International Journal of Analysis and Applications
ISSN 2291-8639
Volume 11, Number 2 (2016), 124-136
http://www.etamaths.com

CATEGORY OF FUZZY TOPOLOGICAL POLYGROUPS

N. ABBASIZADEH AND B. DAVVAZ∗

Abstract. In this paper, the relation between two definitions of a fuzzy topological polygroup is
discussed. The collection of all fuzzy continuous functions from a fuzzy topological space Y to a

fuzzy topological polygroup Z, denoted by FC(Y, Z) induces a polygroup structure from that of Z.

Moreover, we study (fuzzy) topological polygroup of the polygroup FC(Y, Z) when it is equipped
with various known topologies and fuzzy topologies. Also, a few properties of fuzzy topological

polygroups are established and a category CF T P is formed with objects as FTP and morphisms as

the fuzzy topological homomorphisms. The category CT P is seen to be a full subcategory of CF T P .

1. Introduction

The hyperstructure theory was born in 1934 when Marty introduced the notion of hypergroup [20].
In 1979, Foster [13] introduced the concept of fuzzy topological group. Ma and Yu [19] changed the
definition of a fuzzy topological group in order to make sure that an ordinary topological group is a
special case of a fuzzy topological group. On the other hand, in the last few decades, many connections
between hyperstructures and fuzzy sets has been established and investigated. The concept of fuzzy
topological polygroup (in short FTP) was introduced and studied in [1]. In [2] we have observed that
the collection of all fuzzy continuous functions from a fuzzy topological space Y to a fuzzy topological
polygroup Z, denoted by FC(Y,Z) induces a polygroup structure from that of Z. Here we investigate
FC(Y,Z) in presence of various known topologies and fuzzy topologies. Then, we show how a fuzzy
topological polygroup induced by strong homomorphism. Also, we observe that the collection of all
fuzzy topological polygroups and fuzzy topological homomorphisms constitute a category, with we call
CFTP . Moreover, the category CTP of topological polygroups and continuous homomorphisms form a
full subcategory of CFTP . 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 [7, 20].

A special subclass of hypergroups is the class of polygroups. We recall the following definition from
[6]. 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,
(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

2010 Mathematics Subject Classification. 54A40, 20N20, 03E72.
Key words and phrases. polygroup; topological polygroup; fuzzy topological polygroups; category CF T P .

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.

124



CATEGORY OF FUZZY TOPOLOGICAL POLYGROUPS 125

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 [3, 9, 17].

2. Preliminaries

For the sake of convenience and completeness of our study, in this section some basic definition and
results of [4, 5, 13, 19, 21, 22], which will be needed in the sequel are recalled here. Throughout this
paper, the symbol I will denote the unit interval [0, 1].

Let X be a non-empty set. A fuzzy set A in X is characterized by a membership function µA :
X → [0, 1] which associates with each point x ∈ X its grade or degree of membership µA(x) ∈ [0, 1].
That is, an element of IX. We denote by FS(X) the set of all fuzzy sets on X. A family T ⊆ FS(X)
of fuzzy sets is called a fuzzy topology for X if it satisfies the following three 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 .

The pair (X,T ) is called a fuzzy topological space or FTS, for short. The elements of T are called
fuzzy open sets. A fuzzy set is fuzzy closed if and only if its complement is fuzzy open.

A fuzzy set in X is called a fuzzy point if and only if it takes the value 0 for all y ∈ X except one,
say x ∈ X. If its value at x is λ (0 < λ ≤ 1), we denote this fuzzy point by xλ, where the point x
is called its support. The fuzzy point xλ is said to be contained in a fuzzy set A, or to belong to A,
denoted by xλ ∈ A, if and only if λ ≤ µA(x). Evidently, every fuzzy set A can be expressed as the
union of all the fuzzy points which belong to A.

A fuzzy set A in a fuzzy topological space (X,T ) is called a neighborhood of fuzzy point xλ, if
there exists a B ∈T such that xλ ∈ B ≤ A. The family consisting of all neighborhood of xλ is called
the system of neighborhood of fuzzy point xλ. A fuzzy point xλ is said to be quasi-coincident with a
fuzzy set A, denoted by xλqA, if µA(x) + λ > 1. A is said to be quasi-coincident with B, denoted by
AqB, if there exists x ∈ X such that µA(x) + µB(x) > 1. If this is true, we also say that A and B are
quasi-coincident at x. A fuzzy set A in a fuzzy topological space (X,T ) is said to be a Q-neighborhood
of xλ if there exists a B ∈T such that xλqB ≤ A. The family consisting of all Q-neighborhood of xλ
is called the system of Q-neighborhood of fuzzy point xλ. A fuzzy topological space (X,T ) is called a
fully stratified space if T contains all constant fuzzy sets.

Given two topological spaces (X,T ) and (Y,U), a mapping f : X → Y is fuzzy continuous if, for
any fuzzy set B ∈U, the inverse image f−1[B] ∈T . Conversely, f is fuzzy open if, for any open fuzzy
set A ∈T , the image f[A] ∈U (see [4]).

Let A be a fuzzy set in the fuzzy topological space (X,T ). Then the induced fuzzy topology on A
is the family TA of fuzzy subsets of A which are the intersection with A of T -open fuzzy sets in X.
The pair (A,TA) is called a fuzzy subspace of (X,T ). For any fuzzy set A∩Uj of TA, with Uj ∈ T ,
we have µUj∩A(x)=µUj (x) ∧µA(x) (see [13]).

Let X and Y be two non-empty subsets, f : X → Y , A be a fuzzy set in X and B a fuzzy set in Y .
Then, f[A] is the fuzzy set in Y defined by

µf[A](y) =

{ ∨
x∈f−1(y)

µA(x) if f
−1(y) 6= ∅.

0 otherwise.

for all y ∈ Y , where f−1(y) = {x|f(x) = y}.
f−1[B] is the fuzzy set in X defined by µf−1[B](x) = µB (f(x)) for all x ∈ X. Let f be a mapping

of FTS (X,T ) into FTS (Y,U). If for any fuzzy open Q-neighborhood U of f(xλ) = [f(x)]λ there
exists a fuzzy open Q-neighborhood V of xλ such that f(V ) ≤ U, then we say that f is continuous at
xλ with respect to Q-neighborhood (see [21]).

Let f be a function from a fuzzy topological space (X,T ) into a fuzzy topological space (Y,U).
Then the following are equivalent (see [21]):



126 ABBASIZADEH AND DAVVAZ

(1) f is a fuzzy continuous mapping.
(2) f is continuous with respect to Q-neighborhood at any fuzzy point xλ.
(3) f is continuous with respect to neighborhood at any fuzzy point xλ.

3. Topological and fuzzy topological polygroups on FC(Y,Z)

Let P = 〈P,◦,e,−1 〉 be a polygroup, A,B ∈ IP and C,D ⊆ P . We define A•B ∈ IP , A−1 ∈ IP ,
C ◦D ⊆ P and C−1 ⊆ P by the respective formulas: (see [1])

(A•B)(x) =
∨

x∈x1◦x2

(µA(x1) ∧µB(x2))

and
µA−1 (x) = µA(x

−1)

for any x ∈ P . Also,
C ◦D =

⋃
{c◦d : c ∈ C, d ∈ D}

and
C−1 = {c−1 : c ∈ C}.

We denote A•B by AB for short. Then for A,B ∈ IP , we have
(AB)−1 = B−1A−1 and (A−1)−1 = A.

The following definition of a fuzzy topological polygroup was given in [1].

Definition 3.1. 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:

(1) 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-neighborhoods U of xλ and V of yλ such that

U •V ≤ W.
(2) 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.

Now, we give another definition of a fuzzy topological polygroup.

Definition 3.2. 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:

(1′) For all x,y ∈ P and any fuzzy (open) neighborhood W of any fuzzy point zλ of x ◦ y, there
are fuzzy (open) neighborhoods U of xλ and V of yλ such that

U •V ≤ W.
(2′) For all x ∈ P and any fuzzy (open) neighborhood V of x−1λ , there exists a fuzzy (open)

neighborhood U of xλ such that
U−1 ≤ V.

Proposition 3.3. For a same polygroup P and a same fuzzy topology T on P the conditions (2) and
(2′) are equivalent.

Proposition 3.4. For a same polygroup P and a same fuzzy topology T on P we have:
(a) If T is a finite set, then (1) implies (1′).
(b) (1′) implies (1).

Proof. (a) Let W be any fuzzy neighborhood of any fuzzy point zλ of x ◦ y. If µW (z) > λ, then W
is a fuzzy Q-neighborhood of fuzzy point z1−λ. By (1), there exist fuzzy Q-neighborhoods U of x1−λ
and V of b1−λ such that UV ≤ W . Now, it is clear that U and V are fuzzy neighborhoods of xλ and
yλ respectively and the assertion follows from this. Let z ∈ x◦y and µW (z) = λ, choose a decreasing
sequence {εi}∞i=1 of real numbers such that 0 < εi < λ (i = 1, 2, . . . ) and lim

i→∞
εi = 0.

Put λi = λ − εi. Then, lim
i→∞

λi = λ. Now, for any λi, W is a fuzzy neighborhood of zλ and



CATEGORY OF FUZZY TOPOLOGICAL POLYGROUPS 127

µW (z) > λi. Hence, there exist fuzzy open neighborhoods Ui of xλi and Vi of yλi such that UiVi ≤ W .
We assume that U =

n⋃
i=1

Ui and V =
n⋃
i=1

Vi. It is easy to verify that U and V are fuzzy open

neighborhoods of xλ and yλ respectively. Since T is a finite set, we can choose a subsequence {Uil}
∞
l=1

of the sequence {Ui}∞i=1, such that each Uil is a fuzzy neighborhood of xλ. Clearly,
∞⋃
l=1

Vil is a fuzzy

open neighborhood of yλ. It follows from the assumption that there must exist an ik such that Vik is
a fuzzy neighborhood of yλ. Now, Uik and Vik are fuzzy neighborhoods of xλ and yλ respectively and
UikVik ≤ W .

(b) Let x,y ∈ P and W be a fuzzy Q-neighborhood of any fuzzy point zλ of x ◦ y. Choose a
λ1 ∈ (0, 1) such that 1 − λ < λ1 < µW (z). Then W is a fuzzy neighborhood of fuzzy point zλ1 . By
(1′) there exist fuzzy neighborhoods U of xλ1 and V of yλ1 such that UV ≤ W . Now, U and V are
fuzzy Q-neighborhoods of xλ and yλ respectively and UV ≤ W . �

Let FC(Y,Z) be the set of all fuzzy continuous functions from a fuzzy topological space Y into
a fuzzy topological space Z. In this section, we investigate FC(Y,Z) in presence of various known
topologies and fuzzy topologies.

Theorem 3.5. [2] Let (Y,TY ) be an FTS, (Z,◦,TZ) an FTP, and f,g ∈ FC(Y,Z). Then, the maps
f ∗g and f−1 from the fuzzy topological space Y into the fuzzy topological space Z with the types,

(f ∗g)(y) = f(y) ◦g(y)
and

f−1(y) = (f(y))−1

for every y ∈ Y , are fuzzy continuous.

Theorem 3.6. [2] Let (Y,TY ) be a fully stratified fuzzy topological space and (Z,◦,TZ) a fuzzy topo-
logical polygroup. Then, (FC(Y,Z),∗,e′,−1 ) is a polygroup.

Theorem 3.7. Let (FC(Y,Z),∗) be the polygroup of fuzzy continuous functions from a fully stratified
fuzzy topological space (Y,TY ) to a fuzzy topological polygroup (Z,◦,TZ).

(1) If Z is commutative, then FC(Y,Z) is commutative.
(2) If Z contains identity element , then FC(Y,Z) contains identity element.

Proof. (1) For all f,g ∈ FC(Y,Z) and y ∈ Y ,
(f ∗g)(y) = f(y) ◦g(y) = g(y) ◦f(y) = (g ∗f)(y).

(2) For all f ∈ FC(Y,Z) and y ∈ Y ,
(f ∗e′)(y) = f(y) ◦e′(y) = f(y) ◦e = f(y) = e◦f(y) = e′(y) ◦f(y) = (e′ ∗f)(y).

�

Theorem 3.8. [2] Let (Y,TY ) be a fully stratified fuzzy topological space, (Z,◦,TZ) a fuzzy topological
polygroup, and Z1 ∈ IZ a fuzzy polygroup. Then, the fuzzy set µ ∈ IFC(Y,Z) for which µ(f) =∧
y∈Y

Z1(f(y)), f ∈ FC(Y,Z) is a fuzzy polygroup.

Definition 3.9. [14] Let U be a fuzzy open set on an FTS Z and yλ, λ ∈ (0, 1] be a fuzzy point on
an FTS Y . By [yλ,U] we denote the subset of FC(Y,Z) where [yλ,U] = {f ∈ FC(Y,Z) : f(yλ) ≤ U}.
The collection of all such [yλ,U] forms a subbase for some topology on FC(Y,Z), called fuzzy-point
fuzzy-open topology (fp-fo), denoted by T(fp−fo).

Definition 3.10. [16] 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.

Lemma 3.11. [15] Let P be a polygroup. Then, the hyperoperation ◦ : P ×P → ℘∗(P) is continuous
if and only if for every x,y ∈ P and W ∈T such that x◦y ⊆ W then there exist U,V ∈T such that
x ∈ U,y ∈ V and U ◦V ⊆ W .



128 ABBASIZADEH AND DAVVAZ

Theorem 3.12. Let (Y,TY ) be a fully stratified fuzzy topological space, (Z,◦,TZ) a fuzzy topological
polygroup. Then (FC(Y,Z),T(fp−fo)) is a topological polygroup.

Proof. It is clear that FC(Y,Z) is a polygroup and FC(Y,Z) is a topological space with respect to
T(fp−fo). We need to show that the mappings (f,g) 7→ f ∗ g and f 7→ f−1 are continuous. Suppose
that [yλ,U] be a subbasic open set in FC(Y,Z) such that f∗g ⊆ [yλ,U]. So, for any h ∈ f∗g ⊆ [yλ,U],
we have h(yλ) ≤ U, where h(yλ) ∈ (f(y))λ ◦ (g(y))λ. (Z,◦,T ) being an fuzzy topological polygroup ,
then there exist fuzzy open sets V,W of Z such that (f(y))λ ∈ V, (g(y))λ ∈ W and UW ≤ V .

On the other hand, f(yλ) = (f(y))λ ≤ V implies that f ∈ [yλ,V ] and similarly g ∈ [yλ,W]. Now,
we show that f ∗g is continuous. We need to show that

[yλ,V ] ∗ [yλ,W] ⊆ [yλ,U].

Let ξ ∈ [yλ,V ] ∗ [yλ,W]. Then there exist η ∈ [yλ,V ] and ψ ∈ [yλ,W], such that ξ = η ∗ψ.
Since η ∈ [yλ,V ] and ψ ∈ [yλ,W], then we have η(yλ) ≤ V and ψ(yλ) ≤ W . So, (η ∗ ψ)(yλ) ≤

V ∗W ≤ U. Therefore, ξ(yλ) ≤ U and ξ ∈ [yλ,U].
Finally we show that f−1 is continuous. For any subbasic open set [yλ,U] containing f

−1, we get
(f−1)(yλ) ≤ U, so (f−1(y))λ ≤ U and (f(y))λ ≤ U−1. Since U is fuzzy open if and only if U−1 is
fuzzy open and f(yλ) = (f(y))λ, we have f ∈ [yλ,U−1]. Now, we show that [yλ,U]−1 ≤ [yλ,U−1].

Let ψ ∈ [yλ,U]−1. Then, there is some η ∈ [yλ,U] such that ψ = η−1. Since η ∈ [yλ,U] and
η(yλ) ≤ U, so (η−1)−1(yλ) ≤ U, then ψ−1(yλ) ≤ U and ψ(yλ) ≤ U−1, ψ ∈ [yλ,U−1], as desired. �

Definition 3.13. [14, 18] Let Y and Z be two fixed fuzzy topological spaces, U ∈ IZ a fuzzy open
set of Z, and y ∈ Y . Then by yU ∈ IFC(Y,Z) we denote the fuzzy set for which

yU (f) = U(f(y)), for every f ∈ FC(Y,Z).

The fuzzy point open topology TFP on FC(Y,Z) generated by fuzzy sets of the form yU , where y ∈ Y
and U ∈ IZ is a fuzzy open set of Z.

For each fuzzy compact set K in Y and each fuzzy open set U in Z, a fuzzy set KU on FC(Y,Z)
is given by

KU (g) =
∧

x∈supp(K)

U(g(x)).

The collection of all such KU forms a subbase for some fuzzy topology on FC(Y,Z), called fuzzy
compact open topology and it is denoted by ∆co.

Theorem 3.14. [2] Let (Y,TY ) be a fully stratified fuzzy topological space and (Z,◦,TZ) a fuzzy
topological polygroup. Then the triad (FC(Y,Z),∗,TFP ) is a fuzzy topological polygroup.

Theorem 3.15. Let (Y,TY ) be a fully stratified fuzzy topological space and (Z,◦,TZ) a fuzzy topological
polygroup. Then FC(Y,Z) endowed with fuzzy compact-open topology is a fuzzy topological polygroup.

Proof. Clearly, by Theorem 3.6 and Definition 3.13, (FC(Y,Z),∗) is a polygroup and (FC(Y,Z), ∆co)
is a fuzzy topological space. Now, we show that (FC(Y,Z),∗, ∆co) satisfies the conditions (1) and (2)
in Definition 3.1.

(1) Let KU be a fuzzy open subbasic Q-neighborhood of any fuzzy point hλ of f ∗g. We show that
there exist fuzzy open subbasic Q-neighborhoods KV and KW of fλ and gλ, respectively such that

KV •KW ≤ KU.

Since hλqKU , it follows that λ + KU (h) > 1 and λ +
∧

y∈suppK
U(h(y)) > 1. So, for all y ∈ suppK,

h(y)λqU. That is, the fuzzy set U is a fuzzy open Q-neighborhood of h(y)λ.
Now, since (Z,◦,TZ) is a fuzzy topological polygroup, it follows that there exist fuzzy open Q-

neighborhoods V and W of f(y)λ and g(y)λ such that V W ≤ U.
We consider the fuzzy sets KV and KW . Since V is a fuzzy open Q-neighborhood of f(y)λ, then

for all y ∈ suppK, V (f(y)) + λ > 1 and

KV (f) + λ =
∧

x∈suppK
V (f(y)) + λ > 1.



CATEGORY OF FUZZY TOPOLOGICAL POLYGROUPS 129

Hence, KV (f) + λ > 1 and fλqKV . Similarly, it can be proved that gλqKW .
We prove KV •KW ≤ KU . Let f ∈ FC(Y,Z). Then, we have

(KV •KW )(f) =
∨

f∈f1∗f2
[KV (f1) ∧KW (f2)]

=
∨

f∈f1∗f2
[(

∧
x∈suppK

V (f1(x))) ∧ (
∧

x∈suppK
W(f2(x)))]

=
∨

f∈f1∗f2
[
∧

x∈suppK
[V (f1(x)) ∧W(f2(x))]]

≤
∧

x∈suppK
[
∨

f∈f1∗f2
[V (f1(x)) ∧W(f2(x))]]

≤
∧

x∈suppK
[

∨
f(x)∈z1◦z2

[V (z1) ∧W(z2)]]

=
∧

x∈suppK
[(V •W)(f(x))]

≤
∧

x∈suppK
U(f(x))

= KU (f).

Now, the condition (1) in Definition 3.1 follows.
(2) Let f ∈ FC(Y,Z) and KU be a fuzzy open Q-neighborhood of f−1λ . We show that there

exists fuzzy open Q-neighborhood KV of fλ such that K
−1
V ≤ KU . Since f

−1
λ qKU , it follows that

λ + U(f−1(y)) > 1. As (Z,◦,TZ) is a fuzzy topological polygroup then, there exists fuzzy open Q-
neighborhood V of fλ such that V

−1 ≤ U. We prove that K−1V ≤ KU . Let f ∈ FC(Y,Z). Then we
have

K−1V (f) = KV (f
−1)

=
∧

x∈suppK
V (f−1(x))

=
∧

x∈suppK
V −1(f(x))

≤
∧

x∈suppK
U(f(x))

= KU (f).

Hence, K−1V ≤ KU . Now, the condition (2) in Definition 3.1 follows. Therefore, (FC(Y,Z),∗, ∆co) is
a fuzzy topological polygroup. �

4. Fuzzy topological polygroups induced by strong homomorphisms

In this section, we show how to strong homomorphisms induce fuzzy topological polygroup structure
on polygroups.

Definition 4.1. [10] A collection B of fuzzy neighborhoods of xλ, for 0 < λ ≤ 1, is called a fundamental
system of fuzzy neighborhoods of xλ if and only if for any fuzzy neighborhood V of xλ, there exists
U ∈ B such that xλ ≤ U ≤ V .

Definition 4.2. [10] A collection Ω of fuzzy sets in an FTS X is called a prefilterbase on X if 0X /∈ Ω
and for all A,B ∈ Ω, then there exists C ∈ Ω such that C ≤ A∩B.

Proposition 4.3. Let P be a polygroup and A,B,C ∈ IP . Then the following are hold:
(1) If A ≤ B then AC ≤ BC and CA ≤ CB.
(2) If AC = BC for any C ∈ IP , then A = B.
(3) (AB)C = A(BC).
(4) If A ≤ B then A−1 ≤ B−1.

Proof. It is straightforward. �

Theorem 4.4. Let (P,T ) be a fuzzy topological polygroup. Then the mapping φ : P → P−1, x 7→ x−1
is homeomorphic mapping.

Theorem 4.5. Let (P,T ) be a fuzzy topological polygroup. Then, V is fuzzy open if and only if V −1
is fuzzy open.



130 ABBASIZADEH AND DAVVAZ

Proof. For all x ∈ P ,
φ−1(V )(x) = V (φ(x)) = V (x−1) = V −1(x),

φ is fuzzy continuous and V is fuzzy open, so V −1 = φ−1(V ) is fuzzy open.
Converse follows similarly. �

Corollary 4.6. Let (P,T ) be a fuzzy topological polygroup. Then, for each λ with 0 < λ ≤ 1 and
x ∈ P, V is a fuzzy neighborhood of eλ if and only if V −1 is a fuzzy neighborhood of eλ.

Proof. The proof follows from Theorem 4.4 and the fact that eλ ≤ V if and only if eλ ≤ V −1. �

Definition 4.7. A fuzzy open set U of a fuzzy topological polygroup P is called a symmetric neigh-
borhood if U−1 = U.

Theorem 4.8. Every fuzzy topological polygroup has a fuzzy open fundamental system of eλ containing
a symmetric fuzzy open fundamental system of eλ.

Proof. Suppose that B is a fuzzy open fundamental system of eλ. Then, for every U ∈ B put V =
U ∩U−1. So, V = V −1 and V ≤ U. �

Theorem 4.9. If B is a fundamental system of fuzzy neighborhoods of eλ, for 0 < λ ≤ 1, then
D = {U ∩U−1 : U ∈ B} is also a fundamental system of fuzzy neighborhoods of eλ.

Proof. It is obvious. �

Proposition 4.10. Let (P,T ) be a fuzzy topological polygroup. Then, the family B = {Ã ∈ FS(P∗(P)) |
A ∈T}, where µÃ(X) =

∨
x∈X

µA(x), is a base for a fuzzy topology T ∗ on P∗(P).

Proof. B is a base for a 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).

(2) Since 1 ∈T , it follows that µ1̃(X) = 1, for any X ∈P
∗(P) and thus⋃̃

A∈B
= 1.

�

Lemma 4.11. Let U be a fuzzy open subset of a fuzzy topological polygroup P . Then, aλU and Uaλ
are fuzzy open subsets of P for every a ∈ P .

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

(a−1φ
−1(Ũ))(z) = Ũ(a−1φ(z)) = Ũ(a

−1 ◦z)
=

∨
t∈a−1◦z

U(t) =
∨

z∈a◦t
U(t)

= aλU(z).

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

Similarly, we can prove that Uaλ is fuzzy open. �

Lemma 4.12. Let (P,T ) be a fuzzy topological polygroup and B be a fuzzy open fundamental system
of fuzzy neighborhood of eλ. Then, the families {xλU} and {Uxλ}, are fuzzy open fundamental system
of fuzzy neighborhood of xλ.



CATEGORY OF FUZZY TOPOLOGICAL POLYGROUPS 131

Proof. Suppose that W is a fuzzy open subset of P and xλ ≤ W . Since

(xλU)(x) =
∨

x∈x1◦x2
[xλ(x1) ∧U(x2)]

=
∨

x∈x◦x2
[λ∧U(x2)]

≥ λ∧U(e)
= λ,

we conclude that xλ ≤ xλU. Since eλ ≤ x−1λ W , it follows that there exists U ∈ B such that eλ ≤ U ≤
x−1λ W . So, xλU ≤ W . Thus, W is a union of fuzzy open subsets xλU. Therefore, {xλU} is a fuzzy
open fundamental system for P . Similarly, the family {Uxλ} is a fuzzy open fundamental system for
P . �

In the next theorem we characterize a fuzzy topological polygroup via the fundamental system of
fuzzy neighborhoods of eλ.

Theorem 4.13. If P is a fuzzy topological polygroup, then there exists a fundamental system of fuzzy
neighborhoods B of eλ (0 < λ ≤ 1), such that the following conditions hold:

(1) Each member of B is symmetric.
(2) For all U ∈ B, there exists V ∈ B such that V •V ≤ U.
(3) For all U ∈ B, there exists V ∈ B such that V −1 ≤ U.
(4) For all U ∈ B and xλ ≤ U there exists V ∈ B such that xλV ≤ U.

Conversely, given a polygroup P and a prefilterbase B of eλ satisfying the conditions (1)-(4), there
exists a unique fuzzy topology T on P such that (P,T ) forms a fuzzy topological polygroup such that B
forms a fundamental system of fuzzy neighborhoods of eλ.

Proof. (1) Let P be a fuzzy topological polygroup. Consider any fundamental system D of fuzzy
neighborhoods of eλ (viz. consider the fuzzy open sets containing eλ). Then, we get a fundamental
system of fuzzy neighborhoods B of eλ such that each member of B is symmetric.

(2) For any U ∈ B, since P is an FTP, there exist V1,V2 ∈ B such that V1V2 ≤ U. Let V = V1 ∩V2.
So, V V ≤ V1V2 ≤ U.

(3) For any U ∈ B, xλ ≤ P , since P is an FTP, there exists V ∈ B such that V −1 ≤ U.
(4) Let U ∈ B and xλ ≤ U. As xλ = (xe)λ = xλeλ, by fuzzy topological polygroup of P , there exist

fuzzy neighborhoods W1,V1 of xλ and eλ respectively such that W1V1 ≤ U. Since

(xλV1)(z) =
∨

z∈z1◦z2

[xλ(z1) ∧V1(z2)],

it follows that

(xλV1)(z) =

{ ∨
[λ∧V1(z2)] if z ∈ x◦z2,

0 if x 6= z1.
As W1(x) ≥ λ, (W1V1)(z) ≥ (xλV1)(z). Hence, xλV1 ≤ W1V1 ≤ U.

Conversely, suppose B is a prefilterbase at eλ, such that it satisfies (1)-(4). For each x ∈ P , it is
easy to see that Bx = {xλU : U ∈ B} forms a prefilterbase at xλ. Then ∪Bx generates a unique fuzzy
topology on P with B as a fundamental system of fuzzy neighborhoods of eλ. In order to show that P
is a fuzzy topological polygroup, we have to show P satisfies the conditions (1) and (2) in Definition
3.1.

Suppose zλU is a fuzzy open neighborhood of zλ, where U ∈ B and zλ ∈ x ◦ y. Then by (2),
there are V,W ∈ B such that V W ≤ U. It is clear that (xλV )(yλW) = (xy)λV W and consequently
(xλV )(yλW) ≤ zλU.

Suppose xλU is a fuzzy open neighborhood of xλU, where U ∈ B. By the condition (3), there is
V ∈ B such that V −1 ≤ U. It follows from the symmetric of the members of B. �

Definition 4.14. [9] Let 〈P1, ·,e1,−1 〉 and 〈P2,∗,e2,−I 〉 be two polygroups. Let f be a mapping from
P1 to P2 such that f(e1) = e2. Then, f is called a strong homomorphism or a good homomorphism if

f(x ·y) = f(x) ∗f(y), for all x,y ∈ P1.



132 ABBASIZADEH AND DAVVAZ

Since P1 is a polygroup, e1 ∈ a ◦1 a−1 for all a ∈ P1, it follows that f(e1) ∈ f(a) ◦2 f(a−1) or
e2 ∈ f(a) ◦2 f(a−1) which implies f(a−1) ∈ f(a)−1 ◦2 e2. Therefore, f(a−1) = f(a)−1 for all a ∈ P1.
Moreover, if f is a fuzzy topological homomorphism from P1 into P2, then the kernel of f is the set
kerf = {x ∈ P1|f(x) = e2}. It is trivial that kerf is a subpolygroup of P1 but in general is not normal
in P1. As in polygroup, if f is a fuzzy topological homomorphism from P1 into P2, then f is injective
if and only if kerf = {e1}.

Definition 4.15. A fuzzy topology that makes a polygroup FTP is called a fuzzy topology compatible
with the polygroup structure.

Theorem 4.16. Let (P,T ) be a fuzzy topological polygroup. If f : Q → P is a strong homomorphism
from any polygroup Q to P then, f induces a unique compatible fuzzy topology on Q that makes f fuzzy
continuous.

Proof. Let B be a fundamental system of fuzzy neighborhoods of eλ in P . Then it is enough to show
that f−1(B) determines a unique fuzzy topology on Q such that f−1(B) forms a fundamental system
of fuzzy neighborhoods of eλ in Q. It is clear that f

−1(B) is a prefilterbase at eλ in Q. In view of
Theorem 4.13, it is now to verify that f−1(B) satisfies the conditions (1) − (4) of Theorem 4.13.

(1) Any element of f−1(B) is of the form f−1(V ), for some V ∈ B. Now, for all x ∈ P

(f−1(V ))−1(x) = f−1(V )(x−1) = V (f(x−1)) = V (f−1(x))
= V −1(f(x)) = V (f(x)) = f−1(V )(x).

Hence, (f−1(V ))−1 = f−1(V ), showing that each member of f−1(B) is symmetric.
(2) Let f−1(U) ∈ f−1(B), for some U ∈ B. Then as U ∈ B, there exists V ∈ B, such that V V ≤ U.

For any z ∈ Q,

(f−1(V )f−1(V ))(z) =
∨

z∈z1◦z2
[(f−1(V ))(z1) ∧ (f−1(V ))(z2)]

=
∨

z∈z1◦z2
[V (f(z1)) ∧V (f(z2))]

=
∨

f(z)∈f(z1)◦f(z2)
[V (f(z1)) ∧V (f(z2))]

= (V V )(f(z)) ≤ U(f(z)) = f−1(U)(z).

(3) Let f−1(U) ∈ f−1(B), for some U ∈ B. Then as U ∈ B, there exists V ∈ B, such that V −1 ≤ U.
For any z ∈ Q,

f−1(V )(z) = V (f−1(z)) = V −1(f(z))
≤ U(f(z)) = U−1(f(z))
= f−1(U−1)(z).

(4) Let a ∈ Q and f−1(U) ∈ f−1(B). Then f(a) ∈ P and U ∈ B so that there exists V ∈ B such
that f(a)λV ≤ U,

(aλf
−1(V ))(z) =

∨
z∈z1◦z2

[aλ(z1) ∧f−1(V )(z2)].

This implies that

(aλf
−1(V ))(z) =

{ ∨
[λ∧f−1(V )(z2)] if z ∈ a◦z2,

0 if x 6= z1.
≤

∨
z∈a◦z2

[λ∧V (f(z2))]

=
∨

f(z)∈f(a)◦f(z2)
[λ∧V (f(z2))]

=
∨

f(z)∈f(a)◦x
[λ∧V (f(z2))]

= (f(a)λV )(f(z))
≤ U(f(z)) = f−1(U)(z).

�

Corollary 4.17. Any subpolygroup of a fuzzy topological polygroup is a fuzzy topological polygroup.



CATEGORY OF FUZZY TOPOLOGICAL POLYGROUPS 133

Proof. Let (P,T ) be a fuzzy topological polygroup and K be a subpolygroup of P . If B is a fundamental
system of fuzzy neighborhoods of eλ in P , and i : K → P given by i(x) = x, is the inclusion
homomorphism, then by the Theorem 4.16 f−1(B) determines a unique compatible fuzzy topology on
K such that f−1(B) forms a fundamental system of fuzzy neighborhoods of eλ in K. �

Corollary 4.18. Let P be an FTP and f : P → Q is a strong homomorphism from P onto Q with
kernel K such that K is a normal subpolygroup of P . Then, f̄ : P/N → Q induces a compatible fuzzy
topology on P/N that makes f̄ fuzzy continuous.

Proof. It is straightforward. �

Theorem 4.19. Let (P,T ) be an FTP and f : P → Q is a strong homomorphism from P onto
any polygroup Q. Then, P induces a fuzzy topology compatible with the polygroup structure on Q that
makes f fuzzy continuous.

Proof. Let B be a fundamental system of fuzzy neighborhoods of eλ in Q. In view of Theorem 4.13, it
is enough to show that f(B) is a fundamental system of fuzzy neighborhoods of eλ in Q.

(1) For any U ∈ B, U = U−1 and so, f(U) = f(U−1). Now, for all z ∈ Q,
f(U−1)(z) =

∨
f(t)=z

U−1(t) =
∨

f(t)=z

U(t−1)

=
∨

f(t−1)=z−1
U(t−1) = f(U)(z−1)

= f−1(U)(z).

Consequently, f(U) = f−1(U).
(2) If f(U) ∈ f(B), then U ∈ B and so, there exists V ∈ B such that V V ≤ U. For any z ∈ Q,

(f(V )f(V ))(z) =
∨

z∈z1◦z2
[(f(V ))(z1) ∧ (f(V ))(z2)]

=
∨

z∈z1◦z2
[(

∨
f(y1)=z1

V (y1)) ∧ (
∨

f(y2)=z2

V (y2))]

=
∨

z∈z1◦z2
[
∨

f(y1)=z1
f(y2)=z2

[V (y1) ∧V (y2)]]

≤
∨

f(t)=z

(V V )(t) = f(V V )(z) ≤ f(U)(z).

(3) If f(U) ∈ f(B), then U ∈ B and so, there exists V ∈ B, such that V −1 ≤ U. For any z ∈ Q,
(f(V ))−1(z) = (

∨
f(t)=z

V (t))−1 =
∨

f(t)=z

V −1(t)

≤
∨

f(t)=z

U(t) = f(U)(z).

(4) Let b ∈ Q and f(U) ∈ f(B). Then there exists a ∈ P with f(a) = b. So, there exists V ∈ B
such that aλV ≤ U. It is to show that bλf(V ) ≤ f(U). For any z ∈ Q, it is easy to see that

bλf(V )(z) = f(aλ)f(V )(z) ≤ f(aλV )(z) ≤ f(U)(z).
Hence, all the condition (1)-(4) are satisfied proving f(B) to be a fundamental system of fuzzy neighbor-
hoods of eλ in Q. Let U be any fuzzy neighborhood of eλ in Q, by definition of fundamental system of
fuzzy neighborhoods, there exists some B ∈ B such that eλ ≤ f(B) ≤ U. As B ≤ f−1(f(B)) ≤ f−1(U)
and eλ ∈ B it follows that f−1(U) is a fuzzy neighborhood of eλ in Q. Therefore, f is fuzzy continu-
ous. �

Corollary 4.20. Let P be a polygroup and N be a normal subpolygroup of P . The polygroup
epimorphism π : P → P/N given by π(x) = xN induces a fuzzy topology compatible with the
polygroup P/N that makes π fuzzy continuous.

Proof. The proof follows from Corollary 4.18 and Theorem 4.19. �

Theorem 4.21. Let P be an FTP. If Q is an FTP induced from a strong homomorphism f : P → Q
and N = kerf such that N is a normal subpolygroup of P then, the fuzzy topology compatible with P/N
induced from f̄ : P/N → Q and the fuzzy topology compatible with P/N induced from k : P → P/N
are same.



134 ABBASIZADEH AND DAVVAZ

Proof. Let B be a fundamental system of fuzzy neighborhoods of eλ in P . It follows from the Theorem
4.19 that {π(V ) : V ∈ B} is a fundamental system of fuzzy neighborhoods of eλ for the compatible
fuzzy topology on P/N induced by π and {f̄−1(f(V )) : V ∈ B} is a fundamental system of fuzzy
neighborhoods of eλ for the compatible fuzzy topology on P/N induced by f̄. Since, f̄π = f, it follows
that

f̄−1(f(V ))(xN) = f(V )(f̄(xN)) = f(V )(f̄π)(x) = f(V )(f(x))
=

∨
f(y)=f(x)

V (y) =
∨

yx−1∈N
V (y) =

∨
yN=xN

V (y)

=
∨

π(y)=xN

V (y) = π(V )(xN).

Hence, both the fundamental systems are identical leading to the same compatible topology on P/N.
�

5. Fuzzy topological polygroups and the category CFTP

In this section we introduce the category CFTP , which the objects in this category are fuzzy topo-
logical polygroups, morphisms are fuzzy topological homomorphisms and compositions is the usual
composition of functions. Also, we show that the category CTP of topological polygroups and contin-
uous topological homomorphisms form a full subcategory of CFTP .

Theorem 5.1. In a fuzzy topological polygroup P , V is a fuzzy Q-neighborhood of eλ if and only if
V −1 is a fuzzy Q-neighborhood of eλ.

Proof. Let V be a fuzzy Q-neighborhood of eλ. Then there exists fuzzy open set A such that eλqA ≤ V ,
that is, A(e) + λ > 1 and A ≤ V . For all x ∈ P , A(x−1) ≤ V (x−1), so A−1(x) ≤ V −1(x) and
A−1 ≤ V −1. Now, A−1(e) + eλ(e) = A−1(e) + λ > 1. Hence, eλqA−1 and A−1 ≤ V −1. Therefore, V −1
is a fuzzy Q-neighborhood of eλ.

Conversely, let V −1 be a fuzzy Q-neighborhood of eλ. Then there exist fuzzy open set A such that
eλqA ≤ V −1. As above, A−1 ≤ V and eλqA−1. That is, V is a fuzzy Q-neighborhood of eλ. �

Proposition 5.2. [1] Let (P1,T1) and (P2,T2) be two fuzzy topological polygroups and f : P1 → P2
be a homomorphism. Then, f is fuzzy continuous if and only if f is continuous at eλ (here e is the
unit of P1) for any λ ∈ (0, 1].

Definition 5.3. [1] Let 〈P1,◦1,e1,−1 ,T1〉 and 〈P2,◦2,e2,−I ,T2〉 be fuzzy topological polygroups. A
mapping f from P1 into P2 is said to be a fuzzy topological homomorphism if for all a,b ∈ P1:

(1) f(e1) = e2.
(2) f(a◦1 b) = f(a) ◦2 f(b).
(3) f is fuzzy continuous mapping of FTS (P1,T1) into FTS (P2,T2).

Theorem 5.4. The collection of all fuzzy topological polygroups and fuzzy topological homomorphisms
form a category.

Proof. Consider the collection of all FTP as objects, morphisms are fuzzy topological homomorphisms
and compositions is the usual composition of functions. In checking that CFTP is a category, one must
note that for each object P , i : P → P given by i(x) = x is the identity morphism. Consequently, it
forms a category. �

Remark 1. It is well known that corresponding to any topological space (X,T ), one can obtain the
characteristic fuzzy topological space (X,Tf ).

Theorem 5.5. If (P,T ) is a topological polygroup, then (P,Tf ) is a fuzzy topological polygroup.

Proof. Clearly (P,Tf ) is a fuzzy topological space. Now, we show that (P,Tf ) satisfies the conditions
(1) and (2) in Definition 3.1.

(1) Let x,y ∈ P and W be a fuzzy open Q-neighborhood of any fuzzy point zλ of x◦y. We show
that there exist fuzzy open Q-neighborhood U and V of xλ and yλ respectively, such that UV ≤ W .

Let W be a fuzzy open Q-neighborhood on (P,Tf ) with zλqW. Then W = χA for some A ∈ T .
Hence,

zλq χA ⇒ χA(z) + λ > 1 ⇒ z ∈ A.



CATEGORY OF FUZZY TOPOLOGICAL POLYGROUPS 135

Since (P,T ) is a topological polygroup, there exist open sets B,C ∈ T such that x ∈ B, y ∈ C and
BC ⊆ A. Then xλq χB and yλq χC, where χB,χC ∈ Tf . In order to complete the proof, we show
χB χC ≤ χA = W . For all t ∈ P ,

(χB χC)(t) =
∨

t∈t1◦t2
(χB(t1) ∧χC(t2))

=

{
1 if t1 ∈ B,t2 ∈ C,
0 otherwise.

=

{
1 if t ∈ BC,
0 otherwise.

= χBC(t)
≤ χA(t).

So, the condition (1) in Definition 3.1 follows.
(2) Let x ∈ P and U be a fuzzy open Q-neighborhood of x−1λ . We show that there exists fuzzy open

Q-neighborhood V of xλ such that V
−1 ≤ U.

Since x−1qU, it follows that U = χA for some A ∈T . Hence,
x−1q χA ⇒ χA(x−1) + λ > 1 ⇒ x−1 ∈ A.

Since (P,T ) is a topological polygroup, there exists open set B ∈ T such that x ∈ B and B−1 ⊆ A.
Then χB(x) + λ > 1, where χB ∈Tf . In order to complete the proof, we show χ−1B ≤ χA = U. For all
t ∈ P ,

χ−1B (t) =

{
1 if t ∈ B−1,
0 if t /∈ B−1.

≤
{

1 if t ∈ A,
0 if t /∈ A.

= χA(t).

Now, the condition (2) in Definition 3.1 follows. Therefore, (P,Tf ) is a fuzzy topological polygroup. �

Theorem 5.6. If f is a continuous topological homomorphism from a topological polygroup (P1,T1)
to a topological polygroup (P2,T2) then f : (P1,T1f ) → (P2,T2f ) is a fuzzy topological homomorphism
between the corresponding fuzzy topological polygroup.

Proof. The proof is straightforward. �

Theorem 5.7. If CTP is the category of topological polygroups and continuous topological homomor-
phisms, then CTP is a full subcategory of CFTP .

Proof. We know that any object of CTP can be viewed as an object of CFTP and any morphism
between two objects of CTP is a morphism between the corresponding objects of CFTP . Hence,
CTP is a subcategory of CFTP . Let the inclusion functor i : CTP → CFTP that sends (P,T ) to its
characteristic fuzzy topological space (P,Tf ) and f : (P,T ) → (Q,σ) to f∗ : (P,Tf ) → (Q,σf ). To
show that the functor i is full. Let (P,T ) and (Q,σ) be two objects in CTP and f∗ : (P,Tf ) → (Q,σf )
a morphism in CFTP . If U ∈ σ then χU ∈ σf and so, f∗

−1
(χU ) = χf∗−1 (U) ∈Tf , which in turn gives

f∗
−1

(U) ∈ T . Hence, there exist f∗ : (P,T ) → (Q,σ)a morphism in CTP such that i(f∗) = f∗, that
is, i is full. Consequently, CTP is a full subcategory of CFTP . �

Theorem 5.8. Let (P,T ) be a fuzzy topological polygroup. Then, for all 0 ≤ λ < 1, (P,iλ(T )) is a
topological polygroup.

Proof. It is clear that (P,iλ(T )) is a topological space. We need to show that the mappings (x,y) 7→ x◦y
and x 7→ x−1 are continuous.

Let x,y ∈ P and W be any open set in (P,iλ(T )) such that x◦y ⊆ W . There exists a fuzzy open
set γ in (P,T ) such that γλ = W . So, for any zλ ∈ x ◦ y, we have zλ < γ. Since (P,T ) is a fuzzy
topological polygroup, there exist fuzzy open sets U and V such that xλ < U, yλ < V and UV ≤ γ.
Then x ∈ Uλ and y ∈ V λ. We shall show that UλV λ ⊆ W .

If r ∈ UλV λ, then r ∈ st where s ∈ Uλ and t ∈ V λ, that is, U(s) > λ and V (t) > λ. Now,,
(UV )(r) =

∨
r∈r1r2

[U(r1) ∧V (r2)] ≥ U(s) ∧V (t) > λ.



136 ABBASIZADEH AND DAVVAZ

So, γ(r) > λ, r ∈ γλ = W . Hence, UλV λ ⊆ W. This show that the mapping (x,y) 7→ x ◦ y is
continuous. Now, we prove that x 7→ x−1 is continuous. Let x ∈ P and V be an open set of (P,iλ(T ))
containing x−1. There is a fuzzy open set γ on (P,T ) such that γλ = V . So, x−1 ∈ γλ and we have
x−1λ < γ. Since (P,T ) is a fuzzy topological polygroup, there exists fuzzy open set U containing xλ
such that xλ < U and U

−1 ≤ γ. We shall show (U−1)λ ⊆ V .
Let t ∈ (U−1)λ, then γ(t) ≥ U−1(t) > λ. So, t ∈ γλ. Hence, (U−1)λ ⊆ V . �

Theorem 5.9. A function f : (X,T ) → (Y,σ) is fuzzy continuous if and only if f : (X,iλ(T )) →
(Y,iλ(σ)) is continuous for each 0 ≤ λ < 1, where (X,T ), (Y,σ) are fuzzy topological spaces.

Proof. The proof is straightforward. �

Theorem 5.10. A function f : (P1,T1) → (P2,T2) is fuzzy topological homomorphism if and only
if f : (P1, iλ(T1)) → (P2, iλ(T2)) is continuous topological homomorphism for each 0 ≤ λ < 1, where
(P1,T1) and (P2,T2) are fuzzy topological polygroups.

Proof. The proof follows from Theorem 5.9. �

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Department of Mathematics, Yazd University, Yazd, Iran

∗Corresponding author: davvaz@yazd.ac.ir