 1∼(x) = (1, 0) for each x ∈ X, respectively.

We will denote the set of all IFSs in X as IFS(X) (see [5, 7]).
Let X be a non-empty set and let A = (µA,νA) and B = (µB,νB) be IFSs on X. Then (see [5]):

(1) A ⊂ B iff µA ≤ µB and νA ≥ νB,
(2) A = B iff A ⊂ B and B ⊂ A,
(3) Ac = (νA,µA),
(4) A∩B = (µA ∧µB,νA ∨νB),
(5) A∪B = (µA ∨µB,νA ∧νB).

Let {Ai}i∈J be an arbitrary family of IFSs in X, where Ai = (µAi,νAi) for each i ∈ J. Then (see [7]):
(1)

⋂
Ai = (

∧
µAi,

∨
νAi).

(2)
⋃
Ai = (

∨
µAi,

∧
νAi).

Let X and Y be non-empty sets and let f : X −→ Y a mapping. Let A = (µA,νA) be an IFS in X
and B = (µB,νB) be IFS on Y . Then (see [7]):

(1) The preimage of B under f, denoted by f−1(B), is the IFS in X defined by:

f−1(B) = (f−1(µB),f
−1(νB)),

where f−1(µB)(x) = µB(f(x)) and f
−1(νB)(x) = νB(f(x)).

(2) The image of A under f, denoted by f(A), is the IFS in Y defined by:

f(A) = (f(µA),f(νA)),

where

f(µA)(y) =

{ ∨
x∈f−1(y)

µA(x) if f
−1(y) 6= ∅

0 otherwise,



INTUITIONISTIC FUZZY TOPOLOGICAL POLYGROUPS 165

and

f(νA)(y) =

{ ∧
x∈f−1(y)

νA(x) if f
−1(y) 6= ∅

1 otherwise.

Throughout this paper, the symbol I will denote the unit interval [0, 1].
A intuitionistic fuzzy topology (in short, IFT) in Coker’s sense on a non-empty set X is a family T

on IFSs in X satisfying the following axioms:

(1) 0∼, 1∼ ∈T .
(2) For all A,B ∈T , then A∩B ∈T .
(3) For all (Aj)j∈J, then

⋃
j∈J

Aj ∈T .

In this case, the pair (X,T ) is called an intuitionistic fuzzy topological space (in short, IFTS) in the
sense of Coker, and each elements of T is called an intuitionistic fuzzy open set (in short, IFOS) in X.
The complement Ac of an IFOS A in X is called an intuitionistic fuzzy closed set (in short, IFCS) in
X.
We will denote the set of all the IFTs on a set X as IFT(X), and the set of all IFOSs and the set of
all IFCSs in an IFTS(X) as IFO(X) and IFC(X), respectively (see [7]).

Example 1. (1) The family T ={0∼,1∼} is an intuitionistic fuzzy topology on X.
(2) The family of all intuitionistic fuzzy sets in X is an intuitionistic fuzzy topology on X.

Example 2. Let X = {a,b,c} and A =< x, ( a
0.4
, b
0.3
, c
0.2

), ( a
0.6
, b
0.5
, c
0.7

) >. Then, the family T =
{0∼,A, 1∼} of IFS’s in X is an IFT on X.

Example 3. Let X = [0, 1] and consider the IFS A = (µA,νA) as follows:

µA(x) =

{
−3

2
x + 1 if 0 ≤ x ≤ 2

3
0 if 2

3
≤ x ≤ 1,

and

νA(x) =

{
x
4

if 0 ≤ x ≤ 2
3

1
6

if 2
3
≤ x ≤ 1.

Then, T = {0∼,Ac, 1∼} is an intuitionistic fuzzy topology on X.

Let X,Y be non-empty sets and A = (µA,νA), B = (µB,νB) IFSs of X and Y , respectively. Then
A×B is an IFS of X ×Y defined by (see [17]):

(A×B)(x,y) = (µA(x) ∧µB(y),νA(x) ∨νB(y)).
Let X be a non-empty set. An intuitionistic fuzzy point, (in short, IFP) in X denoted by xr,s is an

intuitionistic fuzzy set A = (µA,νA) such that

µA(y) =

{
r if y = x
0 if y 6= x,

and

νA(y) =

{
s if y = x
1 if y 6= x,

where x ∈ X is a fixed point, the constants r ∈ I0, s ∈ I1 and r + s ≤ 1.
The intuitionistic fuzzy point xr,s is said to be contained in an intuitionistic fuzzy set A, denoted by
xr,s ∈ A, if and only if µA(x) ≥ r and νA(x) ≤ s. In particular

xr,s ⊆ ym,n ⇔ x = y and r ≤ m, s ≥ n.
The intuitionistic fuzzy characteristic mapping of a subset A of a set X is denoted by χA is defined as

χA(x) =

{
(1, 0) if x ∈ A
(0, 1) otherwise.

Obviously an intuitionistic characteristic function χA is also an intuitionistic fuzzy set on X and
for any non-empty subsets A and B of a set X, we have A ⊆ B if and only if χA ⊆ χB (see [22]).

Let xr,s be an IFP in X and let A = (µA,νA) be an IFS in X. We say that xr,s is quasi-coincident
with A, written xr,s q A, if µA(x) + r > 1 and νA(x) + s < 1 (see [16]).



166 ABBASIZADEH AND DAVVAZ

Let (X,T ) be an IFTS, and let p be an IFP of X. Say that an IFS N of X is a Q-neighbourhood
of p if there exists an IFOS A of (X,T ) such that p q A and A ⊆ N (see [23]).

Let X,Y be two non-empty sets, let f : X −→ Y be a map, let T be an IFT in X and let σ be an
IFT in Y . Then, f : (X,T ) −→ (Y,σ) is continuous if and only if, for each IFP p of X and for each
Q-neighbourhood V of f(p), there exists a Q-neighbourhood U of p such that f(U) ⊆ V (see [23]).

3. Intuitionistic fuzzy subpolygroups

Definition 3.1. Let P be a polygroup and A ∈ IFS(P). Then A is called intuitionistic fuzzy
subpolygroup (in short, IFSP) of P if it satisfies the following conditions:

(1) µA(z) ≥ µA(x) ∧µA(y) and
νA(z) ≤ νA(x) ∨νA(y) for each z ∈ x◦y and x,y ∈ P .

(2) µA(x
−1) ≥ µA(x) and

νA(x
−1) ≤ νA(x) for each x ∈ P.

We will denote the set of all IFSPs of P as IFSP(P).

Example 4. Let P = {e,a,b}. Then, P together with the following hyperoperation
◦ e a b
e e a b
a a e b
b b b {e,a}

is a polygroup. Let A =< x, ( e
0.7
, a
0.5
, b
0.3

), ( e
0.1
, a
0.3
, b
0.5

) >. Then, A is an IFSP of P .

Definition 3.2. [28] Let P be a polygroup. A fuzzy subset µ of P is called a fuzzy subpolygroup if

(1) min{µ(x),µ(y)}≤ µ(z), for all x,y ∈ P and for all z ∈ x◦y,
(2) µ(x) ≤ µ(x−1), for all x ∈ P .

The following elementary facts about fuzzy subpolygroups follow easily from the axioms: µ(x) =
µ(x−1) and µ(x) ≤ µ(e), for all x ∈ P .

Proposition 3.3. Let A be an IFSP of a polygroup P . Then A(x−1) = A(x), that is, µA(x) =
µA(x

−1), νA(x) = νA(x
−1) and µA(x) ≤ µA(e), νA(x) ≥ νA(e) for each x ∈ P , where e is the identity

element of P .

Proof. By Definition 3.2, we have µA(x) = µA(x
−1) and µA(x) ≤ µA(e) for each x ∈ P . Thus it is

enough to show that νA(x) = νA(x
−1) and νA(x) ≥ νA(e) for each x ∈ P .

Let x ∈ P . Then,
νA(x) = νA((x

−1)−1) ≤ νA(x−1) ≤ νA(x).
On the other hand, for each z ∈ x◦x−1, we have νA(z) ≤ νA(x)∨νA(x−1). Since e ∈ x◦x−1∩x−1◦x, so
νA(e) ≤ νA(x) ∨νA(x−1) = νA(x). Thus νA(e) ≤ νA(x) for each x ∈ P . This complete the proof. �

Proposition 3.4. Let P be a polygroup.

(1) If µA is a fuzzy subpolygroup of P , then A = (µA,µAc) ∈ IFSP(P).
(2) If A ∈ IFSP(P), then µA and νAc are fuzzy subpolygroups of P .
(3) A = (χT ,χTc) ∈ IFSP(P) if and only if T is a subpolygroup of P .

Proof. It is straightforward. �

Proposition 3.5. Let {Aα}α∈J ⊂ IFSP(P). Then
⋂
α∈J

Aα ∈ IFSP(P).

Proof. It is straightforward. �

Proposition 3.6. If A be an IFSP of a polygroup P then,

PA = {x ∈ P : A(x) = A(e), that is, µA(x) = µA(e) and νA(x) = νA(e)}
is a subpolygroup of P .

Proof. We have to show that:

(1) x◦y ⊆ PA for each x,y ∈ PA.



INTUITIONISTIC FUZZY TOPOLOGICAL POLYGROUPS 167

(2) If x ∈ PA then, x−1 ∈ PA.
Let x,y ∈ PA and z ∈ x◦y. Since x,y ∈ PA then, µA(x) = µA(e), νA(x) = νA(e) and µA(y) = µA(e),
νA(y) = νA(e). Since A ∈ IFSP(P) and z ∈ x◦y,

µA(z) ≥ µA(x) ∧µA(y)
= µA(e) ∧µA(e) = µA(e),

and
νA(z) ≤ νA(x) ∨νA(y)

= νA(e) ∨νA(e) = νA(e).
So µA(z) ≥ µA(e) and νA(z) ≤ νA(e). Then µA(z) = µA(e), νA(z) = νA(e) and z ∈ PA, that is,
x◦y ⊆ PA.
Now, if x ∈ PA then, µA(x) = µA(e) and νA(x) = νA(e). Since µA(x−1) = µA(x) = µA(e) and
νA(x

−1) = νA(x) = νA(e), that is, x
−1 ∈ PA. Hence PA is a subpolygroup of P . �

Definition 3.7. [26] Let A be an IFS in a set X and let α,β ∈ I with α + β ≤ 1. Then the set

Cα,β(A) = {x ∈ X : µA(x) ≥ α and νA(x) ≤ β}

is called a (α,β)- cut set of A.

Proposition 3.8. Let A be an IFSP of a polygroup P . Then for each (α,β) ∈ I×I with (α,β) ≤ A(e),
that is, α ≤ µA(e), β ≥ νA(e), Cα,β(A) is a subpolygroup of P , where e is the identity of P .

Proof. Clearly, Cα,β(A) 6= ∅. Let x,y ∈ Cα,β(A). We show that x◦y ⊆ Cα,β(A).
Let z ∈ x ◦ y. Since x,y ∈ Cα,β(A), then A(x) ≥ (α,β) and A(y) ≥ (α,β), that is, µA(x) ≥
α, νA(x) ≤ β and µA(y) ≥ α, νA(y) ≤ β. Since A ∈ IFSP(P), µA(z) ≥ µA(x) ∧ µA(y) ≥ α and
νA(z) ≤ νA(x) ∨νA(y) ≤ β. Thus A(z) ≥ (α,β). So z ∈ Cα,β(A) and x◦y ⊆ Cα,β(A).
On the other hand, µA(x

−1) ≥ µA(x) ≥ α and νA(x−1) ≤ νA(x) ≤ β. Thus A(x−1) ≥ (α,β). So
x−1 ∈ Cα,β(A). Hence Cα,β(A) is a subpolygroup of P . �

Proposition 3.9. Let A be an IFS in a polygroup P such that Cα,β(A) is a subpolygroup of P for
each (α,β) ∈ I × I with (α,β) ≤ A(e). Then A is an IFSP of P .

Proof. For any x,y ∈ P , let A(x) = (t1,s1) and A(y) = (t2,s2). Then clearly x ∈ Ct1,s1 (A) and
y ∈ Ct2,s2 (A). Suppose t1 < t2 and s1 > s2, then Ct1,s1 (A) ⊂ Ct2,s2 (A). Thus y ∈ Ct1,s1 (A). Since
Ct1,s1 (A) is a subpolygroup of P, x ◦ y ⊆ Ct1,s1 (A). Then for each z ∈ x ◦ y, A(z) = (t1,s1), that
is, µA(z) ≥ t1 and νA(z) ≤ s1. So µA(z) ≥ µA(x) ∧µA(y) and νA(z) ≤ νA(x) ∨νA(y). On the other
hand, for each x ∈ P , let A(x) = (α,β). Then x ∈ Cα,β(A). Since Cα,β(A) is a subpolygroup of P ,
x−1 ∈ Cα,β(A). So A(x−1) ≥ (α,β), that is, µA(x−1) ≥ µA(x) and νA(x−1) ≤ νA(x). Hence A is an
IFSP of P . �

Theorem 3.10. Let A and B be two IFSP’s of a polygroup P . Then A∩B is IFSP of polygroup P .

Proof. By Theorems 3.8 and 3.9, A ∩ B is IFSP of polygroup P if and only if Cα,β(A ∩ B) is a
subpolygroup of P . Clearly, Cα,β(A ∩ B) = Cα,β(A) ∩ Cα,β(B) and both Cα,β(A) and Cα,β(B) are
subpolygroups of P and intersection of two subpolygroups of a polygroup is a subpolygroup of P
implies that Cα,β(A∩B) is a subpolygroup of P and hence A∩B is IFSP of polygroup P. �

Theorem 3.11. Let A and B IFSP of polygroups P1 and P2 respectively. Then A×B is also IFSP
of polygroup P1 ×P2.

Proof. Let A and B be IFSP of polygroups P1 and P2 respectively, then, Cα,β(A) and Cα,β(B) are
subpolygroups of polygroups P1 and P2 respectively, for all α,β ∈ I with α + β ≤ 1. So Cα,β(A) ×
Cα,β(B) is subpolygroup of polygroup P1 × P2. Hence Cα,β(A × B) is subpolygroup of polygroup
P1 ×P2. Therefore A×B is an IFSP of polygroup P1 ×P2. �

Proposition 3.12. Let A and B IFS of the polygroups P1 and P2 respectively such that µA(x) ≤ µB(e2)
and νA(x) ≥ νB(e2) hold for x ∈ P1, e2 being the identity element of P2. If A × B is an IFSP of
P1 ×P2, then, A is IFSP of polygroup P1.



168 ABBASIZADEH AND DAVVAZ

Proof. Let x,y ∈ P1 and z ∈ x◦y, we have:
µA(z) = µA(z) ∧µB(e2) = µA×B(z,e2)

≥ µA×B(x,e2) ∧µA×B(y,e2)
= [µA(x) ∧µB(e2)] ∧ [µA(y) ∧µB(e2)]
= µA(x) ∧µA(y).

Then, µA(z) ≥ µA(x) ∧µA(y). Also,
νA(z) = νA(z) ∨νB(e2) = νA×B(z,e2)

≤ νA×B(x,e2) ∨νA×B(y,e2)
= [νA(x) ∨νB(e2)] ∨ [νA(y) ∨νB(e2)]
= νA(x) ∨νA(y).

Hence, νA(z) ≤ νA(x) ∨νA(y). Therefore A is an IFSP of P1. �

Proposition 3.13. Let A and B IFS of the polygroups P1 and P2 respectively such that µB(y) ≤ µA(e1)
and νB(y) ≥ νA(e1) hold for y ∈ P2, e1 being the identity element of P1. If A × B is an IFSP of
P1 ×P2, then, B is IFSP of polygroup P2.
Proof. The proof is similar to the proof of Proposition 3.12. �

Corollary 3.14. Let A and B IFS of the polygroups P1 and P2 respectively. If A×B is an IFSP of
P1 ×P2, then, either A is IFSP of P1 or B is IFSP of polygroup P2.
Definition 3.15. [12] Let 〈P, ·,e,−1 〉 and 〈P ′,∗,e′,−I 〉 be two polygroups. Let f be a mapping from
P to P ′ such that f(e) = e′. Then, f is called a strong homomorphism or a good homomorphism if
f(x ·y) = f(x) ∗f(y), for all x,y ∈ P .
Definition 3.16. [19] Let A ∈ IFS(P). Then A is said to have the sup property if for any T ∈P∗(P),
there exists a t0 ∈ T such that A(t0) =

⋃
t∈T

A(t), that is, µA(t0) =
∨
t∈T

µA(t) and νA(t0) =
∧
t∈T

νA(t).

Proposition 3.17. Let f : P −→ P ′ be a strong polygroup homomorphism and A ∈ IFSP(P),
B ∈ IFSP(P ′). Then, the following hold:

(1) If A has the sup property then, f(A) ∈ IFSP(P ′).
(2) f−1(B) ∈ IFSP(P).

Proof. (1) First, we suppose that A is an IFSP of P . In order to prove that f(A) is an IFSP of
P ′, by proposition 3.8, it is sufficient to show that each non-empty (α,β)-cut set Cα,β(f(A)) is an
subpolygroup of P ′. So, suppose that Cα,β(f(A)) is non-empty set for some (α,β) ∈ I × I with
(α,β) ≤ A(e). Let y1,y2 ∈ Cα,β(f(A)). We show that y1 ∗y2 ⊆ Cα,β(f(A)). We have

µf(A)(y1) ≥ α, νf(A)(y1) ≤ β and µf(A)(y2) ≥ α, νf(A)(y2) ≤ β,
which implies that∨

x∈f−1(y1)
µA(x) ≥ α,

∧
x∈f−1(y1)

νA(x) ≤ β and
∨

x∈f−1(y2)
µA(x) ≥ α,

∧
x∈f−1(y2)

νA(x) ≤ β.

Since A has the sup property, it follows that there exist x1 ∈ f−1(y1) and x2 ∈ f−1(y2) such that
µA(x1) ≥ α, νA(x1) ≤ β and µA(x2) ≥ α, νA(x2) ≤ β.

Since f is strong homomorphism, it follows that y1 ∗y2 = f(x1) ∗f(x2) = f(x1.x2). Let z ∈ y1 ∗y2.
Then, there exists x′ ∈ x1.x2 such that f(x′) = z. Thus, f(x) = f(x′) = z. Since A is an IFSP of P ,
we have

µA(x
′) ≥ µA(x1) ∧µA(x2) ≥ α and νA(x′) ≤ νA(x1) ∨νA(x2) ≤ β.

Therefore, we obtain

µf(A)(z) =
∨

x∈f−1(z)
µA(x) ≥ α and νf(A)(z) =

∧
x∈f−1(z)

νA(x) ≤ β.

Hence z ∈ Cα,β(f(A)). Thus, y1 ∗y2 ⊆ Cα,β(f(A)).
Next, for y ∈ Cα,β(f(A)), we have µf(A)(y) ≥ α and νf(A)(y) ≤ β. Thus,∨

x∈f−1(y)
µA(x) ≥ α and

∧
x∈f−1(y)

νA(x) ≤ β.



INTUITIONISTIC FUZZY TOPOLOGICAL POLYGROUPS 169

Hence, we have ∨
x−1∈f−1(y−1)

µA(x
−1) ≥ α and

∧
x−1∈f−1(y−1)

νA(x
−1) ≤ β.

Therefore, µf(A)(y
−1) ≥ α and νf(A)(y−1) ≤ β implies that y−1 ∈ Cα,β(f(A)).

(2) Let x,y ∈ P and z ∈ x.y. Then,

µf−1(B)(z) = µB(f(z)) ≥ µB(f(x)) ∧µB(f(y))
= µf−1(B)(x) ∧µf−1(B)(y),

and

νf−1(B)(z) = νB(f(z)) ≤ νB(f(x)) ∨νB(f(y))
= νf−1(B)(x) ∨νf−1(B)(y).

Also, for all x ∈ P

µf−1(B)(x
−1) = µB(f(x

−1)) = µB(f(x)
−1) ≥ µB(f(x)) = µf−1(B)(x),

and

νf−1(B)(x
−1) = νB(f(x

−1)) = νB(f(x)
−1) ≤ νB(f(x)) = νf−1(B)(x).

Hence f−1(B) ∈ IFSP(P). �

Definition 3.18. Let A be an IFSP in a polygroup P. Then A is called an intuitionistic fuzzy normal
subpolygroup (in short, IFNSP) of P if for all x,y ∈ P

A(z) = A(z′) i.e µA(z) = µA(z
′) and νA(z) = νA(z

′),

for all z ∈ x◦y and z′ ∈ y ◦x.

It is obvious that if A is an intuitionistic fuzzy normal subpolygroup of P , then for all x,y ∈ P ,

A(z) = A(z′) for all z,z′ ∈ x◦y.

Theorem 3.19. Let A be an intuitionistic fuzzy subpolygroup of P . Then A is an intuitionistic fuzzy
normal subpolygroup if and only if

(1) µA(z) = µA(y) and
(2) νA(z) = νA(y), for all x,y ∈ P and for all z ∈ x◦y ◦x−1.

Proof. It is straightforward. �

Theorem 3.20. Let A be an intuitionistic fuzzy normal subpolygroup of polygroup P . Then Cα,β(A)
is normal subpolygroup of polygroup P , where µA(e) ≥ α, νA(e) ≤ β and e is the identity element of
P .

Proof. Let y ∈ Cα,β(A) and x ∈ P be any element. Then µA(y) ≥ α, νA(y) ≤ β. Since A be
intuitionistic fuzzy normal subpolygroup of polygroup P , so µA(z) = µA(y) and νA(z) = νA(y) for all
x,y ∈ P and for all z ∈ x ◦ y ◦ x−1. Therefore µA(z) = µA(y) ≥ α and νA(z) = νA(y) ≤ β implies
that µA(z) ≥ α, νA(z) ≤ β and so z ∈ Cα,β(A), x ◦ y ◦ x−1 ⊆ Cα,β(A). Hence Cα,β(A) is normal
subpolygroup of P . �

Proposition 3.21. If A is an IFNSP of P then, PA is a normal subpolygroup of P .

Proof. It is straightforward. �

Theorem 3.22. Let A and B IFNSP of polygroups P1 and P2 respectively. Then A×B is also IFNSP
of polygroup P1 ×P2.

Proof. The proof is similar to the proof of Theorem 3.11. �



170 ABBASIZADEH AND DAVVAZ

4. t-Intuitionistic fuzzy subpolygroups and t-intuitionistic fuzzy quotient polygroups

In this section, the notion of t-intuitionistic fuzzy (normal) subpolygroup, t-intuitionistic fuzzy
cosets of an intuitionistic fuzzy normal subpolygroup and t-intuitionistic fuzzy quotient polygroup are
defined and discussed.

Definition 4.1. [27] Let A be an IFS of a polygroup P . Then the IFS At of P is called the t-
intuitionistic fuzzy subset of P and is defined as At = (µAt,νAt), where

µAt(x) = µA(x) ∧ t and νAt(x) = νA(x) ∨ (1 − t),
for all x ∈ P and t ∈ [0, 1].

Definition 4.2. Let A be an IFS of a polygroup P. Then A is called the t-intuitionistic fuzzy
subpolygroup (in short t-IFSP) of P if At is IFSP of P , i.e, the following conditions hold:

(1) µAt(z) ≥ µAt(x) ∧µAt(y) and
νAt(z) ≤ νAt(x) ∨νAt(y) for each z ∈ x◦y and x,y ∈ P .

(2) µAt(x
−1) ≥ µAt(x) and

νAt(x
−1) ≤ νAt(x) for each x ∈ P .

Proposition 4.3. If A is IFSP of a polygroup P , then A is also t-IFSP of P .

Proof. Let x,y ∈ P and z ∈ x◦y. Then, we have:
µAt(z) = µA(z) ∧ t

≥ (µA(x) ∧µA(y)) ∧ t
= (µA(x) ∧ t) ∧ (µA(y) ∧ t)
= µAt(x) ∧µAt(y).

Thus µAt(z) ≥ µAt(x) ∧µAt(y). Similarly we can show that νAt(z) ≤ νAt(x) ∨νAt(y). Also,

µAt(x
−1) = µA(x

−1) ∧ t = µA(x) ∧ t = µAt(x).
Similarly, we can show that νAt(x

−1) = νAt(x). Hence A is t-IFSP of P . �

Definition 4.4. Let P be a polygroup, A be an intuitionistic fuzzy subpolygroup of P and t ∈ [0, 1].
Then, the intuitionistic fuzzy subset Ata of P which is defined by

Ata(x) = (µAta(x),νAta(x)),

where
µAta(x) = (

∧
z∈x◦a

µA(z)) ∧ t and νAta (x) = (
∧

z∈x◦a
µA(z)) ∨ (1 − t),

is called the t- intuitionistic fuzzy right coset of A. The t- intuitionistic fuzzy left coset aA
t of A is

defined similarly.

Proposition 4.5. Let A be an intuitionistic fuzzy normal subpolygroup of P and a an arbitrary element
of P . Then, the t- intuitionistic fuzzy right coset Ata is same as the t- intuitionistic fuzzy left coset

aA
t.

Proof. Let A be an intuitionistic fuzzy normal subpolygroup of P , a ∈ P and t ∈ [0, 1]. Then, for any
x ∈ P and z ∈ x◦a, z′ ∈ a◦x, µA(z) = µA(z′), νA(z) = νA(z′). So,

µAta(x) = (
∧

z∈x◦a
µA(z)) ∧ t

= (
∧

z′∈a◦x
µA(z

′)) ∧ t

= µ
aAt(x),

and
νAta(x) = (

∧
z∈x◦a

νA(z)) ∨ (1 − t)

= (
∧

z′∈a◦x
νA(z

′)) ∨ (1 − t)

= ν
aAt(x).

Hence Ata =a A
t. �



INTUITIONISTIC FUZZY TOPOLOGICAL POLYGROUPS 171

Definition 4.6. Let A be t-IFSP of a polygroup P . Then A is called t-intuitionistic fuzzy normal
subpolygroup (in short t-IFNSP) of P if and only if Ata =a A

t for all a ∈ P .

Lemma 4.7. Let A be t-IFNSP of a polygroup P . Then

Ata = A
t
b ⇔ Na = Nb

for all a,b ∈ P , where N = Ct,1−t(A).

Proof. It is straightforward. �

Consider the set P/At = {Ata | a ∈ P} of all t- intuitionistic fuzzy right coset of A.

Theorem 4.8. Let A be an intuitionistic fuzzy normal subpolygroup of a polygroup P and N =
Ct,1−t(A). Then, there is a bijection between P/A

t and P/N.

Proof. The proof is similar to the proof of Theorem 2.3.8 in [10]. �

Corollary 4.9. Let P be a polygroup, A be an intuitionistic fuzzy normal subpolygroup of P and
a ∈ P . Then, Atz = Ata for all z ∈ Na, where N = Ct,1−t(A) and t ∈ [0, 1].

Proposition 4.10. Let P be a polygroup, A be an intuitionistic fuzzy normal subpolygroup of a poly-
group P . Then, (P/At,⊗) is a polygroup (called the polygroup of t- intuitionistic fuzzy coset induced
by A and t), where the hyperoperation ⊗ is defined as follows:

⊗ : P/At ×P/At −→P∗(P/At)

(Ata , A
t
b) 7→ {A

t
c | c ∈ N.a.b}

and −1 on P/At is defined by (Ata)
−1 = At

a−1
.

Proof. The proof is similar to the proof of Theorem 2.3.10 in [10]. �

Definition 4.11. Let P be a polygroup, A be an intuitionistic fuzzy subset of P and β∗ the funda-
mental relation on P . Define the intuitionistic fuzzy subset Aβ∗ on P/β

∗ as follows:

Aβ∗ : P/β
∗ −→ I × I

Aβ∗(β
∗(x)) = (µAβ∗ (β

∗(x)),νAβ∗ (β
∗(x)))

= (
∨

a∈β∗(x)
µA(a),

∧
a∈β∗(x)

νA(a)).

Theorem 4.12. Let P be a polygroup, A be an intuitionistic fuzzy subpolygroup of P . Then Aβ∗ is
an intuitionistic fuzzy subgroup of the group P/β∗.

Proof. We have

νAβ∗ (β
∗(x)) ∨νAβ∗ (β

∗(y)) = (
∧

a∈β∗(x)
νA(a)) ∨ (

∧
b∈β∗(y)

νA(b))

=
∧

a∈β∗(x)
b∈β∗(y)

[νA(a) ∨νA(b)]

≥
∧

a∈β∗(x)
b∈β∗(y)

(
∨

z∈a◦b
νA(z))

≥
∧

a∈β∗(x)
b∈β∗(y)

(
∧

z∈a◦b
νA(z))

≥
∧

a∈β∗(x)
b∈β∗(y)

(
∧

z∈β∗(a.b)
νA(z))

=
∧

a∈β∗(x)
b∈β∗(y)

(νAβ∗ (β
∗(a.b)))

=
∧

a∈β∗(x)
b∈β∗(y)

(νAβ∗ (β
∗(a) �β∗(a)))

= νAβ∗ (β
∗(x) �β∗(y)).

Similarly we can show that

µAβ∗ (β
∗(x)) ∧µAβ∗ (β

∗(y)) ≤ µAβ∗ (β
∗(x) �β∗(y)).



172 ABBASIZADEH AND DAVVAZ

Now, suppose that β∗(x) is an arbitrary element of P/β∗. Then,

νAβ∗ (β
∗(x)−1) = νAβ∗ (β

∗(x−1))
=

∧
a∈β∗(x−1)

νA(a)

=
∧

a∈β∗(x−1)
νA(a

−1)

=
∧

b∈β∗(x)
νA(b)

= νAβ∗ (β
∗(x)).

Similarly, we can show that µAβ∗ (β
∗(x)−1) = µAβ∗ (β

∗(x)). Thus, the proof is complete. �

5. Intuitionistic fuzzy topological polygroups

In this section, we define and study the concept of intuitionistic fuzzy topological polygroups, and
we prove some properties in this respect.

Definition 5.1. Let (P,T ) be a polygroup and A = (µA,νA), B = (µB,νB) are two intuitionistic
fuzzy sets in P . We define AB and A−1 by the respective formula:

(1)

µAB(x) =

{ ∨
(x1,x2)∈X×X

[µA(x1) ∧µB(x2)] if x ∈ x1 ◦x2.

0 otherwise,

and

νAB(x) =

{ ∧
(x1,x2)∈X×X

[νA(x1) ∨νB(x2)] if x ∈ x1 ◦x2.

1 otherwise.

(2)
µA−1 (x) = µA(x

−1) and νA−1 (x) = νA(x
−1).

Definition 5.2. Let (P,◦) be a polygroup and (P,T ) be an intuitionistic fuzzy topological space. Let
U = (µU,νU ), V = (µV ,νV ) and W = (µW ,νW ) be an intuitionistic fuzzy sets in P. (P,◦,T ) is called
an intuitionistic fuzzy topological polygroup or IFTP for short, if and only if:

(1) For all x,y ∈ P and any fuzzy open Q-neighborhood W of any intuitionistic fuzzy point zr,s
of x◦y, there are fuzzy open Q-neighborhoods U of xr,s and V of yr,s such that:
UV ⊆ W .

(2) For all x ∈ P and any fuzzy open Q-neighborhood V of an intuitionistic fuzzy point (x−1)r,s,
there exists a fuzzy open Q-neighborhood U of xr,s such that:
U−1 ⊆ V .

Evidently, every intuitionistic fuzzy topological group is an intuitionistic fuzzy topological poly-
group. We give some other examples.

Example 5. Let P be a polygroup and T is the collection of all constant intuitionistic fuzzy sets in
P. Then (P,T ) is an intuitionistic fuzzy topological polygroup.

Example 6. Let P = {e,a,b}. Then, P together with the following hyperoperation
◦ e a b
e e a b
a a e b
b b b {e,a}

is a polygroup. It is clear that a−1 = a, b−1 = b. Consider on P the fuzzy topology T = {0∼,A, 1∼},
where A =< x, ( e

0.7
, a
0.5
, b
0.3

), ( e
0.3
, a
0.5
, b
0.7

) >. Then, (P,T ) is an intuitionistic fuzzy topological poly-
group.

Definition 5.3. [25] Let (X,T ) be an intuitionistic fuzzy topological space. Let α,β ∈ [0, 1]. An
intuitionistic fuzzy set (αβ)∗ = (µ(αβ)∗,ν(αβ)∗), where µ(αβ)∗(x) = α and ν(αβ)∗(x) = β, for every
x ∈ X such that µ(αβ)∗(x) + ν(αβ)∗(x) = 1. Then (X,T ) is called a fully stratified space if for every
α,β ∈ [0, 1], (αβ)∗ ∈T .



INTUITIONISTIC FUZZY TOPOLOGICAL POLYGROUPS 173

Proposition 5.4. Suppose (P,T ) is a fully stratified space. Let (P,◦,T ) be an intuitionistic fuzzy
topological polygroup. Then the mapping f : x −→ x−1 is intuitionistic fuzzy homeomorphic function
of (P,T ) onto itself.

Proof. It is seen that f is invertible. Hence the only thing which needs to be proved that f is intu-
itionistic fuzzy continuous. Let (P,◦,T ) be an intuitionistic fuzzy topological polygroup and V be a
fuzzy open Q-neighbourhood of intuitionistic fuzzy point (x−1)r,s. Then, there exists a fuzzy open
Q-neighbourhood U of xr,s such that U

−1 ⊆ V . Since

µU−1 (x) + r = µU (x
−1) + r

> µU (x) + r > 1,

and
νU−1 (x) + s = νU (x

−1) + s
< νU (x) + s < 1.

This implies that xr,sq U
−1. Hence U−1 is a fuzzy open Q-neighbourhood of xr,s. Thus f(U) =

U−1 ⊆ V . Then f is an intuitionistic fuzzy continuous function at the intuitionistic fuzzy point xr,s.
Therefore, f is an intuitionistic fuzzy continuous function. �

Proposition 5.5. Let (P,◦,T ) be an intuitionistic fuzzy topological polygroup.
(1) If U is an intuitionistic fuzzy compact subset of P then, U−1 is an intuitionistic fuzzy compact.
(2) If U is an intuitionistic fuzzy open set in T then, U−1 is an intuitionistic fuzzy open set in T .

Proof. It is straightforward. �

Definition 5.6. [7] Let (X,T ) be an IFTS and A an IFS in X. Then the fuzzy closure is defined by

cl(A) = ∩{F| A ⊆ F, Fc ∈T},

and the fuzzy interior is defined by

int(A) = ∪{G| A ⊇ G, G ∈T}.

Definition 5.7. Let P be a polygroup and A be IFSP of polygroup P . Let a ∈ P be a fixed element.
Then the set aA =< µaA,νaA > where

µaA(x) =
∨

z∈a−1◦x
µA(z) for all x ∈ P,

and
νaA(x) =

∧
z∈a−1◦x

νA(z) for all x ∈ P,

is called intuitionistic fuzzy left coset of P determined by A and a.
Similarly, the set Aa =< µAa,νAa > where

µAa(x) =
∨

z∈x◦a−1
µA(z) for all x ∈ P,

and
νAa(x) =

∧
z∈x◦a−1

νA(z) for all x ∈ P,

is called intuitionistic fuzzy right coset of P determined by A and a.

If A is an intuitionistic fuzzy normal subpolygroup of P and a an arbitrary element of P , then the
intuitionistic fuzzy right coset Aa is same as the intuitionistic fuzzy left coset aA. Consider the set
P/A = {Aa | a ∈ P} of all intuitionistic fuzzy right cosets of A. Now we give a structure on P/A by
defining the operation ⊗ between two intuitionistic fuzzy right cosets as

Aa⊗Ab = {Ac | c ∈ a◦ b}.

If A is an intuitionistic fuzzy normal subpolygroup of a polygroup P , then the operation ⊗ defined on
P/A is well defined. Then, (P/A,⊗) becomes a polygroup and is called the intuitionistic fuzzy quotient
polygroup relative to the intuitionistic fuzzy normal subpolygroup A.



174 ABBASIZADEH AND DAVVAZ

Proposition 5.8. Let (P,T ) be an intuitionistic fuzzy topological polygroup. Then, the family B =
{à ∈ IFS(P∗(P)) | A ∈ T}, where µÃ(X) =

∨
x∈X

µA(x) and νÃ(X) =
∧
x∈X

νA(x) is a base for an

intuitionistic fuzzy topology T ∗ on P∗(P).

Proof. B is a base for an intuitionistic fuzzy topology on P∗(P) because:
(1) For any Ã1, Ã2 ∈B, with A1,A2 ∈T , it follows that Ã1 ∩ Ã2 ∈B, because Ã1 ∩ Ã2 = Ã1 ∩A2

and A1 ∩A2 ∈T .
Indeed, for any X ∈P∗(P), we have

µ
Ã1∩A2

(X) =
∨
x∈X

µ(A1∩A2)(x) =
∨
x∈X

(µA1 (x) ∧µA2 (x))

= (
∨
x∈X

µA1 (x)) ∧ (
∨
x∈X

µA2 (x))

= µÃ1 (X) ∧µÃ2 (X)
= µ(Ã1∩Ã2)(X),

and
ν
Ã1∩A2

(X) =
∧
x∈X

ν(A1∩A2)(x) =
∧
x∈X

(νA1 (x) ∨νA2 (x))

= (
∧
x∈X

νA1 (x)) ∨ (
∧
x∈X

νA2 (x))

= νÃ1 (X) ∨νÃ2 (X)
= ν(Ã1∩Ã2)(X),

(2) Since 1∼ ∈T it follows that 1̃∼(X) = 1 for any X ∈P∗(P) and thus⋃̃
A∈B

= 1. �

Lemma 5.9. Let U be an intuitionistic fuzzy open subset of an intuitionistic fuzzy topological polygroup
P . Then, aU and Ua are intuitionistic fuzzy open subsets of P for every a ∈ P .

Proof. Suppose that U be an intuitionistic fuzzy open subset of P . Then,

µ(
a−1φ

−1(Ũ))(z) = µŨ (a−1φ(z)) = µŨ (a
−1 ◦z)

=
∨

t∈a−1◦z
µU (t) = µaU (z),

and
ν(
a−1φ

−1(Ũ))(z) = νŨ (a−1φ(z)) = νŨ (a
−1 ◦z)

=
∧

t∈a−1◦z
νU (t) = νaU (z).

Since the mapping a−1φ
−1 : P −→ P∗(P),x 7→ a−1 ◦x, is intuitionistic fuzzy continuous, thus aU is

intuitionistic fuzzy open. Similarly, we can prove that Ua is intuitionistic fuzzy open. �

Proposition 5.10. Let (P,T ) be a fully stratified space. Let (P,◦,T ) be an intuitionistic fuzzy topo-
logical polygroup and U = (µU,νU ) be an intuitionistic fuzzy set of P . If IFcl(U) is an intuitionistic
fuzzy closed set , then aIFcl(U), IFcl(U)a are intuitionistic fuzzy closed sets, where a ∈ P is a definite
point.

Proof. It is straightforward. �

Proposition 5.11. Let A be an IFSP of polygroup P . Then for each (r,s) ∈ I ×I with (r,s) ≥ A(e),
xr,sA = xA, where x ∈ P and e is the identity of P .

Proof. We have (xr,sA)(t) = (µxr,sA(t),νxr,sA(t), where

µxr,sA(t) =
∨

t∈t1◦t2
[µxr,s(t1) ∧µA(t2)]

=

{ ∨
t∈t1◦t2

[r ∧µA(t2)] if t1 = x

0 if t1 6= x
=

∨
t2∈x−1◦t

µA(t2)

= µxA(t),



INTUITIONISTIC FUZZY TOPOLOGICAL POLYGROUPS 175

and
νxr,sA(t) =

∧
t∈t1◦t2

[νxr,s(t1) ∨νA(t2)]

=

{ ∧
t∈t1◦t2

[s∨νA(t2)] if t1 = x

1 if t1 6= x
=

∧
t2∈x−1◦t

νA(t2)

= νxA(t).

Hence xr,sA = xA. �

Proposition 5.12. Let (P,T ) be a fully stratified space. Let (P,◦,T ) be an intuitionistic fuzzy topo-
logical polygroup and U = (µU,νU ) be an intuitionistic fuzzy set of P . If IFcl(U) is an intuitionistic
fuzzy closed set , then ar,sIFcl(U), IFcl(U)ar,s and IFcl(U)

−1 are intuitionistic fuzzy closed sets.

Theorem 5.13. In an intuitionistic fuzzy topological polygroup P , V is a Q-neighbourhood of er,s if
and only if V −1 is a Q-neighbourhood of er,s.

Proof. Let V be a Q-neighbourhood of er,s. Then there exists intuitionistic fuzzy open set A such that
er,sqA ⊆ V that is,

µA(e) + r > 1, A ⊆ V,
νA(e) + s < 1, A ⊆ V.

For all x ∈ P , µA(x−1) ≤ µV (x−1) and νA(x−1) ≥ νV (x−1), so µA−1 (x) ≤ µV−1 (x) and νA−1 (x) ≥
νV−1 (x) then, A

−1 ⊆ V −1.
Now,

µA−1 (e) + µer,s(e) = µA−1 (e) + r > 1,

νA−1 (e) + νer,s(e) = νA−1 (e) + s < 1.

Hence, er,sqA
−1 and A−1 ⊆ V −1. Therefore, V −1 is a Q-neighbourhood of er,s.

Conversely, let V −1 be a Q-neighbourhood of er,s. Then there exists intuitionistic fuzzy open set A
such that er,sqA ⊆ V −1. As above, A−1 ⊆ V and er,sqA−1. That is, V is a Q-neighbourhood of
er,s. �

Proposition 5.14. Let (P,T ) be a fully stratified space. Let (P,◦,T ) be an intuitionistic fuzzy topo-
logical polygroup and U = (µU,νU ) be an intuitionistic fuzzy set of P . If U is a Q-neighbourhood of
er,s, then x1,0U is a Q-neighbourhood of xr,s.

Proof. Since U is a Q-neighbourhood of er,s, there exists an intuitionistic fuzzy open set A such that
r + µA(e) > 1 and s + νA(e) < 1, A ⊆ U. So,

µx1,0A(x) =
∨
x∈xy

[µx1,0 (x) ∧µA(y)]

≥ 1 ∧µA(e)
= µA(e),

and

r + µx1,0A(x) ≥ r + µA(e) > 1.
Also

νx1,0A(x) =
∧
x∈xy

[νx1,0 (x) ∨νA(y)]

≥ 0 ∨νA(e)
= νA(e),

and

s + νx1,0A(x) ≤ s + νA(e) < 1.
Thus, for all z ∈ P ,

x1,0U(z) = xU(z) = (
∨

t∈x−1◦z
µU (t),

∧
t∈x−1◦z

νU (t))

⊇ (
∨

t∈x−1◦z
µA(t),

∧
t∈x−1◦z

νA(t))

= x1,0A(z).



176 ABBASIZADEH AND DAVVAZ

Hence xr,sq x1,0A ⊆ x1,0U and since x1,0A is an intuitionistic fuzzy open set, Therefore x1,0U is a
Q-neighbourhood of xr,s. �

Proposition 5.15. [25] An intuitionistic fuzzy point xr,s ⊆ IFcl(A) if and only if each Q-neighbourhood
of xr,s is quasi-coincident with A.

Proposition 5.16. Let (P,T ) be a fully stratified space. Let (P,◦,T ) be an intuitionistic fuzzy topo-
logical polygroup and A = (µA,νA) be an intuitionistic fuzzy subset of P . If xr,s ⊆ IFclA, then
(
⋂

C∈{C}
AC)(x) = (

⋂
C∈{C}

CA)(x) ⊃ 0, where {C} is the system of all Q- neighbourhood of ea,b in P

with a ≤ r and b ≥ s.

Proof. Since xr,s ⊆ IFclA then, each Q- neighbourhood of xr,s is quasi-coincident with A. For any
C ∈{C}, there exists IFOS B such that ea,b q B ⊆ C, that is,

µB(e) + a > 1, B ⊆ C,

νB(e) + b < 1, B ⊆ C.

Hence, x1,0B
−1 is an IFOS. Moreover, we have

µx1,0B−1 (x) =
∨
x∈xy

[µx1,0 (x) ∧µB−1 (y)]

≥ 1 ∧µB−1 (e)
= 1 ∧µB(e−1)
= µB(e),

and

µx1,0B−1 (x) + r ≥ µB(e) + r ≥ µB(e) + a > 1.

Also, we have

νx1,0B−1 (x) =
∧
x∈xy

[νx1,0 (x) ∨νB−1 (y)]

≥ 0 ∨νB−1 (e)
= 0 ∨νB(e−1)
= νB(e),

and

νx1,0B−1 (x) + s ≤ νB(e) + s ≤ νB(e) + b < 1.

Hence, we conclude that

x1,0C
−1(z) = xC−1(z) = (

∨
t∈x−1◦z

µC−1 (t),
∧

t∈x−1◦z
νC−1 (t))

⊇ (
∨

t∈x−1◦z
µB−1 (t),

∧
t∈x−1◦z

νB−1 (t))

= xB−1(z)
= x1,0B

−1(z).

This implies that xr,s q x1,0B
−1 ⊆ x1,0C−1. Since x1,0C−1 and A are quasi-coincident, there exists

y ∈ P such that
µx1,0C−1 (y) + µA(y) > 1 and νx1,0C−1 (y) + νA(y) < 1.

Also

µx1,0C−1 (y) =
∨
y∈xz

[µx1,0 (x) ∧µC−1 (z)]

=
∨
y∈xz

[1 ∧µC−1 (z)]

=
∨

z∈x−1y
µC−1 (z)

= µxC−1 (y),



INTUITIONISTIC FUZZY TOPOLOGICAL POLYGROUPS 177

and
νx1,0C−1 (y) =

∧
y∈xz

[νx1,0 (x) ∨νC−1 (z)]

=
∧
y∈xz

[0 ∨νC−1 (z)]

=
∧

z∈x−1y
νC−1 (z)

= νxC−1 (y).

Thus
µAC(x) =

∨
x∈t1t2

[µA(t1) ∧µC(t2)]

≥ µA(y) ∧ (
∨

z∈y−1x
µC(z))

= µA(y) ∧ (
∨

z−1∈x−1y
µC−1 (z))

= µA(y) ∧µx1,0C−1 (y)
> 0,

and
νAC(x) =

∧
x∈t1t2

[νA(t1) ∨νC(t2)]

≤ νA(y) ∨ (
∧

z∈y−1x
νC(z))

= νA(y) ∨ (
∧

z−1∈x−1y
νC−1 (z))

= νA(y) ∨νx1,0C−1 (y)
< 1.

That is, AC(x) ⊃ 0 for every C ∈ {C}. Hence (∩AC)(x) =
∧

C∈{C}
AC(x) ⊃ 0. It is easy to prove

∩AC = ∩CA. �

Proposition 5.17. Let (P,T ) be a fully stratified space. Let (P,◦,T ) be an intuitionistic fuzzy topolog-
ical polygroup and A = (µA,νA) be an intuitionistic fuzzy subset of P . If xr,s ⊆

⋂
C∈{C}

AC =
⋂

C∈{C}
CA

and r > 0.5, s < 0.5, then xr,s ⊆ IFcl(A), where {C} is the system of all Q- neighbourhood of ea,b in
P with a ≤ r and b ≥ s.

Proof. Let xr,s ⊆ AC for each C ∈ {C}. Then µAC(x) ≥ r and νAC(x) ≤ s. Let D be an arbitrary
Q-neighbourhood of xr,s. Then there exists an IFOS B such that xr,s q B ⊆ D. That is,

µB(x) + r > 1, B ⊆ D,

νB(x) + s < 1, B ⊆ D.
Since µB(x) + r > 1, r > 0.5 and νB(x) + s < 1, s < 0.5, thus D(x) ⊇ B(x) ⊇ 0. Hence, B−1x1,0 is
an IFOS. Moreover we have

µB−1x1,0 (e) =
∨
e∈yx

[µB−1 (y) ∧µx1,0 (x)]

≥ µB−1 (x−1) ∧ 1
= µB−1 (x

−1)
= µB(x).

and

µB−1x1,0 (e) + a ≥ µB(x) + a ≥ µB(x) + r > 1.
Similarly, we have

νB−1x1,0 (e) =
∧
e∈yx

[νB−1 (y) ∨νx1,0 (x)]

≤ νB−1 (x−1) ∨ 0
= νB−1 (x

−1)
= νB(x).

and

νB−1x1,0 (e) + b ≤ νB(x) + b ≤ νB(x) + s < 1.



178 ABBASIZADEH AND DAVVAZ

Hence
B−1x1,0(z) = B

−1x(z) = (
∨

t∈z◦x−1
µB−1 (t),

∧
t∈z◦x−1

νB−1 (t))

⊆ (
∨

t∈z◦x−1
µD−1 (t),

∧
t∈z◦x−1

νD−1 (t))

= D−1x1,0(z).

This implies that ea,b q B
−1x1,0 ⊆ D−1x1,0. So D−1x1,0 ∈ {C}. Thus µAD−1x1,0 (x) ≥ r and

νAD−1x1,0 (x) ≤ s. Moreover, we have

µAD−1x1,0 (x) =
∨
x∈yx

[µAD−1 (y) ∧µx1,0 (x)]

≥ µAD−1 (e) ∧ 1
= µAD−1 (e),

and
νAD−1x1,0 (x) =

∧
x∈yx

[νAD−1 (y) ∨νx1,0 (x)]

≤ νAD−1 (e) ∨ 0
= νAD−1 (e).

µAD−1 (e) =
∨

e∈t1t2
[µA(t1) ∧µD−1 (t2)]

≥ µA(k) ∧µD−1 (k−1)
= µA(k) ∧µD(k),

and
νAD−1 (e) =

∧
e∈t1t2

[νA(t1) ∨νD−1 (t2)]

≤ νA(k) ∨νD−1 (k−1)
= νA(k) ∨νD(k).

Thus there exists z ∈ P such that

µAD−1 (e) ≥ µA(z) ∧µD(z) and νAD−1 (e) ≤ νA(z) ∨νD(z).

Since µAD−1 (e) ≥ r and νAD−1 (e) ≤ s then, µA(z) ≥ r, νA(x) ≤ s and µD(z) ≥ r,νD(x) ≤ s. Since
r > 0.5 and s < 0.5,

µA(z) + µD(z) ≥ r + r = 2r > 1 and νA(z) + νD(z) ≤ s + s = 2s < 1.

That is, D is quasi-coincident with A. Therefore xr,s ⊆ IFcl(A). �

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Department of Mathematics, Yazd University, Yazd, Iran

∗Corresponding author: davvaz@yazd.ac.ir