Studies on fuzzy topological polygroups

Abstract

In this paper, we study fuzzy topological polygroups and prove some results in this respect. Denote by FC (Y, Z) the set of all fuzzy continuous functions from a fuzzy topological space Y into a fuzzy topological space Z. Our aim in this paper is to study a polygroup, fuzzy polygroup, topological polygroup and fuzzy topological polygroup structure on the (fuzzy) function space FC (Y, Z). Also, we study induced fuzzy topological polygroups. It is shown that if $(P, T)$ is a usual topological polygroup, then the induced fuzzy topology $ω (T)$ on I^P makes $(P, ω (T))$ a fuzzy topologiccal polygroup.

Keywords

Polygroup fuzzy polygroup topological polygroup fuzzy topological polygroups function space

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 $\begin{matrix} A \circ B & = & ⋃_{\binom{a \in A}{b \in B}} a \circ b, \\ x \circ A & = & {x} \circ A and A \circ x = A \circ {x} . \end{matrix}$ 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 =x ∘ e,

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 [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 $T \subseteq FS (X)$ of fuzzy sets is called a fuzzy topology for X if it satisfies the following three axioms:

$\underline{0}, \underline{1} \in T$ .

For all $A, B \in T$ , then $A \land B \in T$ .

For all $(A_{j})_{j \in J} \in T$ , then $⋁_{j \in J} A_{j} \in T$ .

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.

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

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

The family of all fuzzy sets in X is called the discrete fuzzy topology on X.

The collection of all constant fuzzy sets in X is a fuzzy topology on X.

Example 2. Consider the set of natural numbers $ℕ$ and let A be a fuzzy set on $ℕ$ defined as $μ_{A} (x) = {\begin{matrix} 0.7 & if x \in 2 ℕ, \\ 0.5 & if x \in 2 ℕ - 1 . \end{matrix}$ Then, $T = {\underline{0}, A, \underline{1}}$ is a fuzzy topology on $ℕ$ .

Example 3. Let A and B be two fuzzy sets on X = [-1, 1] defined as $μ_{A} (x) = {\begin{matrix} 0 & if - 1 \leq x \leq 0, \\ x & if 0 \leq x \leq 1 \end{matrix}$ and $μ_{B} (x) = {\begin{matrix} 1 & if - 1 \leq x \leq - \frac{1}{3}, \\ - 3 x & if - \frac{1}{3} \leq x \leq 0, \\ 0 & if 0 \leq x \leq 1 . \end{matrix}$ Then, $T = {\underline{0}, A, B, A \lor B, \underline{1}}$ 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 $B \in T$ 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_λqA, 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 $B \in T$ such that x_λqB ≤ 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 $B \in U$ , the inverse image $f^{- 1} [B] \in T$ . Conversely, f is fuzzy open if for all open fuzzy set $A \in T$ , the image $f [A] \in U$ (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 $U_{j} \in T$ , 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 $μ_{f [A]} (y) = {\begin{matrix} ⋁_{x \in f^{- 1} (y)} μ_{A} (x) & if f^{- 1} (y) \neq \emptyset \\ 0 & otherwise \end{matrix}$ 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.

f is continuous with respect to a Q-neighborhood in all fuzzy point x_λ.

f is continuous with respect to a neighborhood in all fuzzy point x_λ.

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.

For all a ∈ X and all fuzzy open Q-neighborhood V of $a_{λ}^{- 1}$ , there exists a fuzzy open Q-neighborhood U of a_λ such that U^-1 ≤ V.

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: $(A • B) (x) = ⋁_{x \in x_{1} \circ x_{2}} (μ_{A} (x_{1}) \land μ_{B} (x_{2}))$ and $μ_{A^{- 1}} (x) = μ_{A} (x^{- 1})$ for all x ∈ P. Also $C \circ D = ⋃ {c \circ d : c \in C, d \in D}$ and $C^{- 1} = {c^{- 1} : c \in C} .$

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.

If AC = BC for all C ∈ I^P, then A = B.

(AB) C = A (BC).

If A ≤ B then A^-1 ≤ B^-1.

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 $\begin{matrix} μ_{f (AB)} (y) & = & ⋁_{x \in f^{- 1} (y)} μ_{AB} (x) \\ = & ⋁_{x \in f^{- 1} (y)} {⋁_{x \in x_{1} x_{2}} [μ_{A} (x_{1}) \land μ_{B} (x_{2})]} \\ = & ⋁_{y \in f (x_{1}) f (x_{2})} {μ_{A} (x_{1}) \land μ_{B} (x_{2})} \end{matrix}$ and $\begin{matrix} μ_{f (A) f (B)} (y) \\ = ⋁_{y \in y_{1} y_{2}} {μ_{f (A)} (y_{1}) \land μ_{f (B)} (y_{2})} \\ = ⋁_{y \in y_{1} y_{2}} {(⋁_{x_{1} \in f^{- 1} (y_{1})} μ_{A} (x_{1})) \land (⋁_{x_{2} \in f^{- 1} (y_{2})} μ_{B} (x_{2}))} \\ = ⋁_{y \in f (x_{1}) f (x_{2})} {μ_{A} (x_{1}) \land μ_{B} (x_{2})} . \end{matrix}$ 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 $(P, \circ, T)$ 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.

For all x ∈ P and all fuzzy open Q-neighborhood V of $x_{λ}^{- 1}$ , there exists a fuzzy open Q-neighborhood U of x_λ such that U^-1 ≤ V.

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:

$\begin{matrix} \circ & e & x & y \\ e & e & x & y \\ x & x & e & y \\ y & y & y & {e, x} \end{matrix}$

P = 〈P, ∘ , e, ^-1〉 is a polygroup. It is clear that x^-1 = x, y^-1 = y. Consider on P the fuzzy topology $T = {\underline{0}, A, \underline{1}}$ , where $μ_{\underline{0}} (e) = μ_{\underline{0}} (x) = μ_{\underline{0}} (y) = 0$ , μ_A (e) =0.9, μ_A (x) =0.7, μ_A (y) =0.5, $μ_{\underline{1}} (e) = μ_{\underline{1}} (x) = μ_{\underline{1}} (y) = 1$ . 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

(Y, T_{Y})

be an FTS,

(Z, \circ, T_{Z})

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

$(f * g) (y) = f (y) \circ g (y)$

and

f^{- 1} (y) = (f (y))^{- 1}

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 $(Z, \circ, T_{Z})$ 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 y₁_{λ
₁} ∈ U₁ ∧ V₁. We show that $\begin{matrix} (f * g) ({y_{1}}_{λ_{1}}) & = & (f (y_{1}) \circ g (y_{1}))_{λ_{1}} \\ = & ((f * g) (y_{1}))_{λ_{1}} \in W, \end{matrix}$ i.e., for all z_{λ
₁} ∈ f (y₁) ∘ g (y₁), we check that z_{λ
₁} ∈ W.

We have y₁_{λ
₁} ∈ U₁ and y₁_{λ
₁} ∈ V₁. Therefore, $\begin{matrix} f ({y_{1}}_{λ_{1}}) = {f (y_{1})}_{λ_{1}} \in U and \\ g ({y_{1}}_{λ_{1}}) = {g (y_{1})}_{λ_{1}} \in V . \end{matrix}$ Hence $λ_{1} \leq U (f (y_{1})) and λ_{1} \leq V (g (y_{1})) .$ Also, for all z_{λ
₁} ∈ f (y₁) ∘ g (y₁), $\begin{matrix} (U • V) (z) \\ = ⋁_{z \in z_{1} \circ z_{2}} (U (z_{1}) \land V (z_{2})) \\ \geq min {U (f (y_{1})), V (g (y_{1})) : z \in f (y_{1}) g (y_{2})} \geq λ_{1} . \end{matrix}$ 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 $(Z, \circ, T_{Z})$ 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 y₁_{λ
₁} ∈ U₁. It is sufficient to check that f^-1 (y₁_{λ
₁}) ∈ W, that is λ₁ ≤ W (f^-1 (y₁)) = W ((f (y₁)) ^-1).

We have f (y₁_{λ
₁}) = f (y₁) _{λ
₁} ∈ U. Thus $λ_{1} \leq U (f (y_{1})) = U^{- 1} ((f (y_{1}))^{- 1}) \leq W ((f (y_{1}))^{- 1})$ and therefore, the map f^-1 is fuzzy continuous. □

Theorem 3.5. Let $(Y, T_{Y})$ be a fully stratified fuzzy topological space, $(Z, \circ, T_{Z})$ 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 $U \in T_{Z}$ . We show that $(e^{'})^{- 1} (U) \in T_{Y}$ . 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 $(e^{'})^{- 1} (U) \in T_{Y}$ . Thus, the map e′ is fuzzy continuous. □

Theorem 3.6. Let $(Y, T_{Y})$ be a fully stratified fuzzy topological space and $(Z, \circ, T_{Z})$ a fuzzy topological polygroup. Then (FC (Y, Z) , ∗ , e′, ^-1) is a polygroup.

Proof. For all f, g, h ∈ FC (Y, Z), we have $\begin{matrix} ((f * g) * h) (y) & = & (f * g) (y) \circ h (y) \\ = & (f (y) \circ g (y)) \circ h (y) \\ = & f (y) \circ (g (y) \circ h (y)) \\ = & f (y) \circ ((g * h) (y)) \\ = & (f * (g * h)) (y) . \end{matrix}$ 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 $(Z, \circ, T_{Z})$ 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,

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.

Theorem 3.8. Let $(Y, T_{Y})$ be a fully stratified fuzzy topological space, $(Z, \circ, T_{Z})$ 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 $\begin{matrix} μ (h) & = & inf {Z_{1} (h (y)) : y \in Y} \\ \geq & inf {min {Z_{1} (f (y)), Z_{1} (g (y))} : y \in Y} \\ \geq & min {inf {Z_{1} (f (y)) : y \in Y}, \\ inf {Z_{1} (g (y)) : y \in Y}} \\ = & min {μ (f), μ (g)} . \end{matrix}$ Also, for the map f^-1 ∈ FC (Y, Z) we have $\begin{matrix} μ (f^{- 1}) & = & inf {Z_{1} (f^{- 1} (y)) : y \in Y} \\ = & inf {Z_{1} ((f (y))^{- 1}) : y \in Y} \\ \geq & inf {Z_{1} (f (y)) : y \in Y} \\ = & μ (f) . \end{matrix}$ 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 $(y; U) (f) = U (f (y)), for all f \in FC (Y, Z) .$

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 $(Z, \circ, T_{Z})$ 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 $(y_{1}; U) • (y_{2}; V) \leq (y; W) .$

Since h_λq (y ; W), it follows that λ + W (h (y)) >1 and so h (y) _λqW.

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

Now, since $(Z, \circ, T_{Z})$ 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 $\begin{matrix} ((y; U) • (y; V)) (f) \\ = sup {min {(y; U) (f_{1}), (y; V) (f_{2})} : f \in f_{1} * f_{2}} \\ = sup {min {U (f_{1} (y)), V (f_{2} (y))} : f \in f_{1} * f_{2}} \\ \leq sup {min {U (z_{1}), V (z_{2})} : f (y) \in z_{1} \circ z_{2}} \\ = (U • V) (f (y)) \leq W (f (y)) = (y; W) (f) . \end{matrix}$

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 $(Z, \circ, T_{Z})$ 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 $\begin{matrix} (y; U)^{- 1} (f) & = & (y; U) (f^{- 1}) \\ = & U (f^{- 1} (y)) \\ = & U ((f (y))^{- 1}) \\ = & U^{- 1} (f (y)) \\ \leq & V (f (y)) \\ = & (y; V) (f) . \end{matrix}$

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 $(Z, \circ, T_{Z})$ 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 $(y; λ^{*}) \in T_{FP}$ 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 $(Z, \circ, T_{Z})$ 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: $μ_{t} = {x \in P | μ (x) \geq t}$ 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 $μ_{t}^{>} = {x \in P | μ (x) > t} .$

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.

All nonempty strong level set of μ is a subpolygrop of P.

All nonempty level set of μ is a subpolygrop of P.

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

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

Definition 3.15. Let P = 〈P, ∘ , e, ^-1〉 be a polygroup and $(P, T)$ be a topological space. Then the system $P = 〈 P, \circ, e,^{- 1}, T 〉$ 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 $W \in T$ such that x ∘ y ⊆ W, there exist $U, V \in T$ 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 $W \in J = ω (T)$ is a fuzzy open Q-neighborhood of a fuzzy point c_λ of a ∘ b, then $W = ⋃ {r_{i} . 1_{A_{i}} : i \in I and A_{i} \in T} .$

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 $λ_{i} + r_{j_{i}} > 1, c_{i} \in A_{j_{i}} \in T$ and U_iV_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 $a \circ b = ⋃_{i = 1}^{n} c_{λ_{i}} \subseteq ⋃_{i = 1}^{n} A_{j_{i}} \in T .$

Since $(P, T)$ is a topological polygroup, then there exist $U, V \in T$ 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 $RS = (r_{t} . 1_{U}) (r_{t} . 1_{V}) \leq r_{t} . A \leq W .$

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 $W \in T$ such that a ∘ b ⊆ W. Also suppose λ, r ∈ I such that r + λ > 1.

If T = r . 1_W, then $T \in J = ω (T)$ 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 $R = ⋃ {r_{j} . 1_{U_{j}} : j \in J, r_{j} \in I and U_{j} \in T}$ and $S = ⋃ {r_{k} . 1_{V_{k}} : k \in K, r_{k} \in I and V_{k} \in T}$

Since R is a fuzzy open Q-neighborhood of a_λ, it follows that there exist m ∈ J and $U_{m} \in T$ 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 $V_{n} \in T$ such that b ∈ V_n and r_n + λ > 1. So, we have $(r_{m} . 1_{U_{m}}) (r_{n} . 1_{V_{n}}) \leq RS \leq T = r . 1_{W} .$

This implies that $U_{m} V_{n} \subseteq W .$

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, $μ (z) = μ (z^{'}), for all z \in x \circ y and z^{'} \in y \circ x .$

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 $μ_{a} (x) = ⋁_{z \in x \circ a^{- 1}} μ (z) for all x \in P .$

Similarly, a fuzzy subset _aμ of P is called a fuzzy left coset of μ in P if $_{a} μ (x) = ⋁_{z \in a^{- 1} \circ x} μ (z) for all x \in P .$

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 _aμ.

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 $μ_{a} * μ_{b} = {μ_{c} | c \in a \circ b}$ 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 = {_xμ | x ∈ P} and f : P ⟶ P/μ, x ↦ _xμ. Then, $U = {A | f^{- 1} (A) \in T}$ 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 = {_xμ | 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 $\bar{A}$ . Obviously $\bar{A}$ is the smallest fuzzy closed set containing A and $\bar{(\bar{A})} = \bar{A}$ .

Theorem 4.6. [15] A fuzzy point $x_{λ} \in \bar{A}$ if and only if each Q-neighborhood of x_λ is quasi-coincident with A.

Proposition 4.7. Let

(P, T)

be a FTP. If

(K, T_{K})

is a subpolygroup of

(P, T)

and

μ_{K} (x) = {\begin{matrix} λ & if x \in suppK, \\ 0 & if x \notin suppK, \end{matrix}

then

(\bar{K}, T_{\bar{K}})

is also a subpolygroup of

(P, T)

, where

\bar{K}

the closure of K.

Proof. We have only to prove that $\bar{K}$ is a fuzzy subpolygroup of P. It sufficient to show that

min ${μ_{\bar{K}} (a), μ_{\bar{K}} (b)} \leq μ_{\bar{K}} (c)$ , for all a, b ∈ P and c ∈ a ∘ b .

$μ_{\bar{K}} (a) \leq μ_{\bar{K}} (a^{- 1})$ , for all a ∈ P .

Let $μ_{\bar{K}} (a) = λ_{1}, μ_{\bar{K}} (b) = λ_{2}$ 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 $a_{λ^{'}}, b_{λ^{'}} \in \bar{K}$ , 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 $\begin{matrix} μ_{W} (c) + μ_{K} (c) & \geq & μ_{UV} (c) + μ_{K} (c) \\ \geq & min {μ_{U} (a), μ_{V} (b)} + λ^{'} \\ > & 1 . \end{matrix}$ Hence W is quasi-coincident with K. From Theorem 4.6 it follows that $c_{λ^{'}} \in \bar{K}$ . Hence $μ_{\bar{K}} (c) > λ^{'} = min {μ_{\bar{K}} (a), μ_{\bar{K}} (b)} .$ 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 $(\bar{K}, T_{\bar{K}})$ 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 $\tilde{B}$ of $B$ such that $\tilde{B}$ 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.

If a ∈ K, then a^-1 ∈ K.

Let c ∈ a ∘ b. Since a, b ∈ K = suppA, it follows that μ_A (a) >0, μ_A (b) >0. However A is a fuzzy subpolygroup of P, then min {μ_A (a) , μ_A (b)} ≤ μ_A (c) for all c ∈ a ∘ b. So, μ_A (c) >0 and c ∈ K and the condition (1) holds. Now, suppose that a ∈ K = suppA. Since A is a fuzzy subpolygroup of P, it follows that μ_A (a^-1) = μ_A (a) >0 and a^-1 ∈ K. Therefore K is a subpolygroup of P.

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 ⟶ _xμ 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 $f : (P, T) ⟶ (P / μ, U), x \mapsto_{x} μ = x^{'}$ , 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 $U_{\bar{λ}}^{'} \in U_{\bar{λ}}^{'}$ such that $x_{\bar{λ}}^{'} U_{\bar{λ}}^{'} \leq U^{'}$ , since $x_{\bar{λ}}^{'} U_{\bar{λ}}^{'} = f (x_{\bar{λ}} U_{\bar{λ}})$ , where $x_{\bar{λ}}^{'} U_{\bar{λ}}^{'}$ 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 $A \in T$ , $f (A) \in U$ . For all x_λqA, ${x_{\bar{λ}} U_{\bar{λ}}}$ is a Q-neighborhood base of x_λ. There exists $x_{\bar{λ}} U_{\bar{λ}}$ such that $x_{\bar{λ}} U_{\bar{λ}} \leq A$ . So, we have $\begin{matrix} f (A) & = & f (⋃_{x_{λ} qA} x_{\bar{λ}} U_{\bar{λ}}) \\ = & ⋃ f (x_{\bar{λ}} U_{\bar{λ}}) = ⋃ x_{\bar{λ}}^{'} U_{\bar{λ}}^{'} \in U \end{matrix}$ 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 $T_{N} = {U \cap N : U \in T}$ , $J_{N} = {\tilde{U} \cap N : \tilde{U} \in J}$ and $J = ω (T)$ .

$ω (T^{*}) = J^{*}$ .

Proof. (i) It id obvious.

(ii) Let ${\tilde{A}}^{*} \in ω (T^{*})$ . Then for all r ∈ I, ${x^{*} \in P^{*} : {\tilde{A}}^{*} (x^{*}) \leq r} \in (T^{*})^{'} .$

Since $f : (P, T) ⟶ (P^{*}, T^{*})$ defined by f (x) = xN is continuous, it follows that $\begin{matrix} f^{- 1} ({x^{*} \in P^{*} : {\tilde{A}}^{*} (x^{*}) \leq r}) \\ = {x \in P : {\tilde{A}}^{*} (f (x)) \leq r} \\ = {x \in P : f^{- 1} ({\tilde{A}}^{*}) (x) \leq r} \in T^{'} . \end{matrix}$ This implies that $f^{- 1} ({\tilde{A}}^{*}) \in ω (T) = J$ . Since the map $f : (P, J) ⟶ (P^{*}, J^{*})$ is fuzzy open, it follows that $f (f^{- 1} ({\tilde{A}}^{*})) = {\tilde{A}}^{*} \in J^{*}$ .

Conversely, let ${\tilde{A}}^{*} \in J^{*}$ . Since the map $f : (P, J) ⟶ (P^{*}, J^{*})$ is fuzzy continuous, it follows that $f^{- 1} ({\tilde{A}}^{*}) \in J$ . Hence, for each r ∈ I ${x \in P : f^{- 1} ({\tilde{A}}^{*}) (x) \leq r} \in T^{'}$ and it follows that ${x \in P : f^{- 1} ({\tilde{A}}^{*}) (x) ≰; r} \in T .$ Since the map $f : (P, T) ⟶ (P^{*}, T^{*})$ is open, it follows that $\begin{matrix} f ({x \in P : f^{- 1} ({\tilde{A}}^{*}) (x) ≰; r}) \\ = {f (x) : {\tilde{A}}^{*} (f (x)) ≰; r} \\ = {x^{*} \in X^{*} : {\tilde{A}}^{*} (x^{*}) ≰; r} \in T^{*} . \end{matrix}$ Hence, ${x^{*} \in P^{*} : {\tilde{A}}^{*} (x^{*}) \leq r} \in (T^{*})^{'}$ , which impl-ies that ${\tilde{A}}^{*} \in ω (T^{*})$ . This completes the proof. □

Footnotes

Acknowledgments

The authors would like to thank Center of Excellence for Robust Intelligent Systems of Yazd University for their support.

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