Studies on fuzzy topological polygroups

1 Introduction

The hyperstructure theory was born in 1934 when Marty introduced the notion of a hypergroup [13]. In 1979, Foster [9] introduced the concept of a fuzzy topological group. Ma and Yu [12] 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 have been established and investigated.

We recall some basic definitions and results to be used in what follows.

Let H be a non-empty set. A mapping ∘ : H × H ⟶ ℘ ^∗ (H) is called a hyperoperation, where ℘^∗ (H) is the family of all 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 it 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 [5, 13].

A special subclass of hypergroups is the class of polygroups. We recall the following definition from [4]. A polygroup is a system P = 〈P, ∘ , e, ^-1〉, where ∘ : P × P ⟶ ℘ ^∗ (P), e ∈ P, ^-1 is a unitary operation on P and the following axioms hold for all x, y, z ∈ P:

(x ∘ y) ∘ z =x ∘ (y ∘ z),

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 [1, 11].

2 Preliminaries

For the sake of convenience and completeness of our study, we recall some basic definition and results from [2, 16], which we use in what follows.

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 μ _A  : X ⟶ [0, 1] which associates with each point x ∈ X its grade or degree of membership μ _A  (x) ∈ [0, 1]. In other words, it is an element of I ^X . 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:

0 _ , 1 _ ∈ T.

Example 1. We give some examples of fuzzy topologies on a set X.

The family T={0 _,1 _} is called the indiscrete fuzzy topology on X.

Example 2. Consider the set of natural numbers ℕ and let A be a fuzzy set on ℕ defined as μ A ( x ) = { 0.7 if x ∈ 2 ℕ , 0.5 if x ∈ 2 ℕ - 1 . Then, T = { 0 _ , A , 1 _ } is a fuzzy topology on ℕ.

Example 3. Let A and B be two fuzzy sets on X = [-1, 1] defined as μ A ( x ) = { 0 if - 1 ≤ x ≤ 0 , x if 0 ≤ x ≤ 1 and μ B ( x ) = { 1 if - 1 ≤ x ≤ - 1 3 , - 3 x if - 1 3 ≤ x ≤ 0 , 0 if 0 ≤ x ≤ 1 . Then, T = { 0 _ , A , B , A ∨ B , 1 _ } is a fuzzy topology on X.

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 x _λ , where the point x is called its support.

The fuzzy point x _λ is said to be contained in a fuzzy set A, or to belong to A, denoted by x _λ  ∈ A, if and only if λ ≤ μ _A  (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 x _λ , if there exists a B ∈ T such that x _λ  ∈ B ≤ A. The family consisting of all neighborhoods of x _λ is called the system of neighborhood of fuzzy point x _λ .

A fuzzy point x _λ is said to be quasi-coincident with a fuzzy set A, denoted by x _λ qA, if μ _A  (x) + λ > 1. A is said to be quasi-coincident with B, denoted by AqB, if there exists x ∈ X such that μ _A  (x) + μ _B  (x) >1. If this is true, we also say that A and B are quasi-coincident in x.

A fuzzy set A in a fuzzy topological space ( X , T ) is said to be a Q-neighborhood of x _λ if there exists a B ∈ T such that x _λ qB ≤ A. The family consisting of all Q-neighborhoods of x _λ is called the system of Q-neighborhood of fuzzy point x _λ .

A fuzzy topological space ( X , T ) is called 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 all fuzzy set B ∈ U, the inverse image f - 1 [ B ] ∈ T. Conversely, f is fuzzy open if for all open fuzzy set A ∈ T, the image f [ A ] ∈ U (see [2]).

Let A be a fuzzy set in the fuzzy topological space ( X , T ). Then, the restricted fuzzy topology on A is the family T A of fuzzy subsets of A which are the intersection with A of T-open fuzzy sets in X. The pair ( A , T A ) is called a fuzzy subspace of ( X , T ). For all fuzzy set A ∩ U _j of T A, with U j ∈ T, we have μ_{U j ∩A} (x)=μ _{U j}  (x) ∧ μ _A  (x) (see [9]).

Let X and Y be two non-empty subsets, f: X ⟶ Y. Let 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 μ f [ A ] ( y ) = { ⋁ x ∈ f - 1 ( y ) μ A ( x ) if f - 1 ( y ) ≠ ∅ 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 μ_f^-1[B] (x) = μ _B   (f (x)) for all x ∈ X. Let f be a mapping from a FTS ( X , T ) into a FTS ( Y , U ). If for all fuzzy open Q-neighborhood U of f (x _λ ) = [f (x)]  _λ there exists a fuzzy open Q-neighborhood V of x _λ such that f (V) ≤ U, then we say that f is continuous in x _λ (see [15]).

Let f be a function from a fuzzy topological space ( X , T ) into a fuzzy topological space ( Y , U ). Then the following statements are equivalent (see [15]):

f is a fuzzy continuous mapping.

Let X be a group, ( X , T ) be a fuzzy topological space. ( X , T ) is called a fuzzy topological group or FTG for short, iff: (see [12])

For all a, b ∈ X and all fuzzy open Q-neighborhood W of fuzzy point (ab)  _λ , there are fuzzy open Q-neighborhoods U of a _λ and V of b _λ such that UV ≤ W.

3 Fuzzy topological polygroups

Let P = 〈P, ∘ , e, ^-1〉 be a polygroup, A, B ∈ I ^P and C, D ⊆ P.

We define A • B ∈ I ^P ,  A^-1 ∈ I ^P , C ∘ D ⊆ P and C^-1 ⊆ P as follows: ( A • B ) ( x ) = ⋁ x ∈ x 1 ∘ x 2 ( μ A ( x 1 ) ∧ μ B ( x 2 ) ) and μ A - 1 ( x ) = μ A ( x - 1 ) for all 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 ∈ I ^P , we have ( AB ) - 1 = B - 1 A - 1 and ( A - 1 ) - 1 = A .

Proposition 3.1. Let P be a polygroup and A, B, C ∈ I ^P . Then the following statements hold:

If A ≤ B then AC ≤ BC and CA ≤ CB.

Proof. It is straightforward. □

Proposition 3.2. Let P₁, P₂ be two polygroups and f : P₁ ⟶ P₂ be a strong homomorphism. Then for all A, B ∈ I ^{P ₁} , we have f ( AB ) = f ( A ) f ( B ) .

Proof. We have μ f ( AB ) ( y ) = ⋁ x ∈ f - 1 ( y ) μ AB ( x ) = ⋁ x ∈ f - 1 ( y ) { ⋁ x ∈ x 1 x 2 [ μ A ( x 1 ) ∧ μ B ( x 2 ) ] } = ⋁ y ∈ f ( x 1 ) f ( x 2 ) { μ A ( x 1 ) ∧ μ B ( x 2 ) } and μ f ( A ) f ( B ) ( y ) = ⋁ y ∈ y 1 y 2 { μ f ( A ) ( y 1 ) ∧ μ f ( B ) ( y 2 ) } = ⋁ y ∈ y 1 y 2 { ( ⋁ x 1 ∈ f - 1 ( y 1 ) μ A ( x 1 ) ) ∧ ( ⋁ x 2 ∈ f - 1 ( y 2 ) μ B ( x 2 ) ) } = ⋁ y ∈ f ( x 1 ) f ( x 2 ) { μ A ( x 1 ) ∧ μ B ( x 2 ) } . Therefore, the proof is completed. □

Definition 3.3. 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:

For all x, y ∈ P and all fuzzy open Q-neighborhood W of any fuzzy point z _λ of x ∘ y, there are fuzzy open Q-neighborhood U of x _λ and V of y _λ such that U • V ≤ W.

Evidently, every fuzzy topological group is a fuzzy topological polygroup. We give some other examples.

Example 4. Every polygroup with discrete or indiscrete fuzzy topology is a fuzzy topological polygroup.

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

Example 6. Let P = {e, x, y} with the hyperoperation defined as follows:

∘ e x y e e x y x x e y y y y { e , x }

P = 〈P, ∘ , e, ^-1〉 is a polygroup. It is clear that x^-1 = x,  y^-1 = y. Consider on P the fuzzy topology T = { 0 _ , A , 1 _ }, where μ 0 _ ( e ) = μ 0 _ ( x ) = μ 0 _ ( y ) = 0, μ _A  (e) =0.9,  μ _A  (x) =0.7,  μ _A  (y) =0.5, μ 1 _ ( e ) = μ 1 _ ( x ) = μ 1 _ ( y ) = 1. Then ( P , T ) is a fuzzy topological polygroup.

Let FC (Y, Z) be the set of all fuzzy continuous functions from a fuzzy topological space Y into a fuzzy topological space Z. Our aim in this section is to study the notion of a polygroup, fuzzy polygroup, topological polygroup and fuzzy topological polygroup on the (fuzzy) function space FC (Y, Z).

Theorem 3.4. Let

be an FTS,

be 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

( f ∗ g ) ( y ) = f ( y ) ∘ g ( y )

and

for all y ∈ Y, are fuzzy continuous.

Proof. Let y ∈ Y and y _λ be a fuzzy point. Let W be a fuzzy open Q-neighborhood of (f ∗ g) (y _λ ). Since (f ∗ g) (y) = f (y) ∘ g (y), it follows that W is a fuzzy open Q-neighborhood of any fuzzy point of f (y) ∘ g (y).

Since ( Z , ∘ , T Z ) is an FTP, there exist a fuzzy open Q-neighborhoods U and V of f (y)  _λ and g (y)  _λ respectively, such that UV ≤ W.

Now, since the maps f and g are fuzzy continuous, there exist a fuzzy open Q-neighborhoods U₁ and V₁ of the fuzzy point y _λ in Y such that f (U₁) ≤ U and g (V₁) ≤ V.

Clearly, the fuzzy set U₁ ∧ V₁ ∈ I ^Y is a fuzzy open Q-neighborhood of y _λ in Y. We prove that (f ∗ g) (U₁ ∧ V₁) ≤ W.

Indeed, let y_{1 λ 1}  ∈ U₁ ∧ V₁. We show that ( f ∗ g ) ( y 1 λ 1 ) = ( f ( y 1 ) ∘ g ( y 1 ) ) λ 1 = ( ( f ∗ g ) ( y 1 ) ) λ 1 ∈ W , i.e., for all z _{λ 1}  ∈ f (y₁) ∘ g (y₁), we check that z _{λ 1}  ∈ W.

We have y_{1 λ 1}  ∈ U₁ and y_{1 λ 1}  ∈ V₁. Therefore, f ( y 1 λ 1 ) = f ( y 1 ) λ 1 ∈ U and g ( y 1 λ 1 ) = g ( y 1 ) λ 1 ∈ V . Hence λ 1 ≤ U ( f ( y 1 ) ) and λ 1 ≤ V ( g ( y 1 ) ) . Also, for all z _{λ 1}  ∈ f (y₁) ∘ g (y₁), ( U • V ) ( z ) = ⋁ z ∈ z 1 ∘ z 2 ( U ( z 1 ) ∧ V ( z 2 ) ) ≥ min { U ( f ( y 1 ) ) , V ( g ( y 1 ) ) : z ∈ f ( y 1 ) g ( y 2 ) } ≥ λ 1 . Thus, we have λ₁ ≤ (U • V) (z) ≤ W (z) and so, for all z _{λ 1}  ∈ f (y) ∘ g (y) ,  z _{λ 1}  ∈ W. Hence, (f ∗ g) (U₁ ∧ V₁) ≤ W, and so the map f ∗ g is fuzzy continuous.

Finally, we prove that the map f^-1 is fuzzy continuous. Let y _λ be a fuzzy point of Y and W be a fuzzy open Q-neighborhood of f - 1 ( y λ ) = f - 1 ( y ) λ = ( f ( y ) ) λ - 1 in Z. Since ( Z , ∘ , T Z ) is an FTP, there exists a fuzzy open Q-neighborhood U of f (y)  _λ in Z such that U^-1 ≤ W.

Now, since the map f is fuzzy continuous in the fuzzy point y _λ in Y, there exists a fuzzy open Q-neighborhood U₁ of y _λ such that f (U₁) ≤ U.

For the fuzzy open Q-neighborhood U₁ of y _λ in Y, we have f^-1 (U₁) ≤ W.

Indeed, let y_{1 λ 1}  ∈ U₁. It is sufficient to check that f^-1 (y_{1 λ 1} ) ∈ W, that is λ₁ ≤ W (f^-1 (y₁)) = W ((f (y₁)) ^-1).

We have f (y_{1 λ 1} ) = f (y₁)  _{λ 1}  ∈ U. Thus λ 1 ≤ U ( f ( y 1 ) ) = U - 1 ( ( f ( y 1 ) ) - 1 ) ≤ W ( ( f ( y 1 ) ) - 1 ) and therefore, the map f^-1 is fuzzy continuous. □

Theorem 3.5. Let ( Y , T Y ) be a fully stratified fuzzy topological space, ( Z , ∘ , T Z ) an FTP, and e the identity element of the polygroup (Z, ∘). Then, the map e′ from the fuzzy topological space Y into the fuzzy topological space Z where e′ (y) = e for every y ∈ Y, is fuzzy continuous.

Proof. Let U ∈ T Z. We show that ( e ′ ) - 1 ( U ) ∈ T Y. We have ((e′) ^-1 (U)) (y) = U (e′ (y)) = U (e), for every y ∈ Y. Since the fuzzy space ( Y , T Y ) is fully stratified, it follows that ( e ′ ) - 1 ( U ) ∈ T Y. Thus, the map e′ is fuzzy continuous. □

Theorem 3.6. Let ( Y , T Y ) be a fully stratified fuzzy topological space and ( Z , ∘ , T Z ) a fuzzy topological polygroup. Then (FC (Y, Z) , ∗ , e′, ^-1) is a polygroup.

Proof. For all f, g, h ∈ FC (Y, Z), we have ( ( f ∗ g ) ∗ h ) ( y ) = ( f ∗ g ) ( y ) ∘ h ( y ) = ( f ( y ) ∘ g ( y ) ) ∘ h ( y ) = f ( y ) ∘ ( g ( y ) ∘ h ( y ) ) = f ( y ) ∘ ( ( g ∗ h ) ( y ) ) = ( f ∗ ( g ∗ h ) ) ( y ) . Therefore, ∗ is associative. It is easy to see that e′ is the unit element in FC (Y, Z), and f^-1 is the inverse of the element f. Now, we show that f ∈ g ∗ h implies that g ∈ f ∗ h^-1 and h ∈ g^-1 ∗ f.

We have f ∈ g ∗ h, and hence f (y) ∈ (g ∗ h) (y) = g (y) ∘ h (y). Therefore f (y) ∈ g (y) ∘ h (y). Since ( Z , ∘ , T Z ) is an FTP, it follows that g (y) ∈ f (y) ∘ h^-1 (y) = (f ∗ h^-1) (y), and so g ∈ f ∗ h^-1. Similarly, we obtain h ∈ g^-1 ∗ f. Therefore, (FC (Y, Z) , ∗ , e′, ^-1) is a polygroup. □

Definition 3.7. [17] Let P be a polygroup. A fuzzy subset μ of P is called a fuzzy subpolygroup if

min {μ (x) , μ (y)} ≤ μ (z), for all x, y ∈ P and for all z ∈ x ∘ y,

Theorem 3.8. Let ( Y , T Y ) be a fully stratified fuzzy topological space, ( Z , ∘ , T Z ) a fuzzy topological polygroup, and Z₁ ∈ I ^Z a fuzzy polygroup. Then the fuzzy set μ ∈ I^FC(Y,Z) for which μ (f) = ⋀ _y∈YZ₁ (f (y)) ,  f ∈ FC (Y, Z) is a fuzzy polygroup.

Proof. By Theorem 3.6, (FC (Y, Z) , ∗ , e′, ^-1) is a polygroup. Now, let f, g ∈ FC (Y, Z) and h ∈ f ∗ g. Then, we have μ ( h ) = inf { Z 1 ( h ( y ) ) : y ∈ Y } ≥ inf { min { Z 1 ( f ( y ) ) , Z 1 ( g ( y ) ) } : y ∈ Y } ≥ min { inf { Z 1 ( f ( y ) ) : y ∈ Y } , inf { Z 1 ( g ( y ) ) : y ∈ Y } } = min { μ ( f ) , μ ( g ) } . Also, for the map f^-1 ∈ FC (Y, Z) we have μ ( f - 1 ) = inf { Z 1 ( f - 1 ( y ) ) : y ∈ Y } = inf { Z 1 ( ( f ( y ) ) - 1 ) : y ∈ Y } ≥ inf { Z 1 ( f ( y ) ) : y ∈ Y } = μ ( f ) . Thus, μ ∈ I^FC(Y,Z) is a fuzzy polygroup. □

Definition 3.9. [10] Let Y and Z be two fixed fuzzy topological spaces, U ∈ I ^Z a fuzzy open set of Z, and y ∈ Y. Then by (y ; U) ∈ I^FC(Y,Z) we denote the fuzzy set for which ( y ; U ) ( f ) = U ( f ( y ) ) , for all f ∈ FC ( Y , Z ) .

The fuzzy point-open topology T FP on FC (Y, Z) generated by fuzzy sets of the form (y ; U), where y ∈ Y and U ∈ I ^Z , is a fuzzy open set of Z.

Theorem 3.10. Let ( Y , T Y ) be a fully stratified fuzzy topological space and ( Z , ∘ , T Z ) a fuzzy topological polygroup. Then the triad ( FC ( Y , Z ) , ∗ , T FP ) is a fuzzy topological polygroup.

Proof. Clearly by Theorem 3.6, the pair (FC (Y, Z) , ∗) is a polygroup and ( FC ( Y , Z ) , T FP ) is a fuzzy topological space.

Now, we check that ( FC ( Y , Z ) , ∗ , T FP ) satisfies conditions (i) and (ii) of Definition 3.3.

(1) Let f, g ∈ FC (Y, Z) and (y ; W) be a fuzzy open subbasic Q-neighborhood of a fuzzy point h _λ of f ∗ g. We show that there exist two fuzzy open subbasic Q-neighborhoods (y₁ ; U) and (y₂ ; V) of f _λ and g _λ respectively, such that ( y 1 ; U ) • ( y 2 ; V ) ≤ ( y ; W ) .

Since h _λ q (y ; W), it follows that λ + W (h (y)) >1 and so h (y)  _λ qW.

Thus, the fuzzy set W is a fuzzy openQ-neighborhood of h (y)  _λ .

Now, since ( Z , ∘ , T Z ) is a fuzzy topological polygroup, there exist two fuzzy open Q-neighborhoods U and V of f (y)  _λ and g (y)  _λ , such that UV ≤ W.

We consider the fuzzy sets (y ; U) and (y ; V). Clearly the fuzzy sets (y ; U) and (y ; V) are Q-neighborhoods of f _λ and g _λ respectively.

We prove that (y ; U) • (y ; V) ≤ (y ; W).

Let f ∈ FC (Y, Z). Then we have ( ( y ; U ) • ( y ; V ) ) ( f ) = sup { min { ( y ; U ) ( f 1 ) , ( y ; V ) ( f 2 ) } : f ∈ f 1 ∗ f 2 } = sup { min { U ( f 1 ( y ) ) , V ( f 2 ( y ) ) } : f ∈ f 1 ∗ f 2 } ≤ sup { min { U ( z 1 ) , V ( z 2 ) } : f ( y ) ∈ z 1 ∘ z 2 } = ( U • V ) ( f ( y ) ) ≤ W ( f ( y ) ) = ( y ; W ) ( f ) .

Now, the condition (i) in Definition 3.3 follows.

(2) Let f ∈ FC (Y, Z) and (y ; V) be a fuzzy open Q-neighborhood of f λ - 1. We show that there exists a fuzzy open Q-neighborhood (y ; U) of f _λ such that (y ; U) ^-1 ≤ (y ; V).

Since f λ - 1 q ( y ; V ), it follows that λ + V (f^-1 (y)) >1 .

Since ( Z , ∘ , T Z ) is a fuzzy topological polygroup, it follows that there exists a fuzzy open Q-neighborhood U of f _λ such that U^-1 ≤ V. We prove that (y ; U) ^-1 ≤ (y ; V).

Let f ∈ FC (Y, Z). Then ( y ; U ) - 1 ( f ) = ( y ; U ) ( f - 1 ) = U ( f - 1 ( y ) ) = U ( ( f ( y ) ) - 1 ) = U - 1 ( f ( y ) ) ≤ V ( f ( y ) ) = ( y ; V ) ( f ) .

Hence, (y ; U) ^-1 ≤ (y ; V). Now the condition (ii) in Definition 3.3 follows.

Therefore the triad ( FC ( Y , Z ) , ∗ , T FP ) is a fuzzy topological polygroup. □

Theorem 3.11. Let ( Y , T Y ), ( Z , T Z ) be fully stratified fuzzy topological spaces and ( Z , ∘ , T Z ) a fuzzy topological polygroup. Then, the triad ( FC ( Y , Z ) , ∗ , T FP ) is a fuzzy topological polygroup. Moreover, the pair ( FC ( Y , Z ) , T FP ) is a fully stratified space.

Proof. By Theorem 3.10 the triad ( FC ( Y , Z ) , ∗ , T FP ) is a fuzzy topological polygroup. Now, let λ ∈ [0, 1] and λ^* be the constant fuzzy subset of Z with value λ. If y is a fixed but arbitrary element of Y, then for all f ∈ FC (Y, Z) we have: ( y ; λ * ) ( f ) = λ * ( f ( y ) ) = λ .

So (y ; λ^*) is the constant fuzzy subset of FC (Y, Z) with value λ. Since ( y ; λ * ) ∈ T FP it follows that ( FC ( Y , Z ) , T FP ) is a fully stratified space. □

Theorem 3.12. Let ( Y , T Y ), ( Z , T Z ) be fully stratified fuzzy topological spaces and ( Z , ∘ , T Z ) a fuzzy topological polygroup. Then, the mapping F : FC (Y, Z) → FC (Y, Z), F (f) = f^-1 is homomorphism.

Proof. It is straightforward. □

Definition 3.13. Let μ be a fuzzy subset of a fuzzy set P. For t ∈ [0, 1], we define μ _t as follows: μ t = { x ∈ P | μ ( x ) ≥ t } μ _t is called a level set of μ.

A strong level set μ t > of a fuzzy subset μ in P is defined by μ t > = { x ∈ P | μ ( x ) > t } .

Theorem 3.14. [6, 17] Let P be a polygroup and μ be a fuzzy subset of P. Then, the following statementsare equivalent:

μ is a fuzzy subpolygroup of P.

Remark 1. Recall first that if ( X , T ) is a topological space, then ω ( T ) = { μ ∈ I X : μ t > ∈ T { for each t ∈ [ 0 , 1 ] } is a fuzzy topology on X, where μ t > = { x ∈ X | μ ( x ) > t }, and ( X , ω ( T ) ) is called an induced fuzzy topological space on ( X , T ).

Recall also that for the induced fuzzy topological space ( X , ω ( T ) ), β = { t . 1 A : A ∈ T , t ∈ [ 0 , 1 ] } is a base for the fuzzy topology ω ( T ), where t . 1 _A  ∈ I ^X is defined by ( t . 1 A ) ( x ) = { t if x ∈ A , 0 if x ∉ A .

Definition 3.15. 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.

Lemma 3.16. Let P be a polygroup. Then the hyperoperation ∘ : P × P ⟶ ℘ ^∗ (P) is continuous if and only if for all x, y ∈ P and W ∈ T such that x ∘ y ⊆ W, there exist U , V ∈ T for which x ∈ U, y ∈ V and U ∘ V ⊆ W.

Theorem 3.17. Let P be a polygroup, ( P , T ) be topological space and ( P , ω ( T ) ) be an induced fuzzy topological space. Then ( P , T ) is a topological polygroup if and only if ( P , ω ( T ) ) is a fuzzy topological polygroup.

Proof. (Necessity) Let ( P , T ) be a topological space. We show that ( P , ω ( T ) ) satisfies the conditions (i) and (ii) of Definition 3.3.

If a, b ∈ P and W ∈ J = ω ( T ) is a fuzzy open Q-neighborhood of a fuzzy point c _λ of a ∘ b, then W = ⋃ { r i . 1 A i : i ∈ I and A i ∈ T } .

So there exists j ∈ I such that λ + r _j  > 1 and c ∈ A _j . Since a ∘ b is the union of all fuzzy points which belong to a ∘ b, if W _i , i = 1, …, n are fuzzy open Q-neighborhoods of c _{λ i} , then we have:

There exist j _i  ∈ I such that λ i + r j i > 1 , c i ∈ A j i ∈ T and U _i V _i  ≤ W _i , where U _i and V _i are Q-neighborhoods of a _{λ i} and b _{λ i} respectively, and i = 1, …, n. Hence a ∘ b = ⋃ i = 1 n c λ i ⊆ ⋃ i = 1 n A j i ∈ T .

Since ( P , T ) is a topological polygroup, then there exist U , V ∈ T such that a ∈ U, b ∈ V and U ∘ V ⊆ A, where A = ⋃ i = 1 n A j i.

Let R = r _t  . 1 _U and S = r _t  . 1 _V . Then R and S are fuzzy open Q-neighborhoods of fuzzy points a _λ and b _λ respectively, where r _t  = max {r₁, …, r _n }. So RS = ( r t . 1 U ) ( r t . 1 V ) ≤ r t . A ≤ W .

Therefore the condition (i) of Definition 3.3 holds.

A similar argument can be done for the condition (ii) of Definition 3.3. Hence ( P , ω ( T ) is a fuzzy topological polygroup.

(Sufficiency) Let a, b ∈ P and W ∈ T such that a ∘ b ⊆ W. Also suppose λ, r ∈ I such that r + λ > 1.

If T = r . 1 _W , then T ∈ J = ω ( T ) and λ + T (c) >1, where c is a fixed but arbitrary element of a ∘ b ⊆ W. This means that W is a fuzzy open Q-neighborhood of a fuzzy point c _λ of a ∘ b.

Since ( P , ω ( T ) ) is an FTP, it follows that there exist two fuzzy open Q-neighborhoods R and S of a _λ and b _λ respectively, such that RS ≤ T.

Let R = ⋃ { r j . 1 U j : j ∈ J , r j ∈ I and U j ∈ T } and S = ⋃ { r k . 1 V k : k ∈ K , r k ∈ I and V k ∈ T }

Since R is a fuzzy open Q-neighborhood of a _λ , it follows that there exist m ∈ J and U m ∈ T such that a ∈ U _m and r _m  + λ > 1. Also, since S is a fuzzy open Q-neighborhood of b _λ , then there exist n ∈ K and V n ∈ T such that b ∈ V _n and r _n  + λ > 1. So, we have ( r m . 1 U m ) ( r n . 1 V n ) ≤ RS ≤ T = r . 1 W .

This implies that U m V n ⊆ W .

Therefore ( P , T ) is a topological polygroup. □

4 On subpolygroups of a fuzzy topological polygroup

Definition 4.1. [6] Let μ be a fuzzy subpolygroup of a polygroup P. Then μ is said to be normal if for all x, y ∈ P, μ ( z ) = μ ( z ′ ) , for all z ∈ x ∘ y and z ′ ∈ y ∘ x .

Let μ be a fuzzy subpolygroup of a polygroup P. For all a ∈ P, a fuzzy subset μ _a of P is called a fuzzy right coset of μ in P if μ a ( x ) = ⋁ z ∈ x ∘ a - 1 μ ( z ) for all x ∈ P .

Similarly, a fuzzy subset _a μ of P is called a fuzzy left coset of μ in P if a μ ( x ) = ⋁ z ∈ a - 1 ∘ x μ ( z ) for all x ∈ P .

If μ is a fuzzy normal subpolygroup of P and a an arbitrary element of P, then the fuzzy right coset μ _a coincides with the fuzzy left coset _a μ.

Consider the set P/μ = {μ _a  | a ∈ P} of all fuzzy right cosets of μ. Now, we define an operation ∗ on P/μ as follows μ a ∗ μ b = { μ c | c ∈ a ∘ b } If μ is a fuzzy normal subpolygroup of a polygroup P, then the operation ∗ is well defined.

Then (P/μ, ∗) becomes a polygroup and it is called the fuzzy quotient polygroup relative to the fuzzy normal subpolygroup μ.

Definition 4.2. Let ( P , T ) be an FTP and μ a fuzzy subpolygroup of P. Then ( μ , T μ ) is called a subpolygroup of FTP ( P , T ). If μ is a fuzzy normal subpolygroup, then ( μ , T μ ) is called a normal subpolygroup.

Proposition 4.3. Let ( μ , T μ ) be a subpolygroup of FTP ( P , T ) and let P/μ = { _x μ | x ∈ P} and f : P ⟶ P/μ,  x ↦  _x μ. Then, U = { A | f - 1 ( A ) ∈ T } is a fuzzy topology for P/μ. We call FTS ( P / μ , U ) a fuzzy left coset space.

Similarly the right coset space can be defined.

Proof. It is straightforward. □

Definition 4.4. Let μ be a fuzzy normal subpolygroup of P. We call P/μ = { _x μ | x ∈ P} a fuzzy quotient polygroup of P.

Definition 4.5. [15] Let A be a fuzzy set in ( X , T ). The intersection of all fuzzy closed sets containing A is called the closure of A, denoted by A ¯. Obviously A ¯ is the smallest fuzzy closed set containing A and ( A ¯ ) ¯ = A ¯.

Theorem 4.6. [15] A fuzzy point x λ ∈ A ¯ if and only if each Q-neighborhood of x _λ is quasi-coincident with A.

Proposition 4.7. Let

be a FTP. If

is a subpolygroup of

and

then

is also a subpolygroup of

, where

the closure of K.

Proof. We have only to prove that K ¯ is a fuzzy subpolygroup of P. It sufficient to show that

min { μ K ¯ ( a ) , μ K ¯ ( b ) } ≤ μ K ¯ ( c ), for all a, b ∈ P and c ∈ a ∘ b .

Let μ K ¯ ( a ) = λ 1 , μ K ¯ ( b ) = λ 2 and λ′ = min {λ₁, λ₂}. Now, suppose that W is a fuzzy open Q-neighborhood of c_λ′. Since ( P , T ) is an FTP, it follows that there exist two fuzzy open Q-neighborhoods U of a_λ′ and V of b_λ′ such that UV ≤ W.

Since a λ ′ , b λ ′ ∈ K ¯, there exist x, y ∈ suppK such that μ _U  (x) + μ _K  (x) >1 and μ _V  (y) + μ _K  (y) >1 . Now, we have μ W ( c ) + μ K ( c ) ≥ μ UV ( c ) + μ K ( c ) ≥ min { μ U ( a ) , μ V ( b ) } + λ ′ > 1 . Hence W is quasi-coincident with K. From Theorem 4.6 it follows that c λ ′ ∈ K ¯. Hence μ K ¯ ( c ) > λ ′ = min { μ K ¯ ( a ) , μ K ¯ ( b ) } . So, the condition (1) holds. Similarly, we can show (2) and the proof is completed. □

Theorem 4.8. Let the conditions of Proposition 4.7 be satisfied. If K is a fuzzy normal subpolygroup of P, then ( K ¯ , T K ¯ ) is a normal subpolygroup of ( P , T ).

Proof. It is strightforward. □

Definition 4.9. [15] Let A be a fuzzy set in X. The set {x ∈ X | μ _A  (x) >0} is called the support of A and is denoted by suppA.

Definition 4.10. [14] Let ( X , T ) be a fuzzy topological space. A fuzzy set B in X is said to be Q-compact if for all open Q-cover B of B, there exists a finite subset B ˜ of B such that B ˜ is an open Q-subcover of B. A fuzzy topological space ( X , T ) is said to be Q-compact if X is Q-compact.

Proposition 4.11. Let ( P , T ) be a FTP and let T contain all constant fuzzy sets. If A is a fuzzy subpolygroup of P and K = suppA, then K is an ordinary subpolygroup of P and ( K , T K ) is Q-compact.

Proof. First we show that K is a subpolygroup of P. It is sufficient to show that

If a, b ∈ K, then a ∘ b ⊆ K.

Now, we show that ( K , T K ) is Q-compact. Since K is an open set, it follows that K is also a closed set. So, ( K , T K ) is Q-compact. □

Proposition 4.12. Let ( P , T ) be a FTP and ( P / μ , U ) be its quotient polygroup. Then the mapping f : x ⟶  _x μ of ( P , T ) into ( P / μ , U ) is an open homomorphism.

Proof. It is straightforward. □

Proposition 4.13. Let ( μ , T μ ) be a normal subpolygroup of FTP ( P , T ) and ( P / μ , U ) its quotient subpolygroup. Then the mapping f : ( P , T ) ⟶ ( P / μ , U ) , x ↦ x μ = x ′, is both continuous and open.

Proof. First we show that f is continuous. In order to prove that f is continuous, it is sufficient to show that for all fuzzy point x _λ  ∈ P and all Q-neighborhood U′ of x λ ′ = f ( x λ ), there exists a Q-neighborhood U of x _λ such that f (U) ≤ U′. So, for all x _λ  ∈ P and any Q-neighborhood U′ of x λ ′ = f ( x λ ), there exists a U λ ¯ ′ ∈ U λ ¯ ′ such that x λ ¯ ′ U λ ¯ ′ ≤ U ′, since x λ ¯ ′ U λ ¯ ′ = f ( x λ ¯ U λ ¯ ), where x λ ¯ ′ U λ ¯ ′ is a Q-neighborhood of x _λ . Thus, f is continuous in all fuzzy point x _λ  ∈ P. Therefore, f is continuous.

Now we show that f is open. In order to prove that f is open, it is sufficient to show that for all A ∈ T, f ( A ) ∈ U. For all x _λ qA, { x λ ¯ U λ ¯ } is a Q-neighborhood base of x _λ . There exists x λ ¯ U λ ¯ such that x λ ¯ U λ ¯ ≤ A. So, we have f ( A ) = f ( ⋃ x λ qA x λ ¯ U λ ¯ ) = ⋃ f ( x λ ¯ U λ ¯ ) = ⋃ x λ ¯ ′ U λ ¯ ′ ∈ U Therefore, the proof is completed. □

Proposition 4.14. Let ( P , T ) be an usual topological polygroup, and let ( P , ω ( T ) ) be an induced fuzzy topological polygroup on ( P , T ). If N is a normal subpolygroup of P and ( P * , T * ) and ( P * , J * ) are the usual quotient polygroup and quotient fuzzy topological polygroup respectively, then we have

ω ( T N ) = J N, where T N = { U ∩ N : U ∈ T }, J N = { U ˜ ∩ N : U ˜ ∈ J } and J = ω ( T ).

Proof. (i) It id obvious.

(ii) Let A ˜ * ∈ ω ( T * ). Then for all r ∈ I, { x * ∈ P * : A ˜ * ( x * ) ≤ r } ∈ ( T * ) ′ .

Since f : ( P , T ) ⟶ ( P * , T * ) defined by f (x) = xN is continuous, it follows that f - 1 ( { x * ∈ P * : A ˜ * ( x * ) ≤ r } ) = { x ∈ P : A ˜ * ( f ( x ) ) ≤ r } = { x ∈ P : f - 1 ( A ˜ * ) ( x ) ≤ r } ∈ T ′ . This implies that f - 1 ( A ˜ * ) ∈ ω ( T ) = J. Since the map f : ( P , J ) ⟶ ( P * , J * ) is fuzzy open, it follows that f ( f - 1 ( A ˜ * ) ) = A ˜ * ∈ J *.

Conversely, let A ˜ * ∈ J *. Since the map f : ( P , J ) ⟶ ( P * , J * ) is fuzzy continuous, it follows that f - 1 ( A ˜ * ) ∈ J. Hence, for each r ∈ I { x ∈ P : f - 1 ( A ˜ * ) ( x ) ≤ r } ∈ T ′ and it follows that { x ∈ P : f - 1 ( A ˜ * ) ( x ) ≰; r } ∈ T . Since the map f : ( P , T ) ⟶ ( P * , T * ) is open, it follows that f ( { x ∈ P : f - 1 ( A ˜ * ) ( x ) ≰; r } ) = { f ( x ) : A ˜ * ( f ( x ) ) ≰; r } = { x * ∈ X * : A ˜ * ( x * ) ≰; r } ∈ T * . Hence, { x * ∈ P * : A ˜ * ( x * ) ≤ r } ∈ ( T * ) ′, which impl-ies that A ˜ * ∈ ω ( T * ). This completes the proof. □