Fuzzy topological F -polygroups

Abstract

This paper deal with certain algebraic systems called F-polygroups. We introduce and study the concept of fuzzy topological F-polygroups and prove some related properties. In particular, the fuzzy topological isomorphism theorems of fuzzy topological F-polygroups are proved.

Keywords

isomorphisms theorems

1 Introduction

Hyperstructure theory has been introduced by Marty in [28]. Hypergroups are the largest class of multi-valued systems that satisfy the axioms similar to group. Since then many researchers have studied in this field and developed it, for example see [4–9 , 26]. One of the motivations for the study of hyperstructures comes from chemical reactions. In [17], Davvaz and Dehghan-Nezhad provided examples of hyperstructures associated with chain reactions. In [14], Davvaz et al. introduced examples of weak hyperstructures associated with dismutation reactions. In [18], Davvaz et al. investigated the examples of hyperstructures and weak hyperstructures associated with redox reactions. Also, see [15 , 19] for more applications of hyperstructures in chemistry.

An important special kind of hypergroups are polygroups. This class of hypergroups were studied by Comer [6]. There exists a rich bibliography: publications appeared within 2012 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.

The notion of fuzzy set was introduced by Zadeh in 1965 [31]. Rosenfeld defined the concept of a fuzzy subgroup of a given group [29]. Since then many researchers worked in this area, for example see [35 –38]. The connections between the fuzzy sets and algebraic hyperstructures have been considered by Corsini, Davvaz, Leoreanu, Zahedi and others. Zahedi et al. [34] introduced the notion of fuzzy subpolygroups of a polygroup, also see [13, 20]. Also, the notion of fuzzy polygroup (F-polygroup), has been introduced and studied by Zahedi and Hasankhani [32, 33].

In this paper, first we introduce the notions of fuzzy topological F-polygroup. Then, we state isomorphism theorems for fuzzy topological F-polygroups.

In [24], Heidari et al. introduced the notion of topological polygroups. Then in [1 –3] Abbasizadeh et al. investigated the notion of fuzzy topological polygroups.

We recall some basic definitions and results to be used in the sequel.

Let H be a non-empty set. Then a mapping $\circ : H \times H ⟶ P^{*} (H)$ is called a hyperoperation, where $P^{*} (H)$ is the family of non-empty subsets of H. The couple (H, ∘) is called a hypergroupoid. In the above definition, if A and B are two non-empty subsets of H and x ∈ H, then we define $\begin{matrix} A \circ B & = & ⋃_{\binom{a \in A}{b \in B}} a \circ b, \\ x \circ A & = & {x} \circ A and A \circ x = A \circ {x} . \end{matrix}$

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 [8, 28].

A special subclass of hypergroups is the class of polygroups. We recall the following definition from [6]. A polygroup is a system P = 〈P, ∘, e, ^-1〉, where $\circ : P \times P ⟶ P^{*} (P)$ , 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.

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 [10 , 23].

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 $\circ : P \times P ⟶ P^{*} (P)$ and ^-1 : P ⟶ P are continuous (see [24, 25]).

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 (see [1 –3]):

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: $U • V \leq W .$

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: $U^{- 1} \leq V .$

2 Preliminaries

For the sake of convenience and completeness of our study, in this section some basic definition and results of [12 , 33], which will be needed in the sequel are recalled here.

Throughout this paper, I is the unit interval [0, 1] and we denote the set of all fuzzy subsets of A by I^A. If μ ∈ I^A, then by supp (μ) we mean the set {x ∈ A | μ (x) ≠0}. For any subset A of X, we denote χ_A the characteristic function of A. Let A be a non-empty set and $I_{*}^{A} = I^{A} \ {0}$ . Then, by an F-hyperoperation “*” on A we mean a function from A × A to $I_{*}^{A}$ , in other words for any a, b ∈ A, a * b is a non-empty fuzzy subset of A. If $μ, η \in I_{*}^{A}$ , then μ * η is defined by $μ * η = ⋃_{\binom{x \in supp (μ)}{y \in supp (η)}} x * y .$

Let $μ \in I_{*}^{A}$ , $B \in P^{*} (A)$ and a ∈ A. Then,

a * μ and μ * a denote χ_{a} * μ and μ * χ_{a}, respectively.

a * B, B * a, μ * B and B * μ denote χ_{a} * χ_B, χ_B * χ_{a}, μ * χ_B and χ_B * μ, respectively (see [12]).

Definition 2.1. [32] Let P be a non-empty set and * be an F-hyperoperation on P. Then, (P, *) is called an F-polygroup if

(x * y) * z = x * (y * z), for all x, y, z ∈ P.

There exists an element e ∈ P such that x ∈ supp (x * e ∩ e * x), for all x ∈ P (in this case we say that e is an F-identity element of P).

For each x ∈ P, there exists a unique element x′ ∈ P such that e ∈ supp (x * x′ ∩ x′ * x) (x′ is called the F-inverse of x and is denoted by x^-1).

z ∈ supp (x * y) ⇒ x ∈ supp (z * y^-1) ⇒ y ∈ supp (x^-1 * z), for all x, y, z ∈ P.

Definition 2.2. [21] Let (P, *) be an F -polygroup and let S be a non-empty subset of P. Then, S is called an F-subpolygroup if

e ∈ S.

supp (a * b) ⊆ S for all a, b ∈ S.

x ∈ S implies x^-1 ∈ S.

Notice that condition (2) of Definition 2.2 is equivalent to a * b ≤ χ_S for all a, b ∈ S.

Definition 2.3. [21] An F-subpolygroup S of (P, *) is said to be normal F- subpolygroup in P if for every x ∈ P, supp (x * (S * x^-1)) ⊆ S.

Corollary 2.4. [12] Let $μ \in I_{*}^{P}$ and z ∈ P. Then, $\begin{matrix} supp (μ * z) & = & ⋃_{t \in supp (μ)} supp (t * z), \\ supp (z * μ) & = & ⋃_{t \in supp (μ)} supp (z * t) . \end{matrix}$

Corollary 2.5. [12] Let $μ_{1}, μ_{2} \in I_{*}^{P}$ . Then, $supp (μ_{1} * μ_{2}) = ⋃_{\binom{x \in supp (μ_{1})}{y \in supp (μ_{2})}} supp (x * y) .$

Corollary 2.6. [12] Let (P, *) be an F-polygroup. Then,

e^-1 = e and e is unique. Moreover, supp (e * e) = {e}.

(x^-1) ^-1 = x, for all x ∈ P.

⋃_{x∈supp(μ₁)}x * μ₂ = μ₁ * μ₂ = ⋃ _{y∈supp(μ₂)}μ₁ * y, for all $μ_{1}, μ_{2} \in I_{*}^{P}$ .

(μ₁ * μ₂) * μ₃ = μ₁ * (μ₂ * μ₃), for all $μ_{1}, μ_{2}, μ_{3} \in I_{*}^{P}$ .

Definition 2.7. [12] Let P₁, P₂ be two F-polygroups and f : P₁ ⟶ P₂ be a function such that f (e) = e′.

f is called a homomorphism if f (x * y) ≤ f (x) * ′f (y), for all x, y ∈ P₁.

f is called a strong homomorphism if f (x * y) = f (x) * ′f (y), for all x, y ∈ P₁.

If f is a homomorphism, then we have (f (x)) ^-1 = f (x^-1), for all x ∈ P₁.

Definition 2.8. 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∈J, 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.

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 neighborhood of x_λ is called the system of neighborhood of fuzzy point x_λ.

3 Fuzzy topological F-polygroup

First let us recall that a base for a fuzzy topological space $(X, T)$ is a sub-collection $B$ of $T$ such that each member A of $T$ can be written as the union of members of $B$ .

Proposition 3.1. [27] A family $B$ of fuzzy sets in X is a base for a fuzzy topology $T$ on X if and only if it satisfies the following conditions:

For any $A_{1}, A_{2} \in B$ , we have $A_{1} \cap A_{2} \in B$ .

$⋃_{A \in B} A = \underline{1}$ .

Conversely, any system of sets fulfilling (B1) and (B2) determines a unique fuzzy topology $T$ on X such that this system is a base of $T$ .

Lemma 3.2.Let $(P, T)$ be a fuzzy topological space. Then, the family $U$ consisting of all sets $S_{V} = {U \in I^{P} | U \leq V, V \in T}$ is a base for a fuzzy topology on I^P. This fuzzy topology is denoted by $T^{*}$ .

Let $(P, T)$ be a fuzzy topological space. Then, we consider the product fuzzy topology on P × P and the fuzzy topology $T^{*}$ on I^P.

Proof. Let $S_{V_{1}}, S_{V_{2}} \in U$ , $V_{1}, V_{2} \in T$ . Evidently, S_{V
₁} ∩ S_{V
₂} = S_V₁∩V₂ as $V_{1} \cap V_{2} \in T$ . Thus axiom (B1) of the base is fulfilled. Further, $\underline{1} \in T$ and $S_{P} = I_{*}^{P}$ , so axiom (B2) of the base is fulfilled, too. □

Definition 3.3. Let (P, *) be an F-polygroup and $(P, T)$ be a fuzzy topological space. Then, the system $(P, *, T)$ is called a fuzzy topological F- polygroup if the following conditions hold:

The mapping (x, y) ⟶ x * y from P × P to $I_{*}^{P}$ is fuzzy continuous.

The mapping x ⟶ x^-1 from P onto itself is fuzzy continuous.

Lemma 3.4. Let (P, *) be an F-polygroup and $T$ be a fuzzy topology on P. Then, the following assertions hold:

The mapping (x, y) ⟶ x * y is fuzzy continuous if and only if for every x, y ∈ P and $U \in T$ such that x * y ≤ U, there exist $V, W \in T$ such that x ∈ suppV, y ∈ suppW and V * W ≤ U.

The mapping x ⟶ x^-1 is fuzzy continuous if and only if whenever $U \in T$ is fuzzy open, then U^-1 is fuzzy open.

Proof. (1) Suppose that the fuzzy hyperoperation * : P × P ⟶ I^P is fuzzy continuous and x * y ≤ U for x, y ∈ P and $U \in T$ . Then, *^-1 (S_U) is a fuzzy open subset of P × P. So there exist fuzzy open subsets V and W of P such that (x, y) ∈ supp (V × W) and V × W ≤ * ^-1 (S_U). Thus, * (V × W) ≤ * (* ^-1 (S_U)) ≤ S_U. Therefore, we have $V * W = ⋃_{\binom{v \in supp (V)}{w \in supp (W)}} v * w = ⋃_{\binom{v \in supp (V)}{w \in supp (W)}} * (v, w) \leq U .$

For the converse, suppose that $U \in T$ and (x, y) ∈ supp (* ^-1 (S_U)). We show that *^-1 (S_U) (x, y) >0. Since x * y ≤ U, it follows that there exist $V, W \in T$ such that V * W ≤ U where x ∈ suppV and y ∈ suppW. So, we have (x, y) ∈ supp (V × W) and V × W ≤ * ^-1 (S_U). Thus, *^-1 (S_U) is fuzzy open in P. Now, if $A \in T^{*}$ , then A = ⋃ _U∈ΛS_U, where Λ is a non-empty subset of $T$ . Thus, $*^{- 1} (A) = *^{- 1} (⋃_{U \in Λ} S_{U}) = ⋃_{U \in Λ} *^{- 1} (S_{U}) .$ Hence, *^-1 (A) is fuzzy open in P. Therefore, the mapping * is fuzzy continuous. (2) It is straightforward. □

Let (P, *) be an F-polygroup and let θ be an equivalence relation on P. If A and B are non-empty subsets of P, then

we write $A \bar{θ} B$ if for every a ∈ A, there exists b ∈ B such that aθb and for every b ∈ B there exists a ∈ A such that aθb.

$A \bar{\bar{θ}} B$ means that for every a ∈ A and for every b ∈ B, we have aθb.

An equivalence relation θ defined on an F-polygroup (P, *) is called regular if for every x₁, x₂, y₁, y₂ ∈ P, x₁θy₁, x₂θy₂ implies that $supp (x_{1} * x_{2}) \bar{θ} supp (y_{1} * y_{2})$ and θ is called strongly regular if x₁θy₁, x₂θy₂ implies that $supp (x_{1} * x_{2}) \bar{\bar{θ}} supp (y_{1} * y_{2})$ .

Theorem 3.5. Let (P, *) be an F-polygroup and let θ be a regular relation on P. Then, P/θ = {θ [x] | x ∈ P} is an F-polygroup with the F-hyperoperation *_θ and unitary operation ^-1 on P defined as follows: $\begin{matrix} *_{θ} : P / θ \times P / θ ⟶ I_{*}^{P / θ} \\ (θ [a], θ [b]) \mapsto χ_{{θ [c] | c \in supp (a * b)}} \end{matrix}$ and $\begin{matrix} ^{- 1} : P / θ ⟶ P / θ \\ θ [a] \mapsto θ [a^{- 1}], \end{matrix}$ for all a, b ∈ P.

Proof. The proof is similar to the proof of Proposition 3.1 in [21]. □

Let S be an F-subpolygroup of an F-polygroup (P, *). We define the relation S^* on P as follows: $x S^{*} y if and only if x \in supp (S * y) .$

Lemma 3.6. [21] The relation S^* is a strongly regular relation.

Let S be a normal F-subpolygroup of an F-polygroup (P, *). Since S^* is a strongly regular relation, it follows that P/S^* is an F-polygroup with F-operation *_{S
^*} defined as follows: $S^{*} [a] *_{S^{*}} S^{*} [b] = χ_{{S^{*} [c]}} {for all c \in supp (a * b)$ and so P/S^* is a quotient F-group.

Definition 3.7. [30] Let X be a fuzzy topological space and ∼ be an equivalence relation on X. For every x ∈ X, denote by [x] its equivalence class. Then quotient space of X modulo ∼ is given by the set X/∼ = {[x] | x ∈ X}. We have the projection map p : X⟶ X/∼, x ↦ [x], and we equip X/∼ by the topology: U is a fuzzy open set of X/∼ if and only if p^-1 (U) is a fuzzy open subset of X.

Example 1. Let P be a F-polygroup with the fuzzy hyperoperation x * y = χ_P for all x, y ∈ P. Then, P with every arbitrary fuzzy topology $T$ is a fuzzy topological F-polygroup. We have the projection map p : P ⟶ P/S^* by x ↦ [x]. Let $U$ be the family of fuzzy sets in P/S^* defined by $U = {B ↾ p^{- 1} (B) \in T}$ . Then $U$ is a fuzzy topology, called the quotient topology for P/S^*, and $(P / S^{*}, U)$ is called the quotient fuzzy topological space.

Definition 3.8. Let (P, *) be an F-polygroup and A be a non-empty subset of P. We say that A is a complete part of P if for any non-zero natural number n and for all a₁, …, a_n of P, the following implication holds: $A \cap supp (\prod_{i = 1}^{n} a_{i}) \neq \emptyset \Rightarrow supp (\prod_{i = 1}^{n} a_{i}) \subseteq A .$

Example 2. If (P, *) is an F-polygroup and S is a strongly regular relation on P, then for all z of P, the equivalence class of z is a complete part of P.

Lemma 3.9. Let $(P, *, T)$ be a fuzzy topological F-polygroup such that supp every fuzzy open subset of P is a complete part. Then, the natural mapping π : P ⟶ P/S^* is a fuzzy open mapping.

Proof. Suppose that V is fuzzy open in P. We should prove that π^-1 (π (V)) is a fuzzy open subset in P. Let π^-1 (π (V)) (x) >0, then π (V) (π (x)) >0. So there exists v ∈ suppV such that π (x) = π (v). Since V is fuzzy open, it follows that there exists a fuzzy open subset U of P such that U (v) >0, U ≤ V. On the other hand, π (x) = π (v) implies that xS^*v, so x ∈ supp (S ∗ v) and {x, v} ⊆ supp (S ∗ v).

Now, we have v ∈ supp (S ∗ v) ∩ suppU and suppU is complete part so x ∈ supp (S ∗ v) ⊆ suppU. Hence, U (x) >0, U ≤ π^-1 (π (V)). Thus, π^-1 (π (V)) (x) >0 so π^-1 (π (V)) is fuzzy open in P. Therefore, π (V) is fuzzy open in P/S^*. □

Theorem 3.10. Let $(P, *, T)$ be a fuzzy topological F-polygroup such that supp every fuzzy open subset of P is a complete part. Then, $(P / S^{*}, *_{S^{*}}, \bar{T})$ is a fuzzy topological F-group, where S^* [a] * _{S
^*}S^* [b] = χ_{S^*[c]} for every c ∈ supp (a * b).

Proof. By Lemma 3.6, (P/S^*, ∗ _{S
^*}) is an F-group. We show that the mappings (π (x), π (y)) ↦ π (x) ∗ _{S
^*}π (y) and π (x) ↦ π (x) ^-1 are fuzzy continuous.

Suppose that A is fuzzy open in P/S^* such that π (x) ∗ _{S
^*}π (y) ∈ A. So x ∗ y ≤ π^-1 (A). Since π^-1 (A) is fuzzy open in P, it follows that there exist fuzzy open subsets V and W of P such that x ∈ suppV, y ∈ suppW and V ∗ W ≤ π^-1 (A). Thus, π (V) ∗ _{S
^*}π (W) ≤ A. Also, π (V) and π (W) are fuzzy open in P/S^*. Hence, ∗_{S
^*} is fuzzy continuous.

Now, we prove that the inverse mapping is fuzzy continuous. Suppose that A is fuzzy open in P/S^* such that A^-1 (π (x)) >0, and let π (e) be the identity element of P/S^*. So, there exists fuzzy open subset V in P such that V (x) >0. Thus π (V) (π (x)) >0 and π (V) ≤ A^-1. Therefore, A^-1 is fuzzy open in P/S^*. □

4 Isomorphism theorems

In this section we state and prove the isomorphism theorems for fuzzy topological F-polygroups.

Let $(P, *, T)$ be a fuzzy topological F-polygroup and S be a normal F-subpolygroup of P. Let π be the natural mapping x ↦ S * x of P onto P/S. Then, $(P / S, \bar{T})$ is a fuzzy topological space, where $\bar{T}$ is the quotient fuzzy topology induced by π. That is for every subset X of P we have {S * x | x ∈ X} is a fuzzy open subset of P/S if and only if π^-1 ({S * x | x ∈ X}) is a fuzzy open subset of P. In the following, the notation X/S is used for {S * x | x ∈ X} for every subset X of P.

Definition 4.1. Let $(P_{1}, *_{1}, T_{1})$ and $(P_{2}, *_{2}, T_{2})$ be fuzzy topological F-polygroups. A mapping φ from P₁ into P₂ is said to be a fuzzy strong topological homomorphism if for all a, b ∈ P₁,

φ (e₁) = e₂,

φ (supp (a * ₁b)) = supp (φ (a) * ₂φ (b)),

φ is fuzzy continuous,

φ is fuzzy open.

Clearly, a fuzzy strong topological homomorphism φ is a fuzzy topological isomorphism if φ is injective and onto homomorphism. We say that P₁ is fuzzy topological isomorphic to P₂, denoted by P₁ ≅ P₂, if there exists a fuzzy topological isomorphic from P₁ to P₂.

Because P₁ is an F-polygroup, e₁ ∈ supp (a * a^-1) for all a ∈ P₁, then we have φ (e₁) ∈ supp (φ (a) * φ (a^-1)) or e₂ ∈ supp (φ (a) * φ (a^-1)) which implies φ (a^-1) ∈ supp (φ (a) ^-1 * e₂), therefore, φ (a^-1) = φ (a) ^-1 for all a ∈ P₁. Moreover, if φ is a fuzzy strong topological homomorphism from P₁ into P₂, then the kernel of φ is the set kerφ = {x ∈ P₁ | φ (x) = e₂}. It is trivial that kerφ is an F-subpolygroup of P₁ but in general is not normal in P₁.

As in F-polygroups, if φ is a fuzzy strong topological homomorphism from P₁ into P₂, then, φ is injective if and only if kerφ = {e₁}.

Example 3. Let P = {a, b} with the F-hyper- operation $\begin{matrix} * & a & b \\ a & \frac{a}{0.7}, \frac{b}{0} & \frac{a}{0}, \frac{b}{0.3} \\ b & \frac{a}{0}, \frac{b}{0.3} & \frac{a}{0.7}, \frac{b}{0} \end{matrix}$

be an F-polygroup. Consider on P the discrete fuzzy topology $T$ . Clearly, the map $φ : (P, T) ⟶ (P, T)$ by φ (x) = a where x ∈ P, is a fuzzy strong topological homomorphism.

Theorem 4.2. Let S be a normal F-subpolygroup of fuzzy topological F-polygroup P and let P/S = {x * S | x ∈ P}. We define the F-hyperoperation ⊙ on P/S as follows: $\begin{matrix} ⊙ : P / S \times P / S ⟶ I_{*}^{P / S} \\ (x * S, y * S) \mapsto x * S ⊙ y * S, \end{matrix}$ where (x ∗ S ⊙ y ∗ S) (z ∗ S) = (x ∗ y ∗ S) (z), for all z ∗ S ∈ P/S. Then, (P/S, ⊙) is a fuzzy topological F-polygroup.

Proof. We prove that the F-hyperoperation ⊙ and the unitary operation ^-1 are fuzzy continuous. Suppose x ∗ S, y ∗ S ∈ P/S and A is a fuzzy open subset of P/S such that x ∗ S ⊙ y ∗ S ≤ A. Then x ∗ y ≤ π^-1 (A). Since π^-1 (A) is fuzzy open in P, it follows that there exist fuzzy open subsets V and W of P such that x ∈ suppV, y ∈ suppW and V ∗ W ≤ π^-1 (A). It follows that π (V) and π (W) are fuzzy open in P/S such that x ∗ S ∈ supp π (V), y ∗ S ∈ supp π (W) and π (V) ⊙ π (W) ≤ A. Therefore, the F-hyperoperation ⊙ is fuzzy continuous.

Suppose that A (x^-1 ∗ S) >0. Then, π^-1 (A) (x^-1) >0. Thus, there exists a fuzzy open subset U of P such that U^-1 (x^-1) >0 and U^-1 ≤ π^-1 (A). So π (U^-1) (x^-1 ∗ S) >0 and π (U^-1) ≤ A. Therefore, π (U^-1) is fuzzy open in P/S. □

Example 4. We consider a group G with order greater than 1 and x² = e, for all x ∈ G. Let t ∈ (0, 1). We define the F-hyperoperation * on G by $(x * y) (z) = {\begin{matrix} e_{1} (xyz) & if z = e, \\ e_{t} (xyz) & if z \neq e . \end{matrix}$ where e₁, e_t are fuzzy points of G. Clearly, $(G, *, T)$ where $T$ is a discrete fuzzy topology, is a fuzzy topological F-polygroup. Now, suppose that N = {e}. It is easy to see that N is a normal F-subpolygroup. Let P/N = {x * N | x ∈ P} and let ⊙ be the F-hyperoperation on P/N by Theorem 4.2. Then, (P/N, ⊙) is a fuzzy topological F-polygroup.

The isomorphism theorems of Fⁿ-polygroups are presented in [21]. In the following we prove them for fuzzy topological F-polygroups.

Theorem 4.3. Let $(P_{1}, *_{1}, T_{1})$ and $(P_{2}, *_{2}, T_{2})$ be fuzzy topological F-polygroups such that supp every fuzzy open subset of P₁ is a complete part. Let φ be a fuzzy open and continuous fuzzy strong topological homomorphism from P₁ onto P₂ such that K = kerφ is a normal F-subpolygroup of P₁. Then, P₁/K^* and P₂ are fuzzy topologically isomorphic.

Proof. We consider the map ψ : P₂ ⟶ P₁/K^* by ψ (x₂) = supp (K ∗ x₁) where, φ (x₁) = x₂, for all x₂ ∈ P₂. Since φ is onto, it follows that φ^-1 (x₂)≠ ∅. If x₁, y₁ ∈ φ^-1 (x₂), then φ (x₁) = x₂ = φ (y₁). Thus, $e_{2} \in supp (φ (x_{1}) *_{2} φ (y_{1}^{- 1}))$ , hence there exists $k \in supp (x_{1} * y_{1}^{- 1})$ such that φ (k) = e₂. Now, we have $\begin{matrix} supp (K *_{1} x_{1}) & \subseteq & supp (K *_{1} (k *_{1} y_{1})) \\ = & supp (K *_{1} y_{1}) \\ \subseteq & supp (K *_{1} k^{- 1} *_{1} x_{1}) \\ = & supp (K *_{1} x_{1}) . \end{matrix}$

Therefore, ψ is well-defined. Obviously, ψ is onto and an algebraic homomorphism. If ψ (x₂) = supp (K ∗ ₁x₁) = ψ (y₂) = supp (K ∗ ₁y₁), then, x₁ ∈ supp (k ∗ ₁y₁) for some k ∈ K. Thus, x₂ = φ (x₁) ∈ supp (φ (k) ∗ ₂φ (y₁)) = y₂, hence ψ is one- to-one. Therefore, ψ is an algebraic isomorphism.

Now, we show that ψ is fuzzy open and fuzzy continuous. Suppose that U₂ is a fuzzy open subset of P₂. Then, ψ (U₂) = supp (K ∗ ₁u₁) = φ^-1 (U₂)/K^* where u₁ ∈ φ^-1 (U₂). Since φ is fuzzy continuous, it follows that φ^-1 (U₂)/K^* is fuzzy open in P₁/K^*. Therefore, ψ is fuzzy open.

If U₁/K^* is a fuzzy open subset of P₁/K^*, then ψ^-1 (U₁/K^*) is fuzzy open in P₂ since φ is fuzzy open and we have $\begin{matrix} ψ^{- 1} (U_{1} / K^{*}) \\ = {u_{2} | ψ (u_{2}) \in U_{1} / K^{*}} \\ = {u_{2} | supp (K *_{1} u_{1}) \in U_{1} / K^{*}, φ (u_{1}) = u_{2}} \\ = φ (U_{1}) . \end{matrix}$

Therefore, ψ is fuzzy continuous and the proof is complete. □

Theorem 4.4. Let S₁ and S₂ be F-subpolygroups of a polygroup P, S₂ normal F-subpolygroup and S₁ fuzzy open in P. If supp every fuzzy open subset of P is a complete part, then, supp (S₁ ∗ e)/(supp (S₁ ∗ e) ∩ S₂) ^* and $supp (S_{1} * S_{2}) / S_{2}^{*}$ are fuzzy topologically isomorphic.

Proof. We consider the map $φ : S_{1} ⟶ P / S_{2}^{*}$ by φ (s₁) = supp (S₂ ∗ s₁). Then, φ is a fuzzy strong homomorphism and kerφ = (supp (S₁ ∗ e) ∩ S₂) ^*. Since S₁ ⊆ supp (S₂ ∗ S₁) and φ is the restriction of π on S₁, it follows that φ is fuzzy open and continuous. It remains to show that $Im (φ) = supp (S_{1} * S_{2}) / S_{2}^{*}$ . If x ∈ supp (S₂ ∗ S₁), then x ∈ supp (s₂ ∗ s₁), for some s₂ ∈ supp (S₂) and s₁ ∈ supp (S₁). Hence, φ (s₁) = supp (S₂ ∗ s₁) = supp (S₂ ∗ s₂ ∗ s₁) = supp (S₂ ∗ x₁). So $supp (S_{2} * S_{1}) / S_{2}^{*} \subseteq Im (S_{1}) \subseteq supp (S_{2} * S_{1}) / S_{2}^{*}$ . Therefore, by Theorem 4.3, $\begin{matrix} supp (S_{1} * e) / (supp (S_{1} * e) \cap S_{2})^{*} \\ ≅ supp (S_{1} * S_{2}) / S_{2}^{*} . □ \end{matrix}$

Theorem 4.5. Let S, J be normal F-subpolygroups of (P, ∗) such that supp every fuzzy open subset of P is a complete part and S ⊆ J. Then, (P/S^*)/(J/S^*) ^* and P/J^* are fuzzy topological isomorphic.

Proof. We consider the map φ : P/S^* ⟶ P/J^* by φ (supp (S ∗ x)) = supp (J ∗ x). Then, φ is a fuzzy strong homomorphism and kerφ = J/S^*. If U is a fuzzy open subset of P, then we have φ (U/S^*) = U/J^*. Therefore, φ is fuzzy open and continuous. So by Theorem 4.3 we conclude that (P/S^*)/(J/S^*) ^* and P/J^* are fuzzy topological isomorphic. □

Theorem 4.6. If S₁ and S₂ are normal F-subpolygroups of P₁ and P₂ respectively, then S₁ × S₂ is a normal F-subpolygroup of P₁ × P₂ and (P₁ × P₂)/(S₁ × S₂) ^* and $P_{1} / S_{1}^{*} \times P_{2} / S_{2}^{*}$ are fuzzy topological isomorphic.

Proof. It is straightforward. □

5 Conclusion

This paper contributed to the study of the concept of fuzzy topological F-polygroups by extending the traditional concept and several properties are proved. We hope that this paper would offer foundation for further study of the theory on fuzzy topological polygroups.

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