Topological polygroups in the framework of fuzzy sets

Abstract

This paper provides a new connection between algebraic hyperstructures and fuzzy sets. The concept of fuzzy topological polygroups is a generalization of the concept of fuzzy topological groups. In this paper, we present the concept of fuzzy topological polygroups and prove some results in this respect. By considering the relative fuzzy topology on fuzzy subpolygroups we give some characterization of them. Finally, we prove that the product of a finite family of fuzzy topological polygroups is a fuzzy topological polygroup

Keywords

Polygroup fuzzy topological polygroups direct product

1 Introduction and preliminaries

Topological groups have the algebraic structure of a group and the topological structure of a topological space and they are linked by the requirement that multiplication and inversion are continuous functions. Fuzzy topological groups a generalization of topological groups. Zadeh in his basic paper [30] introduced the notion of fuzzy sets and fuzzy operations. Subsequently, Chang [3], Wong [29], Lowen [22] and others applied some basic concepts from general topology to fuzzy sets and developed a theory of fuzzy topological spaces. Rosenfeld [28] formulated the elements of a theory of fuzzy groups. In [14], Foster presented together the structure of a fuzzy topological space and that of a fuzzy group to form the structure of fuzzy topological group. Ma and Yu [23] 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.

The hyperstructure theory was born in 1934 when Marty introduced the notion of hypergroup [25]. The hypergroup theory is a natural generalization of the group theory. In a group the composition of two elements is an element, while in a hypergroup the composition of two elements is a set. Let H be a non-empty set. Then, a mapping ∘ : H × H ⟶ ℘ ^∗ (H) is called a hyperoperation, where ℘^∗ (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 $\begin{matrix} A \circ B = ⋃_{\binom{a \in A}{b \in B}} a \circ b, \\ x \circ A = {x} \circ A and A \circ x = A \circ {x} . \end{matrix}$ A hypergroupoid (H, ∘) is called a semihypergroup if for every x, y, z ∈ H, we have x ∘ (y ∘ z)=(x ∘ y) ∘ z and 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 [6, 25]. Applications of hypergroups have mainly appeared in special subclasses. One of the important subclasses is the class of polygroups. A polygroup is a completely regular, reversible in itself multigroup. We recall the following definition from [5]. A polygroup is a system P = 〈P, ∘ , e, ^-1〉, where ∘ : P × P ⟶ ℘ ^∗ (P), e ∈ P, ^- 1 is a unitary operation 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. Let 〈P, · , e, ^-1〉 and 〈P′, * , e′, ^-I〉 be two polygroups. Let f be a mapping from P to P′ such that f (e) = e′. Then, f is called (1) an inclusion homomorphism if f (x · y) ⊆ f (x) * f (y), for all x, y ∈ P; (2) a strong homomorphism or a good homomorphism if f (x · y) = f (x) * f (y), for all x, y ∈ P. Polygroups have been applied in many areas, such as geometry, lattices, combinatorics and color scheme. There exists a rich bibliography: publications appeared within 2013 can be found in “Polygroup Theory and Related Systems” by B. Davvaz [11]. This book contains the principal definitions endowed with examples and the basic results of the theory. Polygroups were studied by Comer [5], also see [1 , 20]. Specially, Comer and Davvaz developed the algebraic theory for polygroups.

On the other hand, in the last few decades, many connections between hyperstructures and fuzzy sets have been established and investigated. Zahedi, Bolurian and Hasankhani in 1995 [30] introduced the concept of a fuzzy subpolygroup. Also, see [9 , 32]. Till now, only a few papers treated the notion of topological hyperstructures, in the classical and fuzzy case, see [2 , 17–19]. In [17, 18], Heidari et al. introduced the concept of topological hypergroups and topological polygroups as a generalization of topological groups. A topological hypergroup (respectively, polygroup) is a non-empty set endowed with two structures, that of a topological space and that of a hypergroup (respectively, polygroup). Note that applications of hypergroups have mainly appeared in special subclasses. One of the important subclasses is the class of polygroups. For this reason in this paper we study the notion of fuzzy topological polygroups

The rest of this paper is organized as follows. In the second section we review basic concepts regarding the fuzzy topology. The third section is dedicated to the fuzzy topological polygroups. In Section 4, we introduce the concept of (normal) subpolygroups of a fuzzy topological polygroups and we investigate their properties.

2 Basic concepts

For the sake of convenience and completeness of our study, in this section some basic definition and results of [3 , 27], 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 μ_A : X ⟶ [0, 1] which associates with each point x ∈ X its grade or degree of membership μ_A (x) ∈ [0, 1]. That is, an element of I^X. We denote by FS (X) the set of all fuzzy sets on X.

A fuzzy topology on a set X is a collection $T$ of fuzzy sets in X satisfying (see [3]):

$\underline{0} \in T$ and $\underline{1} \in T$ , where $μ_{\underline{0}}, μ_{\underline{1}} : X ⟶ [0, 1], μ_{\underline{0}} (x) = 0, μ_{\underline{1}} (x) = 1$ for any x ∈ X.

If $A_{1}, A_{2} \in T$ , then $A_{1} \cap A_{2} \in T$ .

If $A_{i} \in T$ for any i ∈ I, then $⋃_{i \in I} A_{i} \in T$ , where μ_A₁∩A₂ (x) = μ_A
₁ (x) ∧ μ_A
₂ (x) and μ_{⋃_i∈IA_i} (x) = ⋁ _i∈Iμ_{A
_i} (x).

The pair (X, $T$ ) is called a fuzzy topological space and is denoted by FTS for short. Every member of $T$ is called $T$ -open fuzzy set. A fuzzy set is $T$ -closed if and only if its complement is $T$ -open.

Example 1. We present 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.

If $T$ is a topology on X, then the collection $T$ = { $A_{O} | O \in T$ }, where μ_{A
_O} is the characteristic function of the open set O, is a fuzzy topology on X.

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

Example 2. Let X = {a, b, c} and A be a fuzzy set on X defined as μ_A (a) =0.8, μ_A (b) =0.6, μ_A (c) =0.4. Then, $T = {\underline{0}, A, \underline{1}}$ is a fuzzy topology on X.

Example 3. Let A and B be two fuzzy sets on I = [0, 1] defined as $μ_{A} (x) = {\begin{matrix} 0 & if 0 \leq x \leq \frac{3}{5}, \\ \frac{5}{2} x - \frac{3}{2} & if \frac{3}{5} \leq x \leq 1 . \end{matrix}$ $μ_{B} (x) = {\begin{matrix} 1 & if 0 \leq x \leq \frac{2}{5}, \\ - 5 x + 3 & if \frac{2}{5} \leq x \leq \frac{3}{5}, \\ 0 & if \frac{3}{5} \leq x \leq 1 . \end{matrix}$ Then, $T = {\underline{0}, A, B, A \lor B, \underline{1}}$ is a fuzzy topology on I.

A fuzzy set in X is called a fuzzy point 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 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 neighborhoods 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 at 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-neighborhoods of fuzzy point x_λ. 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 $B \in U$ , the inverse image $f^{- 1} [B] \in T$ . Conversely, f is fuzzy open if, for any open fuzzy set $A \in T$ , the image $f [A] \in U$ (see [3]). Let A be a fuzzy set in the fuzzy topological space $(X, T)$ . Then, the induced 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 any 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 [14]). Let f be a mapping of FTS $(X, T)$ into FTS $(Y, U)$ . If for any 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 at x_λ with respect to Q-neighborhood (see [26]).

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 [26]):

f is a fuzzy continuous mapping.

f is continuous with respect to Q-neighborhood at any fuzzy point x_λ.

f is continuous with respect to neighborhood at any 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 [23])

For all a, b ∈ X and any 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 any 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

In this section, we introduce the notion of fuzzy topological polygroups, and investigate some of their properties.

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 by the following respective formulas: $(A • B) (x) = sup {min {μ_{A} (x_{1}), μ_{B} (x_{2})} | x \in x_{1} \circ x_{2}}$ and μ_{A
^-1} (x) = μ_A (x^-1), for any x ∈ P. Also, $\begin{matrix} C \circ D = ⋃ {c \circ d | c \in C, d \in D} and \\ C^{- 1} = {c^{- 1} | c \in C} . \end{matrix}$

Definition 3.1. 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 any 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 any 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.

Thereafter U • V is denoted by UV for short.

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

Example 4. Let P be a polygroup and $T$ be the indiscrete fuzzy topology on P. Then, $(P, T)$ is a fuzzy topological polygroup.

Example 5. Let P be a polygroup and $T = FS (P)$ be the discrete fuzzy topology on P. Then, $(P, T)$ is a fuzzy topological polygroup.

Example 6. Let P be a polygroup and $T = {λ^{*} | λ \in [0, 1], μ_{λ^{*}} (x) = λ}$ , i.e., $T$ is the collection of all constant fuzzy sets in P. Then, $(P, T)$ is a fuzzy topological polygroup.

Example 7. Let (P, ∘) be a finite polygroup and α be a fixed real number in the interval (1/2, 1). Consider on P the fuzzy topology $T = {A | μ_{A} (z) \in [0, α] \cup {1}, \forall z \in P}$ . Then, $(P, T)$ is a fuzzy topological polygroup.

Example 8. Let P = {x, y} with the hyperoperation $\begin{matrix} \circ & x & y \\ x & x & y \\ y & y & {x, y} \end{matrix}$ is a polygroup. It is clear that x^-1 = x, y^-1 = y, e = x. Consider on P the fuzzy topology $T = {\underline{0}, A, \underline{1}}$ , where $μ_{\underline{0}} (x) = μ_{\underline{0}} (y) = 0$ , μ_A (x) =0.8, μ_A (y) =0.6, $μ_{\underline{1}} (x) = μ_{\underline{1}} (y) = 1$ . Then, $(P, T)$ is a fuzzy topological polygroup.

Proposition 3.2If FTS $(P, T)$ is a fully stratified space, then conditions (i), (ii) in Definition 3.1 are satisfied if and only if for any fuzzy openQ-neighborhoodWof any fuzzy pointz_λofx ∘ y^-1, there are fuzzy openQ-neighborhoodsUofx_λandVofy_λsuch thatUV^-1 ≤ W.

Proof. Suppose that condition (i) is satisfied. Then, for all x, y ∈ P and any fuzzy open Q-neighborhood W of any fuzzy point z_λ of x ∘ y^-1, there are a fuzzy open Q-neighborhood U of x_λ and a fuzzy open Q-neighborhood V′ of $y_{λ}^{- 1}$ such that UV′ ≤ W. By (ii) there is a fuzzy open Q-neighborhood V of y_λ such that V^-1 ≤ V′, whence UV^-1 ≤ UV′ ≤ W.

Conversely, let FTS $(P, T)$ be a fully stratified space, i.e., $λ^{*} \in T$ for all λ ∈ [0, 1]. We show that conditions (i) and (ii) in Definition 3 hold. By supposition for any fuzzy open Q-neighborhood V of $y_{λ}^{- 1}$ of e ∘ y^-1 = y^-1, there are two fuzzy open Q-neighborhoods U′ of e_λ and U^′′ of y_λ such that U′U^{′′^-1} ≤ V, where e is the unit of the polygroup P. From μ_U′ (e) + λ > 1 we have μ_U′ (e) >1 - λ = λ′. Assume that $\bar{λ} = μ_{U^{'}} (e), U = U^{''} \cap {\bar{λ}}^{*}$ . Then, $U \in T, y_{λ} qU$ and U′U^-1 ≤ U′U^{′′^-1} ≤ V. Since $\begin{matrix} μ_{e_{\bar{λ}} U^{- 1}} (a) \\ = sup {min {μ_{e_{\bar{λ}}} (a_{1}), μ_{U^{- 1}} (a_{2})} | a \in a_{1} \circ a_{2}} \end{matrix}$ it follows that $\begin{matrix} μ_{e_{\bar{λ}} U^{- 1}} (a) & = & {\begin{matrix} 0 & if a_{1} \neq e, \\ min {\bar{λ}, μ_{U^{- 1}} (a)} & if a_{1} = e, \end{matrix} \\ = & μ_{U^{- 1}} (a) . \end{matrix}$ because μ_{U
^-1} (a) = μ_U (a^-1) and $U = U^{''} \cap {\bar{λ}}^{*}$ . It follows that $U^{- 1} = e_{\bar{λ}} U^{- 1} \leq U^{'} U^{- 1} \leq V$ . Therefore, condition (ii) in Definition 3 is satisfied.

Now, for any fuzzy open Q-neighborhood W of any fuzzy point z_λ of x ∘ y = x ∘ (y^-1) ^-1, there are two fuzzy open Q-neighborhoods U of x_λ and V′ of $y_{λ}^{- 1}$ such that UV^{′^-1} ≤ W, so from the above, there must be a fuzzy open Q-neighborhood V of y_λ such that V^-1 ≤ V′. Hence, UV = U (V^-1) ^-1 ≤ UV^{′^-1} ≤ W. □

Definition 3.3. In FTP $(P, T)$ , we call e_λ, a fuzzy λ unit or F-λ unit for short, where e is the unit of polygroup P.

Proposition 3.4In FTP $(P, T)$ , let FTS $(P, T)$ be a fully stratified space. If for anyλ ∈ (0, 1], $U = {U}$ is a fuzzy openQ-neighborhood base ofe_λ, then

$U_{\bar{λ}} = {U_{\bar{λ}} | U_{\bar{λ}} = U \cap {\bar{λ}}^{*}, \bar{λ} = ⋁_{U \in U} μ_{U} (e)}$ is also a fuzzy openQ-neighborhood base ofe_λ. It is called an F- $\bar{λ}$ Q-neighborhood base ofe_λ.

Ifμ_U (e) ≥ μ_U (x) for allx ∈ Pand $U \in U$ , then ${x_{\bar{λ}} U_{\bar{λ}}}$ is a fuzzyQ-neighborhood base ofx_λ.

Proof. (i) It is obvious.

(ii) Since $\begin{matrix} μ_{x_{\bar{λ}} U_{\bar{λ}}} (x) \\ = sup {min {μ_{x_{\bar{λ}}} (x_{1}), μ_{U_{\bar{λ}}} (x_{2})} | x \in x_{1} \circ x_{2}}, \end{matrix}$ it follows that $\begin{matrix} μ_{x_{\bar{λ}} U_{\bar{λ}}} (x) & = {\begin{matrix} 0 & if x_{1} \neq x, \\ sup {min {\bar{λ}, μ_{U_{\bar{λ}}} (x_{2})} & if x_{1} = x . \end{matrix} \\ = ⋁_{x \in x \circ x_{2}} μ_{U_{\bar{λ}}} (x_{2}) \\ = ⋁_{x \in x \circ x_{2}} μ_{U} (x_{2}) \\ = μ_{U} (e) . \end{matrix}$

So, we have $μ_{x_{\bar{λ}} U_{\bar{λ}}} (x) + λ > 1$ and we obtain $x_{λ} q x_{\bar{λ}} U_{\bar{λ}}$ . Since $(P, T)$ is an FTP, for any fuzzy open Q-neighborhood $U_{\bar{λ}}$ of e_λ ∈ x^-1 ∘ x there are fuzzy open Q-neighborhoods V of $x_{λ}^{- 1}$ and W of x_λ such that $VW \leq U_{\bar{λ}}$ . Let $V_{\bar{λ}} = V \cap {\bar{λ}}^{*}$ and $W_{\bar{λ}} = W \cap {\bar{λ}}^{*}$ . Then, $V_{\bar{λ}}$ and $W_{\bar{λ}}$ are also Q-neighborhoods of $x_{λ}^{- 1}$ and x_λ, respectively. Let $μ_{V_{\bar{λ}}} (x^{- 1}) = \tilde{λ}$ . Then, $1 - λ < \tilde{λ} \leq \bar{λ}$ . From $x_{\bar{λ}}^{- 1} W_{\bar{λ}} \leq V_{\bar{λ}} W_{\bar{λ}} \leq VW \leq U_{\bar{λ}}$ , we have $x_{\bar{λ}} x_{\bar{λ}}^{- 1} W_{\bar{λ}} \leq x_{\bar{λ}} U_{\bar{λ}}$ . Since e_λ ∈ x^-1 ∘ x, it follows that $e_{\bar{λ}} W_{\bar{λ}} \leq x_{\bar{λ}} x_{\bar{λ}}^{- 1} W_{\bar{λ}} \leq x_{\bar{λ}} U_{\bar{λ}}$ . Let $W_{\tilde{λ}} = W \cap {\tilde{λ}}^{*}$ . Then, $W_{\tilde{λ}} = e_{\tilde{λ}} W_{\tilde{λ}} \leq e_{\bar{λ}} W_{\bar{λ}} \leq x_{\bar{λ}} U_{\bar{λ}}$ . Obviously, $W_{\tilde{λ}}$ is a Q-neighborhood of x_λ. Hence, $x_{\bar{λ}} U_{\bar{λ}}$ is also a Q-neighborhood of x_λ and ${x_{\bar{λ}} U_{\bar{λ}}}$ is a family of Q-neighborhood of x_λ.

Now we prove that ${x_{\bar{λ}} U_{\bar{λ}}}$ is a Q-neighborhood base of x_λ. Since x_λ ∈ x ∘ e, for any fuzzy open Q-neighborhood U of x_λ there are a fuzzy open Q-neighborhoods V of x_λ and a fuzzy open Q-neighborhood $W_{\bar{λ}} \in U_{\bar{λ}}$ of e_λ such that $V W_{\bar{λ}} \leq U$ . Let $V_{\bar{λ}} = V \cap {\bar{λ}}^{*}, μ_{V_{\bar{λ}}} (x) = \hat{λ}$ and $W_{\hat{λ}} = W \cap {\hat{λ}}^{*}$ . It is obvious that $W_{\hat{λ}}$ is also a Q-neighborhood of e_λ and $V_{\bar{λ}} W_{\hat{λ}} \leq V_{\bar{λ}} W_{\bar{λ}} \leq V W_{\bar{λ}} \leq U$ . Since $\begin{matrix} μ_{x_{\bar{λ}} W_{\hat{λ}}} (z) \\ = sup {min {μ_{x_{\bar{λ}}} (a), μ_{W_{\hat{λ}}} (b)} | z \in a \circ b}, \end{matrix}$ it follows that $\begin{matrix} μ_{x_{\bar{λ}} W_{\hat{λ}}} (z) \\ = {\begin{matrix} 0 & if a \neq x, \\ sup {min {\bar{λ}, μ_{W_{\hat{λ}}} (b)} | z \in a \circ b} & if a = x . \end{matrix} \\ = ⋁_{z \in a \circ b} μ_{W_{\hat{λ}}} (b) \end{matrix}$ and $\begin{matrix} μ_{V_{\bar{λ}} W_{\hat{λ}}} (z) & = sup {min {μ_{V_{\bar{λ}}} (a), μ_{W_{\hat{λ}}} (b)} | z \in a \circ b} \\ \geq sup {min {\hat{λ}, μ_{W_{\hat{λ}}} (b)} z \in a \circ b,} \\ = ⋁_{z \in a \circ b} μ_{W_{\hat{λ}}} (b) \end{matrix}$ Thus, $x_{\bar{λ}} W_{\hat{λ}} \leq V_{\bar{λ}} W_{\hat{λ}} \leq U$ . Moreover, $W_{\hat{λ}}$ is a Q-neighborhood of e_λ, and there must be $U_{\bar{λ}}^{'} \in U_{\bar{λ}}$ such that $U_{\bar{λ}}^{'} \leq W_{\hat{λ}}$ , thus $x_{\bar{λ}} U_{\bar{λ}}^{'} \leq x_{\bar{λ}} W_{\hat{λ}} \leq U$ . This shows that ${x_{\bar{λ}} U_{\bar{λ}}}$ is a Q-neighborhood base of e_λ. □

Proposition 3.5Let the conditions of Proposition 3.4be satisfied. Then, the following properties are hold.

If $U_{\bar{λ}}, V_{\bar{λ}} \in U_{\bar{λ}}$ , then there exists $W_{\bar{λ}} \in U_{\bar{λ}}$ such that $W_{\bar{λ}} \leq U_{\bar{λ}} \cap V_{\bar{λ}}$ .

If $U_{\bar{λ}} \in U_{\bar{λ}}$ , then there exists $V_{\bar{λ}} \in U_{\bar{λ}}$ such that $V_{\bar{λ}} V_{\bar{λ}} \leq U_{\bar{λ}}$ .

If $U_{\bar{λ}} \in U_{\bar{λ}}$ , then there exists $V_{\bar{λ}} \in U_{\bar{λ}}$ such that $V_{\bar{λ}}^{- 1} \leq U_{\bar{λ}}$ .

For any $U_{\bar{λ}} \in U_{\bar{λ}}$ andx ∈ P, there exists $V_{\bar{λ}} \in U_{\bar{λ}}$ such that $x_{\bar{λ}}^{- 1} V_{\bar{λ}} x_{\bar{λ}} \leq U_{\bar{λ}}$ .

For any $U_{\bar{λ}} \in U_{\bar{λ}}$ if $x_{ν} q U_{\bar{λ}}$ , then there exists $V_{\bar{ν}} \in U_{\bar{ν}}$ such that $x_{\bar{ν}} V_{\bar{ν}} \leq U_{\bar{λ}}$ .

Proof. (1) If $U_{\bar{λ}}, V_{\bar{λ}} \in U_{\bar{λ}}$ , then $μ_{U_{\bar{λ}}} (e) + λ > 1, μ_{V_{\bar{λ}}} (e) + λ > 1$ and $μ_{U_{\bar{λ}} \cap V_{\bar{λ}}} (e) + λ = min {μ_{U_{\bar{λ}}} (e), μ_{V_{\bar{λ}}} (e)} + λ > 1 .$ So, $U_{\bar{λ}} \cap V_{\bar{λ}}$ is a Q-neighborhood of e_λ, and then there is $W_{\bar{λ}} \in U_{\bar{λ}}$ such that $W_{\bar{λ}} \leq U_{\bar{λ}} \cap V_{\bar{λ}}$ .

(2) If $U_{\bar{λ}} \in U_{\bar{λ}}$ , since $(P, T)$ is an FTP, then from e_λ ∈ e ∘ e, there are fuzzy open Q-neighborhoods $V_{\bar{λ}}^{'}$ and $V_{\bar{λ}}^{''}$ of e_λ such that $V_{\bar{λ}}^{'} V_{\bar{λ}}^{''} \leq U_{\bar{λ}}$ . Let $V_{\bar{λ}} = V_{\bar{λ}}^{'} \cap V_{\bar{λ}}^{''}$ , then $V_{\bar{λ}} V_{\bar{λ}} \leq V_{\bar{λ}}^{'} V_{\bar{λ}}^{''} \leq U_{\bar{λ}}$ .

(3) If $U_{\bar{λ}} \in U_{\bar{λ}}$ , since $(P, T)$ is an FTP, $e_{λ}^{- 1} = e_{λ}$ and $U_{\bar{λ}}$ is an open Q-neighborhood base of e_λ, then there exists $V_{\bar{λ}} \in U_{\bar{λ}}$ of e_λ such that $V_{\bar{λ}}^{- 1} \leq U_{\bar{λ}}$ .

(4) Since $(P, T)$ is an FTP and e_λ ∈ x^-1 ∘ x, it follows that for any fuzzy open Q-neighborhood of e_λ such as $U_{\bar{λ}} \in U_{\bar{λ}}$ , there are fuzzy open Q-neighborhoods V₁ of $x_{λ}^{- 1}$ and V₂ of x_λ such that $V_{1} V_{2} \leq U_{\bar{λ}}$ .

By (ii) of Proposition 3.4 we may suppose that $V_{1} = x_{\bar{λ}}^{- 1} V_{\bar{λ}}^{'}$ , where $V_{\bar{λ}}^{'} \in U_{\bar{λ}}$ . Similar to the proof of (ii) we can show that ${U_{\bar{λ}} x_{\bar{λ}} | U_{\bar{λ}} \in U_{\bar{λ}}}$ is also a Q-neighborhood base of x_λ. Hence, we may suppose that $V_{2} = V_{\bar{λ}}^{''} x_{\bar{λ}}$ , where $V_{\bar{λ}}^{''} \in U_{\bar{λ}}$ . Since $\begin{matrix} V_{\bar{λ}}^{'} V_{\bar{λ}}^{''} (e) \\ = sup {min {μ_{V_{\bar{λ}}^{'}} (x_{1}), μ_{V_{\bar{λ}}^{''}} (x_{2})} | e \in x_{1} \circ x_{2}} \\ \geq min {μ_{V_{\bar{λ}}^{'}} (e), μ_{V_{\bar{λ}}^{''}} (e)} \\ > λ + 1, \end{matrix}$ it follows that $e_{λ} q V_{\bar{λ}}^{'} V_{\bar{λ}}^{''}$ .

Let $min {μ_{V_{\bar{λ}}^{'}} (e), μ_{V_{\bar{λ}}^{''}} (e)} = \bar{\bar{λ}}, V_{\bar{\bar{λ}}} = V_{\bar{λ}}^{''} \cap {\bar{\bar{λ}}}^{*}$ . Then, $e_{\bar{\bar{λ}}} \leq V_{\bar{λ}}^{'}$ and $e_{\bar{\bar{λ}}} V_{\bar{\bar{λ}}} \leq V_{\bar{λ}}^{'} V_{\bar{λ}}^{''}$ . But $e_{\bar{\bar{λ}}} V_{\bar{\bar{λ}}} = V_{\bar{\bar{λ}}}$ is a Q-neighborhood of e_λ, so that $V_{\bar{λ}}^{'} V_{\bar{λ}}^{''}$ is also a Q-neighborhood of e_λ. Therefore, there exists $V_{\bar{λ}} \in U_{\bar{λ}}$ such that $V_{\bar{λ}} \leq V_{\bar{λ}}^{'} V_{\bar{λ}}^{''}$ . Consequently, we obtain $x_{\bar{λ}}^{- 1} V_{\bar{λ}} x_{\bar{λ}} \leq x_{\bar{λ}}^{- 1} V_{\bar{λ}}^{'} V_{\bar{λ}}^{''} x_{\bar{λ}} = V_{1} V_{2} \leq U_{\bar{λ}} .$

(5) Let $U_{\bar{λ}} \in U_{\bar{λ}}$ and $x_{ν} q U_{\bar{λ}}$ . Since ${x_{\bar{ν}} U_{\bar{ν}}}$ is a Q-neighborhood base of x_ν. So $x_{\bar{ν}} V_{\bar{ν}} \leq U_{\bar{λ}}$ where $V_{\bar{ν}} \in U_{\bar{ν}}$ .□

Proposition 3.6.LetPbe a polygroup andeits unit. For eachλ ∈ (0, 1], $U_{\bar{λ}} = {U_{\bar{λ}}}$ is a family of fuzzy sets inPsatisfying the following conditions:

For all $U_{\bar{λ}} \in U_{\bar{λ}}, e_{λ} q U_{\bar{λ}}$ .

If $1 \geq \bar{λ} \geq ⋁ μ_{U_{\bar{λ}}} (e)$ , then for any $U_{\bar{λ}} \in U_{\bar{λ}}$ and allx ∈ P, $μ_{U_{\bar{λ}}} (x) \leq \bar{λ}$ .

$U_{\bar{λ}}$ satisfies conditions (1) - (5) in Proposition 3.5.

We denote $\begin{matrix} U_{x_{\bar{λ}}} \\ = {U | there is U_{\bar{λ}} \in U_{\bar{λ}} such that x_{\bar{λ}} U_{\bar{λ}} \leq U} . \end{matrix}$ Then, $T = {U | U \in U_{x_{\bar{λ}}} whenever x_{λ} q U}$ is a fuzzy topology for P and $(P, T)$ is an FTP.

Proof. First we show that $U_{x_{\bar{λ}}}$ satisfies conditions (1) - (3) in Proposition 2.2 in [26].

(1) If $U \in U_{x_{\bar{λ}}}$ , by assumption, there is $U_{\bar{λ}} \in U_{\bar{λ}}$ such that $x_{\bar{λ}} U_{\bar{λ}} \leq U$ . Since $x_{λ} q x_{\bar{λ}} U_{\bar{λ}}$ and $x_{\bar{λ}} U_{\bar{λ}} \leq U$ , it follows that x_λ q U.

(2) If $U, V \in U_{x_{\bar{λ}}}$ , by assumption, we have $U_{\bar{λ}}, V_{\bar{λ}} \in U_{\bar{λ}}$ such that $x_{\bar{λ}} U_{\bar{λ}} \leq U, x_{\bar{λ}} V_{\bar{λ}} \leq V$ . Since $U_{\bar{λ}}$ satisfies conditions (1) - (5) in Proposition 3.5,there is $W_{\bar{λ}} \in U_{\bar{λ}}$ such that $W_{\bar{λ}} \leq U_{\bar{λ}} \cap V_{\bar{λ}}$ . Consequently we have $x_{\bar{λ}} W_{\bar{λ}} \leq x_{\bar{λ}} U_{\bar{λ}} \cap x_{\bar{λ}} V_{\bar{λ}} \leq U \cap V$ . Therefore, $U \cap V \in U_{x_{\bar{λ}}}$ .

(3) If $U \in U_{x_{\bar{λ}}}$ , by assumption, there is $U_{\bar{λ}} \in U_{\bar{λ}}$ such that $x_{\bar{λ}} U_{\bar{λ}} \leq U$ . Since $x_{\bar{λ}} U_{\bar{λ}} \leq U$ and U ≤ V, it follows that $V \in U_{x_{\bar{λ}}}$ .

So, by Proposition 2.2 in [26], we obtain that $T$ is a fuzzy topology for P.

Now, we prove that $(P, T)$ satisfies conditions (i) and (ii) in Definition 3.1.

(1) For any Q-neighborhood W of any fuzzy point z_λ of x ∘ y, there exists a $U_{\bar{λ}} \in U_{\bar{λ}}$ such that $z_{\bar{λ}} U_{\bar{λ}} \leq W$ . By assumption there is $V_{\bar{λ}}^{'} \in U_{\bar{λ}}$ and consequently $V_{\bar{λ}} \in U_{\bar{λ}}$ such that $V_{\bar{λ}}^{'} V_{\bar{λ}}^{'} \leq U_{\bar{λ}}$ and $y_{\bar{λ}}^{- 1} V_{\bar{λ}} y_{\bar{λ}} \leq V_{\bar{λ}}^{'}$ . Hence, we have $\begin{matrix} (x_{\bar{λ}} V_{\bar{λ}}) (y_{\bar{λ}} V_{\bar{λ}}^{'}) & = x_{\bar{λ}} y_{\bar{λ}} (y_{\bar{λ}}^{- 1} V_{\bar{λ}} y_{\bar{λ}}) V_{\bar{λ}}^{'} \\ \leq x_{\bar{λ}} y_{\bar{λ}} V_{\bar{λ}}^{'} V_{\bar{λ}}^{'} \\ \leq x_{\bar{λ}} y_{\bar{λ}} U_{\bar{λ}} \leq W . \end{matrix}$

Obviously, $x_{\bar{λ}} V_{\bar{λ}}$ and $y_{\bar{λ}} V_{\bar{λ}}^{'}$ are Q-neighborhood of x_λ and y_λ respectively. Then, condition (i) in Definition 3.1 follows.

(2) For any Q-neighborhood U of $x_{λ}^{- 1}$ there is a $U_{\bar{λ}} \in U_{\bar{λ}}$ such that $x_{\bar{λ}}^{- 1} U_{\bar{λ}} \leq U$ . But $U_{\bar{λ}}$ satisfies conditions (1) - (5) in Proposition 3.5, so there exists $V_{\bar{λ}}^{'} \in U_{\bar{λ}}$ such that $V_{\bar{λ}}^{'^{- 1}} \leq U_{\bar{λ}}$ . For $V_{\bar{λ}}^{'}$ there is also $V_{\bar{λ}} \in U_{\bar{λ}}$ such that $x_{\bar{λ}}^{- 1} V_{\bar{λ}} x_{\bar{λ}} \leq V_{\bar{λ}}^{'}$ .

Thus, there is a Q-neighborhood $x_{\bar{λ}} V_{\bar{λ}}$ of x_λ such that $\begin{matrix} (x_{\bar{λ}} V_{\bar{λ}})^{- 1} & = V_{\bar{λ}}^{- 1} x_{\bar{λ}}^{- 1} \\ = x_{\bar{λ}}^{- 1} (x_{\bar{λ}} V_{\bar{λ}}^{- 1} x_{\bar{λ}}^{- 1}) \\ \leq x_{\bar{λ}}^{- 1} V_{\bar{λ}}^{'^{- 1}} \\ \leq x_{\bar{λ}}^{- 1} U_{\bar{λ}} \leq U . \end{matrix}$

Hence, condition (ii) in Definition 3.1 follows. Therefore, $(P, T)$ is an FTP.□

4 (Normal) Subpolygroups of a fuzzy topological polygroups

We recall the following definition from [30].

Definition 4.1. 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.

Definition 4.2. [9] 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 any 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 is same as the fuzzy left coset _aμ. Consider the set P/μ = {μ_a | a ∈ P} of all fuzzy right cosets of μ. Now we give a structure on P/μ by defining the operation ∗ between two fuzzy right cosets as $μ_{a} * μ_{b} = {μ_{c} | c \in a \circ b} .$ If μ is a fuzzy normal subpolygroup of a polygroup P, then the operation ∗ defined on P/μ is well-defined. Then, (P/μ, ∗) becomes a polygroup and is called the fuzzy quotient polygroup relative to the fuzzy normal subpolygroup μ.

Definition 4.3. 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 4.4Let $(μ, T_{μ})$ be a subpolygroup of FTP $(P, T)$ , assume thatP/μ = {_xμ | x ∈ P} andf : P ⟶ P/μ, x ↦ _xμ. Then, $U = {A | f^{- 1} (A) \in T}$ is a fuzzy topology forP/μ. We call FTS $(P / μ, U)$ a fuzzy left coset space.

Proof. It is straightforward. □

Similarly the right coset space may be defined.

Definition 4.5. Let μ be a fuzzy normal subpolygroup of P. We call P/μ = {_xμ | x ∈ P} a fuzzy quotient polygroup of P.

Proposition 4.6Let $(μ, T_{μ})$ be a normal subpolygroup of FTP $(P, T)$ andf : x ↦ _xμ = x′ a mapping ofPintoP/μ. Assume for any $λ \in (0, 1], U_{\bar{λ}} = {U_{\bar{λ}}}$ is an $F - \bar{λ}$ openQ-neighborhood base ofe_λand $U_{\bar{λ}}^{'} = {U_{\bar{λ}}^{'} | U_{\bar{λ}}^{'} = f (U_{\bar{λ}}), U_{\bar{λ}} \in U_{\bar{λ}}}$ . If $U_{x_{\bar{λ}}^{'}} = {U^{'} | there is U_{\bar{λ}}^{'} \in U_{\bar{λ}}^{'} such that x_{\bar{λ}}^{'} U_{\bar{λ}}^{'} \leq U^{'}}$ and $U = {U^{'} | U^{'} \in U_{x_{\bar{λ}}^{'}} whenever x_{λ}^{'} q U^{'}}$ . Then, $(P / μ, U)$ is a fuzzy topological polygroup. It is called the quotient polygroup of FTP $(P, T) .$

Proof. By Proposition 3.6 it is easy to show. □

Proposition 4.7Let $(μ, T_{μ})$ 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 any fuzzy point x_λ ∈ P and any 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 any x_λ ∈ P and any Q-neighborhood U′ of $x_{λ}^{'} = f (x_{λ})$ , by Proposition 3.6, 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. So f is continuous at any 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 any $A \in T$ , $f (A) \in U$ . In order to show this, we have, for any x_λqA, ${x_{\bar{λ}} U_{\bar{λ}}}$ is a Q-neighborhood base of x_λ, there exists a $x_{\bar{λ}} U_{\bar{λ}}$ such that $x_{\bar{λ}} U_{\bar{λ}} \leq A$ . So $\begin{matrix} f (A) & = f (⋃_{x_{λ} qA} x_{\bar{λ}} U_{\bar{λ}}) \\ = ⋃ f (x_{\bar{λ}} U_{\bar{λ}}) = ⋃ x_{\bar{λ}}^{'} U_{\bar{λ}}^{'} \in U \end{matrix}$ Therefore, the proof is completed.□

Definition 4.8. Suppose that $〈 P_{1}, \circ_{1}, e_{1},^{- 1}, T_{1} 〉$ and $〈 P_{2}, \circ_{2}, e_{2},^{- I}, T_{2} 〉$ are fuzzy topological polygroups. A mapping f from P₁ into P₂ is said to be a fuzzy topological homomorphism if for all a, b ∈ P₁:

f (e₁) = e₂;

f (a ∘ ₁b) = f (a) ∘ ₂f (b);

f is fuzzy continuous mapping of FTS $(P_{1}, T_{1})$ into FTS $(P_{2}, T_{2})$ .

Definition 4.9. Suppose that $〈 P_{1}, \circ_{1}, e_{1},^{- 1}, T_{1} 〉$ and $〈 P_{2}, \circ_{2}, e_{2},^{- I}, T_{2} 〉$ are fuzzy topological polygroups. A mapping f from P₁ into P₂ is said to be a fuzzy topological isomorphism if for all a, b ∈ P₁:

f (e₁) = e₂;

f (a ∘ ₁b) = f (a) ∘ ₂f (b);

f is fuzzy continuous mapping of FTS $(P_{1}, T_{1})$ into FTS $(P_{2}, T_{2})$ ;

f is fuzzy open mapping of FTS $(P_{1}, T_{1})$ into FTS $(P_{2}, T_{2})$ ;

f is bijective.

If there exists a fuzzy topological isomorphism f of FTP $(P_{1}, T_{1})$ with FTP $(P_{2}, T_{2})$ , then FTP $(P_{1}, T_{1})$ and FTP $(P_{2}, T_{2})$ are said to be isomorphic.

Since P₁ is a polygroup, e₁ ∈ a ∘ ₁a^-1 for all a ∈ P₁, and hence we have f (e₁) ∈ f (a) ∘ ₂f (a^-1) or e₂ ∈ f (a) ∘ ₂f (a^-1) which implies f (a^-1) ∈ f (a) ^-1 ∘ ₂e₂, 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}.

Proposition 4.10.Let $(P_{1}, T_{1})$ and $(P_{2}, T_{2})$ be two fuzzy topological polygroups andf : P₁ ⟶ P₂be a homomorphism. Then,fis fuzzy continuous if and only iffis continuous ate_λ(hereeis the unit ofP₁) for anyλ ∈ (0, 1] .

Proof. Obviously, if f is fuzzy continuous, then f is continuous at fuzzy point e_λ for any λ ∈ (0, 1].

Conversely, suppose that f is continuous at fuzzy point e_λ for any λ ∈ (0, 1]. Let $f (x_{λ}) = x_{λ}^{'}$ , $U_{\bar{λ}} = {U_{\bar{λ}}}$ be a Q-neighborhood base of e_λ and $U_{\bar{λ}}^{'} = {U_{\bar{λ}}^{'}}$ be a Q-neighborhood base of $e_{λ}^{'}$ , where e′ is the unit of polygroup P₂. For any Q-neighborhood W of $x_{λ}^{'} = f (x_{λ})$ , by Proposition 3.4, there is a $U_{\bar{λ}}^{'} \in U_{\bar{λ}}^{'}$ such that $x_{\bar{λ}}^{'} U_{\bar{λ}}^{'} \leq W$ . Since f is continuous at e_λ, so $f (U_{\bar{λ}}) \leq U_{\bar{λ}}^{'}$ . Also $x_{\bar{λ}} U_{\bar{λ}}$ is a Q-neighborhooh of x_λ and $f (x_{\bar{λ}} U_{\bar{λ}}) = f (x_{\bar{λ}}) f (U_{\bar{λ}}) \leq x_{\bar{λ}}^{'} U_{\bar{λ}}^{'} \leq W,$ i.e., f is continuous at x_λ, and hence f is a fuzzy continuous mapping. □

Proposition 4.11Let $(P, T)$ be an FTP and $(P / μ, U)$ its quotient polygroup. Then, the mappingf : x → _xμof $(P, T)$ into $(P / μ, U)$ is an open homomorphism.

Proof. It is straightforward. □

Theorem 4.12.Let $(P_{1}, T)$ and $(P_{2}, U)$ be fuzzy topological polygroups andgbe a fuzzy open, surjective and fuzzy topological homomorphism from FTP $(P_{1}, T)$ onto FTP $(P_{2}, U)$ such that $μ = g^{- 1} (e_{λ}^{'})$ , (e′ is the unit of polygroupP₂) is a normal subpolygroup of FTP $(P_{1}, T)$ . Then, the quotient polygroup of FTP (P/μ, γ) is isomorphic to FTP $(P_{2}, U)$ .

Proof. By Definition 4.3, $(μ, T_{μ})$ is obviously a normal subpolygroup of FTP $(P, T)$ . For any y ∈ P₂, by assumption, there must be an x ∈ P₁ such that g (x) = y. We define the mapping $φ : (P_{2}, U) ⟶ (P_{1} / μ, γ)$ by setting φ (y) = _xμ, where y = g (x). Obviously, φ is an isomorphism of polygroup P₂ with polygroup P₁/μ.

Suppose that f : x ↦ _xμ is the mapping of $(P_{1}, T)$ into (P₁/μ, γ). Then, f is both fuzzy open and continuous, and φg = f is fuzzy continuous and for any B ∈ γ, $(φ g)^{- 1} (B) = g^{- 1} [φ^{- 1} [B]] \in T$ . Since g is fuzzy open, we have $g [g^{- 1} (φ^{- 1} [B])] = φ^{- 1} (B) \in U$ . Therefore, φ is fuzzy continuous.

Now, for any $A \in U$ , since g is continuous, we have $g^{- 1} (A) \in T$ . From the fact that f is fuzzy open, it follows that f [g^-1 (A)] ∈ γ, i.e., φg [g^-1 (A)] = φ (A) ∈ γ. This shows that φ is fuzzy open. Obviously, φ is bijective and consequently φ^-1 is fuzzy continuous. So, φ is a homeomorphism of FTP $(P_{2}, U)$ with FTP (P₁/μ, γ) and then φ is an isomorphism of FTP $(P_{2}, U)$ with FTP (P₁/μ, γ).

In particular, if μ = {e_λ}, then φ = g and $(P_{2}, U)$ is isomorphic to (P₁, γ). □

Now, we briefly discuss products of fuzzy topological polygroups. Let {P_i}, i = 1, …, n be a finite family of polygroups and $P = \prod_{i = 1}^{n} P_{i}$ the product polygroup. For each i = 1, …, n, let P_i have fuzzy topology $T_{i}$ and let A_i be a set in P_i. The product fuzzy set $A = \prod_{i = 1}^{n} A_{i} \equiv A_{1} \times \dots \times A_{n} \equiv (A_{1}, \dots, A_{n})$ in P has membership function μ_A given by $μ_{A} (x) = min {μ_{A_{1}} (x_{1}), \dots, μ_{A_{n}} (x_{n})},$ where x = (x₁, …, x_n).

In the following we prove that the product of fuzzy topological polygroups is a fuzzy topological polygroup.

Let $U_{{x_{i}}_{λ}}$ be a fuzzy open Q-neighborhood system of x_{i
_λ} relative to $T_{i}$ for each i = 1, 2, …, n. Obviously, for any $U_{i} \in U_{{x_{i}}_{λ}}$ , x_λ = (x₁, …, x_n) is quasi-coincident with $\prod_{i = 1}^{n} U_{i}$ , i.e., $x_{λ} q \prod_{i = 1}^{n} U_{i}$ . In fact, $\begin{matrix} μ_{\prod_{i = 1}^{n} U_{i}} (x_{1}, \dots, x_{n}) + λ \\ = min {μ_{U_{1}} (x_{1}), \dots, μ_{U_{n}} (x_{n})} + λ \\ = min {μ_{U_{1}} (x_{1}) + λ, \dots, μ_{U_{n}} (x_{n}) + λ} \\ > 1 . \end{matrix}$

Let $\begin{matrix} U_{x_{λ}} = & {U_{x_{λ}} | there exists U_{i} \in U_{{x_{i}}_{λ}} i = 1, \dots, n \\ such that \prod_{i = 1}^{n} U_{i} \leq U_{x_{λ}}} . \end{matrix}$

Then, for any x ∈ P and λ ∈ (0, 1], $U_{x_{λ}}$ is a family of fuzzy sets in P (see [23]).

Proposition 4.13. [24] Every $U_{x_{λ}}$ defined above satisfies the condition (1) - (3) in Proposition 2.2 in [26]. Consequently, we have $\begin{matrix} T = {U \in U_{x_{λ}} whenever x_{λ} qU} \end{matrix}$ is a fuzzy topology for $P = \prod_{i = 1}^{n} P_{i}$ .

Definition 4.14. [24] Let ${(P_{i}, T_{i}) | i = 1, \dots, n}$ be a family of FTS’s. We call the topology $T$ defined in Proposition 4 the product fuzzy topology for $P = \prod_{i = 1}^{n} P_{i}$ . The pair $(P, T)$ is called the product space of the $(P_{i}, T_{i})$ ’s.

Theorem 4.15.Let $(P_{i}, T_{i})$ , i = 1, …, nbe a finite family of fuzzy topological polygroups. Let the product polygroup $P = \prod_{i = 1}^{n} P_{i}$ and let $T$ be the product fuzzy topology inPby Definition 4. Then, $(P, T)$ is a fuzzy topological polygroup.

Proof. Let P_i be a fuzzy topological polygroup for any i = 1, …, n. That is,

For all a_i, b_i ∈ P_i and any fuzzy open Q-neighborhoods W_i of any fuzzy point c_{i
_λ} of a_i ∘ b_i, there are fuzzy open Q-neighbourhood U_i of a_{i
_λ} and V_i of b_{i
_λ} such that U_iV_i ≤ W_i.

For all a_i ∈ P_i and any fuzzy open Q-neighborhood V_i of $a_{i_{λ}}^{- 1}$ , there exists a fuzzy open Q-neighborhood U_i of a_{i
_λ} such that $U_{i}^{- 1} \leq V_{i}$ .

We show that

(P, T)

satisfies conditions (i) and (ii) in Definition 3.1.

(1) For all a = (a₁, …, a_n) , b = (b₁, …, b_n) in P and any fuzzy open Q-neighborhood W of any fuzzy point c_λ = (c₁, …, c_n) _λ = (c_{1_λ}, …, c_{n
_λ}) of a ∘ b = (a₁, …, a_n) ∘ (b₁, …, b_n) = (a₁ ∘ b₁, …, a_n ∘ b_n), where c_{1_λ} ∈ a₁ ∘ b₁, …, c_{n
_λ} ∈ a_n ∘ b_n, from the definition of product topology there are fuzzy open Q-neighborhoods W₁ of c_{1_λ}, …, W_n of c_{n
_λ} such that (W₁, …, W_n) ≤ W. Since $(P_{i}, T_{i})$ , i = 1, …, n are fuzzy topological polygroups, there must be fuzzy open Q-neighborhoods U_i of a_{i
_λ} and V_i of b_{i
_λ} such that U_iV_i ≤ W_i. It is obvious that (U₁, …, U_n) and (V₁, …, V_n) are fuzzy open Q-neighborhoods of (a₁, …, a_n) _λ and (b₁, …, b_n) _λ, respectively. We can easily verify that (U₁, …, U_n) (V₁, …, V_n) ≤ (W₁, …, W_n). There-fore, condition (i) in Definition 3.1 is satisfied.

(2) For all a = (a₁, …, a_n) ∈ P and any fuzzy open Q-neighborhood V of $a_{λ}^{- 1} = (a_{1}^{- 1}, \dots, a_{n}^{- 1})_{λ}$ , from the definition of product space there exist fuzzy open Q-neighborhoods V_i of $a_{i_{λ}}^{- 1}$ such that (V₁, …, V_n) ≤ V.

Since $(P_{i}, T_{i})$ , i = 1, …, n are fuzzy topological polygroups, it follows that there exists fuzzy open Q-neighborhoods U_i of a_{i
_λ} such that $U_{i}^{- 1} \leq V_{i}$ . Hence, $(U_{1}^{- 1}, \dots, U_{n}^{- 1}) \leq (V_{1}, \dots, V_{n}) \leq V$ . Since $(U_{1}^{- 1}, \dots, U_{n}^{- 1}) = (U_{1}, \dots, U_{n})^{- 1}$ and (U₁, …, U_n)is a fuzzy open Q-neighborhood of (a₁, …, a_n) _λ. Hence, condition (ii) in Definition 3.1 holds. Therefore, $(P, T)$ is a fuzzy topological polygroup. □

Example 9. Let ${(P_{i}, T_{i}) | i = 1, \dots, n}$ be a finite family of fuzzy topological polygroups, where $T_{i}$ , i = 1, …, n are discrete fuzzy topology on P_i, i = 1, …, n. Let $P = \prod_{i = 1}^{n} P_{i}$ and let $T$ be the product fuzzy topology in P by Definition 4.14. Then $(P, T)$ is a fuzzy topological polygroup.

Example 10. Let FTP $(P_{1}, T_{1})$ , where P₁ = {x, y} $\begin{matrix} \circ & x & y \\ x & x & y \\ y & y & {x, y} \end{matrix}$ $T_{1} = {\underline{0}, A, \underline{1}}$ such that $μ_{\underline{0}} (x) = μ_{\underline{0}} (y) = 0$ , μ_A (x) =0.8, μ_A (y) =0.6, $μ_{\underline{1}} (x) = μ_{\underline{1}} (y) = 1$ and FTP $(P_{2}, T_{2})$ , where P₂ = {e, a, b} $\begin{matrix} * & e & a & b \\ e & e & a & b \\ a & a & e & b \\ b & b & b & {e, a} \end{matrix}$ $T_{2} = {\underline{0}, B, \underline{1}}$ such that $μ_{\underline{0}} (e) = μ_{\underline{0}} (a) = μ_{\underline{0}} (b) = 0$ , μ_B (e) =0.9, μ_B (a) =0.7, μ_B (b) =0.5, $μ_{\underline{1}} (e) = μ_{\underline{1}} (a) = μ_{\underline{1}} (b) = 1$ . Then, $(P, T)$ is a fuzzy topological polygroup, where $P = \prod_{i = 1}^{2} P_{i} = P_{1} \times P_{2}$ the product polygroup and $T$ the product fuzzy topology in P by Definition 4.14.

Example 11. Let ${(P_{i}, T_{i}) | i = 1, \dots, n}$ be a finite family of fuzzy topological polygroups, where $T_{i}$ ,i = 1, …, n are the collection of all constant fuzzy sets in P_i, i = 1, …, n. Let $P = \prod_{i = 1}^{n} P_{i}$ and let $T$ be the product fuzzy topology in P by Definition 4.14. Then, $(P, T)$ is a fuzzy topological polygroup.

5 Conclusion

In this paper we presented together the structure of a fuzzy topological space and that of a fuzzy polygroup to form a combined structure, that of a fuzzy topological polygroup. In a group the composition of two elements is an element, meanwhile in a polygroup the composition of two elements is a non-empty set. So, the study of fuzzy topological polygroups is different than fuzzy topological groups.

Acknowledgment

The authors are highly grateful to referees for their valuable comments and suggestions for improving the paper.

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