A new model of fuzzy topology: I -fuzzy topological polygroups

Abstract

This paper provides a new connection between algebraic hyperstructures and fuzzy sets. We present the concept of I-fuzzy topological polygroups and prove some properties. The concept of I-fuzzy topological polygroups is a generalization of the concept of I-fuzzy topological groups. By considering the relative fuzzy topology on fuzzy subpolygroups we prove some related properties.

Keywords

Polygroup

1 Introduction

The hyperstructure theory was born in 1934 when Marty introduced the notion of hypergroup [24]. In 1971, Rosenfeld [27] introduced the notion of fuzzy groups. Then this concept is studied by many authors, for example see [2–4 , 33]. In 1979, Foster [14] introduced the concept of fuzzy topological group. Ma and Yu [21] 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, also see [16 , 22]. On the other hand, in the last few decades, many connections between hyperstructures and fuzzy sets has been established and investigated. Inspired and motivated by the above achievements, in this paper we shall introduce and study the notion of I-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 $\circ : H \times H \to P^{*} (H)$ 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, 24].

For all n > 1, we define the relation β_n on a semihypergroup H, as follows: a β_n b ⇔ ∃ (x₁, …, x_n) ∈ Hⁿ: ${a, b} \subseteq \prod_{i = 1}^{n} x_{i}$ , and $β = ⋃_{n \geq 1} β_{n},$ where β₁ = {(x, x) | x ∈ H} is the diagonal relation on H. This relation was introduced by Koskas [19] and studied mainly by Corsini, Davvaz, Freni, Leoreanu, Vougiouklis and many others. Suppose that β* is the smallest equivalence relation on a hypergroup (semihypergroup) H such that the quotient H/β* is a group (semigroup). If H is a hypergroup, then β=β* [15]. The relation β* is called the fundamental relation on H and H/β* is called the fundamental groups. A special subclass of hypergroups is the class of polygroups. We recall the following definition from [7]. A polygroup is a system P = 〈P, ∘, e, ⁻¹〉, where $\circ : P \times P \to P^{*} (P)$ , 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 [1 , 18].

2 Preliminaries

For the sake of convenience and completeness of our study, in this section some basic definition and results of [5 , 32], 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 A: X → [0, 1] which associates with each point x ∈ X its grade or degree of membership A (x) ∈ [0, 1]. That is, an element of I^X. We denote by I^X the set of all fuzzy sets on X.

Definition 2.1. [28] An I-fuzzy topology on a set X is a map $T : I^{X} \to I$ such that:

$T (\underline{1}) = T (\underline{0}) = 1$ .

For all U, V ∈ I^X, $T (U \land V) \geq T (U) \land T (V)$ .

For all U_j ∈ I^X, j ∈ Λ, $T (\underset{j \in Λ}{\lor} U_{j}) \geq \underset{j \in Λ}{\land} T (U_{j})$ .

The pair (X,

T

) is called an I-fuzzy topological space and is denoted by I - FTS for short.

T : I^{X} \to I

is called stratified if, in addition to the above,

T (\underline{λ}) = 1

for all λ ∈ I and

(X, T)

is called a stratified I-fuzzy topological space. Throughout the next we consider stratified I-fuzzy topology.

Example 1. Let X be a non-empty set. For any A ∈ I^X, define $T (A) = 1$ , then $T$ is an I-fuzzy topology on X.

Example 2. Let X be a non-empty set. Define $T : I^{X} \to I$ by $T (A) = {\begin{matrix} 1 & if A = \underline{0} or \end{matrix} A = \underline{1}, 0.7 otherwise .$ Then, $T$ is an I-fuzzy topology on X.

A fuzzy set in X is called a fuzzy point iff 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 ≤ A (x). Evidently, every fuzzy set A can be expressed as the union of all the fuzzy points which belong to A. A fuzzy point x_λ is said to be quasi-coincident with a fuzzy set A, denoted by $x_{λ} \hat{q} A$ , if A (x) + λ > 1. Relation "does not quasi-coincide with" or "is not quasi-coincident with" is denoted by $\neg \hat{q}$ . A map $f : (X, T) \to (Y, δ)$ is called fuzzy continuous with respect to I-fuzzy topology $T$ and δ if $T (f^{\leftarrow} (U)) \geq δ (U)$ for all U ∈ I^Y, where f^← is denoted by f^← (U) (x) = U (f (x)). A map $f : (X, T) \to (Y, δ)$ is called fuzzy open if $T (U) \leq δ (f^{\to} (U))$ for all U ∈ I^X, where f^→ (U) (y) = ⋁ _f(x)=yU (x). A function $f : (X, T) \to (Y, δ)$ is called fuzzy homeomorphism if and only if f is bijective and f is both fuzzy continuous and fuzzy open.

Definition 2.2. [12] An I-fuzzy quasi-coincident neighborhood system on X is a set $Q = {Q_{x_{λ}} : x_{λ} \in p t (I^{X})}$ of maps $Q_{x_{λ}} : I^{X} \to I$ such that for all U, V ∈ I^X,

(I-FQ1) $Q_{x_{λ}} (1_{X}) = 1, Q_{x_{λ}} (0_{X}) = 0$ ,

(I-FQ2) $Q_{x_{λ}} (U) > 0 \Rightarrow x_{λ} \hat{q} U$ ,

(I-FQ3) $Q_{x_{λ}} (U \land V) = Q_{x_{λ}} (U) \land Q_{x_{λ}} (V)$ ,

(I-FQ4) $Q_{x_{λ}} (U) = \underset{_{x_{λ} \hat{q} V \leq U}}{\lor} \underset{_{y_{μ} \hat{q} V}}{\land} Q_{y_{μ}} (V)$ .

A set X equipped with an I-fuzzy quasi-coincident neighborhood system

Q = {Q_{x_{λ}} : x_{λ} \in pt (I^{X})}

, denoted

(X, Q)

, is called an I-fuzzy quasi-coincident neighborhood space (I - fqn, in short).

A continuous mapping between two I-fuzzy quasi-coincident neighbohood spaces $(X, P) \to (Y, Q)$ is a map f: X → Y such that for all x_λ ∈ pt (I^X), U ∈ I^Y, $Q_{f^{\to} (x_{λ})} (U) \leq P_{x_{λ}} (f^{\leftarrow} (U))$ , where f^→ (x_λ) = f (x) _λ. The category of I-fuzzy quasi-coincident neighborhood spaces and its continuous maps is denoted by I - FQN. Suppose that $T : I^{X} \to I$ is an I-fuzzy topology on X. For all x_λ ∈ pt (I^X), U ∈ I^X, let $Q_{x_{λ}}^{T} (U) = {\begin{matrix} ⋁_{x_{λ} \hat{q} V \leq U} T (V) & if x_{λ} \hat{q} U, \\ 0 & if x_{λ} \neg \hat{q} U . \end{matrix}$ Then, we have

Lemma 2.3. [12] The set of $Q^{T} = {Q_{x_{λ}}^{T} : x_{λ} \in pt (I^{X})}$ is an I-fuzzy quasi-coincident neighborhood system on X, called I-fuzzy quasi-coincident neighborhood system by $T$ .

Lemma 2.4. [12] Suppose a function $f : (X, T) \to (Y, δ)$ is a fuzzy continuous mapping between I-fuzzy topological spaces, then $f : (X, Q^{T}) \to (Y, Q^{δ})$ is also fuzzy continuous with respect to the induced I-fuzzy quasi-coincident neighborhood system.

Let $Q = {Q_{x_{λ}} : x_{λ} \in pt (I^{X})}$ be an I-fuzzy quasi-coincident neighborhood system on X. Define a mapping $T^{Q} : I^{X} \to I$ such that for every U ∈ I^X, $T^{Q} (U) = ⋀_{x_{λ} \hat{q} U} Q_{x_{λ}} (U)$ . Then, we have

Lemma 2.5. [12] The map $T^{Q} : I^{X} \to I$ defined above is an I-fuzzy topology on X, called the induced I-fuzzy topology by $Q = {Q_{x_{λ}} : x_{λ} \in pt (I^{X})}$ .

Lemma 2.6. [12] If $f : (X, P) \to (Y, Q)$ is a fuzzy continuous map between I-fuzzy quasi-coincident neighborhood spaces, then f is fuzzy continuous with respect to the induced I-fuzzy topologies.

Definition 2.7. [29] Let $(X_{j}, T_{j})_{j \in varLambda}$ be a family of I-fuzzy topological spaces, $X = \prod_{j \in varLambda} X_{j}$ and π_j: X → X_j the j-th projection. The product I-fuzzy topology $\prod T_{j}$ on X is the weakest I-fuzzy topology on X for which each π_j is fuzzy continuous.

Theorem 2.8. [29] Let X_j, X, π_j, j ∈ varLambda be as in the preceding definition. Then,

If $T = \prod_{j \in Λ} T_{j}$ , then $T =_{\in varLambda} π_{\leftarrow} (T)$ .

For x_λ ∈ pt (I^X), A ∈ I^X, we have $\begin{matrix} Q_{x_{λ}}^{T} (A) = ⋁ {⋀_{j \in J} Q_{x_{λ}^{j}} (A^{j}) : \\ J \subset varLambda is a finite set, A^{j} \in I^{X_{j}}, ⋀_{j \in J} π_{j}^{\leftarrow} (A^{j}) \leq A} . \end{matrix}$

If $(Y, T^{*})$ is an I-fuzzy topological space, then a function g: Y → X is fuzzy continuous if and only if each composition π_j ∘ g, j ∈ varLambda is fuzzy continuous.

If J is a finite subset of varLambda and $A = \prod_{j \in varLambda} A^{j}$ and $A^{j} = (\underline{1})^{j}$ when $j \bar{\in} J$ . Then, for x_λ ∈ pt (I^X), $\begin{matrix} Q_{x_{λ}}^{T} (A) = ⋀_{j \in varLambda} Q_{x_{λ}^{j}} (A^{j}) and \\ T (A) = ⋀_{j \in J} T_{j} (A^{j}) = ⋀_{j \in varLambda} T_{j} (A^{j}) . \end{matrix}$

Definition 2.9. [29] An I-fuzzy topological group is a group G together with an I-fuzzy topology on G that satisfies the following two properties:

The mapping f: G × G → G defined by f (x, y) = xy is fuzzy continuous when G × G is endowed with the product I-fuzzy topology.

The mapping g: G → G defined by g (x) = x⁻¹ is fuzzy continuous.

We remark that the item (1) is equivalent to the statement that, for all x, y ∈ G, W ∈ I^G and λ ∈ I₀, there exist U, V ∈ I^P such that $\begin{matrix} ⋁_{UV \leq W} [Q_{x_{λ}} (U) \land Q_{y_{λ}} (V)] \geq Q_{(xy)_{λ}} (W) \end{matrix}$ holds. Also, the item (2) is equivalent to the statement that, for all x ∈ G, U ∈ I^G and λ ∈ I₀, $Q_{x_{λ}} (U^{- 1}) \geq Q_{x_{λ}^{- 1}} (U)$ holds. Where, $Q = {Q_{x_{λ}} : x_{λ} \in pt (I^{X})}$ is an I-fuzzy quasi-coincident neighborhood system induced by $T$ .

3 I-fuzzy topological polygroups

In this section, we define and study the concept of I-fuzzy topological polygroups, and we prove some properties in this respect.

Definition 3.1. 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 respective formulas: $\begin{matrix} (A • B) (x) = ⋁_{x \in x_{1} \circ x_{2}} (A (x_{1}) \land B (x_{2})) \end{matrix}$ and A⁻¹ (x) = A (x⁻¹), for any x ∈ P. Also, $\begin{matrix} C \circ D = ⋃ {c \circ d : c \in C, d \in D} \end{matrix}$ and C⁻¹ = {c⁻¹: c ∈ C}. We denote A • B by AB for short. Then, for A, B ∈ I^P, we have (AB) ⁻¹ = B⁻¹A⁻¹ and (A⁻¹) ⁻¹ = A.

Example 3. Let P = {a, b, c, d} with a⁻¹ = a, b⁻¹ = b, c⁻¹ = c, d⁻¹ = d and the following hyperoperation: $\begin{matrix} \circ & a & b & c & d \\ a & a & b & c & d \\ b & b & a & c & d \\ c & c & c & {a, b, d} & {c, d} \\ d & d & d & {c, d} & {a, b, c} \end{matrix}$ Let C = {a, b} and D = {b, c, d}. Then C ∘ D = P and C⁻¹ = C.

Example 4. Let a polygroup P = {e, a, b} has the following multiplication table: $\begin{matrix} \circ & e & a & b \\ e & e & a & b \\ a & a & e & b \\ b & b & b & {e, a} \end{matrix}$ Let U = {e, a}. Then a ∘ U = {a, e} and b ∘ U = {b}.

Example 5. Let a polygroup P = {e, a, b} has the following multiplication table: $\begin{matrix} \circ & e & a & b \\ e & e & a & b \\ a & a & e & b \\ b & b & b & {e, a} \end{matrix}$ Define fuzzy subsets A and B of P by A (e) =1, A (a) =1/3, A (b) =1/4 and B (e) =1, B (a) =1/2, B (b) =1/5. Then A • B is a fuzzy set by (A • B) (e) =1, (A • B) (a) =1/2 and (A • B) (b) =1/4. Also A⁻¹ is a fuzzy set by A⁻¹ (e) =1, A⁻¹ (a) =1/3 and A (b) =1/4.

Proposition 3.2. Let P be a polygroup and A, B, C ∈ I^P. Then, the following hold:

If A ≤ B, then AC ≤ BC and CA ≤ CB.

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

(AB) C = A (BC).

If A ≤ B, then A⁻¹ ≤ B⁻¹.

Proof. It is straightforward. □

Let $(P, T)$ be an I-fuzzy topological space. In order to construct an I-fuzzy topological polygroup, we need an I-fuzzy topology on $P^{*} (P)$ . The following lemma gives us an I-fuzzy topology on $P^{*} (P)$ induced by $T$ .

Lemma 3.3. Let $(P, T)$ be an I-fuzzy topological space. For any $A^{★} \in I^{P^{*} (P)}$ , define $T^{★} (A^{★}) = (T \times T) (f^{\leftarrow} (A^{★}))$ and A^★ (X) = ⋀ _x∈XA (x). Then, $T^{★}$ is an I-fuzzy topology on $P^{*} (P)$ .

Proof. It is straightforward. □

Now, we show that f^← (A^★) = A_★, where A_★ ∈ I^P×P and A_★ (x, y) = ⋀ _u∈x∘yA (u), $\begin{matrix} (f^{\leftarrow} (A^{★})) (x, y) & = A^{★} (f^{\to} (x, y)) \\ = A^{★} (x \circ y) \\ = ⋀_{u \in x \circ y} A (u) \\ = A_{★} (x, y) . \end{matrix}$ So, f^← (A^★) = A_★. Let $Q^{★} = {Q_{x_{λ}^{★}}^{★} : x_{λ}^{★} \in pt (I^{P^{*} (P)})}$ be an I- fuzzy quasi-coincident neighborhood system on $P^{*} (P)$ , that is, for any $x_{λ}^{★} \in pt (I^{P^{*} (P)})$ , $A^{★} \subseteq P^{*} (P)$ , $\begin{matrix} Q_{x_{λ}^{★}}^{★} (A^{★}) = ⋁_{x_{λ}^{★} \hat{q} B^{★} \leq A^{★}} T^{★} (B^{★}) . \end{matrix}$ Also, we can easily verify that $\begin{matrix} Q_{x_{λ}^{★}}^{★} (A^{★}) = Q_{x_{λ}} (f^{\leftarrow} (A^{★})) = Q_{x_{λ}} (A_{★}) . \end{matrix}$

Definition 3.4. Let (P, ∘) be a polygroup and $(P, T)$ be an I-fuzzy topological space. Then, the system $(P, \circ, T)$ is called an I-fuzzy topological polygroup if the following conditions hold:

The mapping $f : P \times P \to P^{*} (P)$ , (x, y) ↦ x ∘ y is fuzzy continuous.

The mapping g: P → P, x ↦ x⁻¹ is fuzzy continuous.

Example 6. Obviously, every I-fuzzy topological group is an I-fuzzy topological polygroup.

Example 7. Every polygroup equipped with I-fuzzy topology $T$ , where $T (A) = 1$ for all $A \in T$ is an I-fuzzy topological polygroup.

Theorem 3.5. Let (P, ∘) be a polygroup and $T$ be an I-fuzzy topology on P. Then, the mapping $f : P \times P \to P^{*} (P)$ , (x, y) ↦ x ∘ y is fuzzy continuous if and only if for every x, y ∈ P, W ∈ I^P, λ ∈ I₀ and z_λ ∈ x ∘ y, $z_{λ} \hat{q} W$ , there exist U, V ∈ I^P such that $\begin{matrix} ⋁_{UV \leq W} [Q_{x_{λ}} (U) \land Q_{y_{λ}} (V)] \geq Q_{z_{λ}} (W) \end{matrix}$ holds, where $Q = {Q_{x_{λ}} : x_{λ} \in pt (I^{X})}$ is an I-fuzzy quasi-coincident neighborhood system induced by $T$ .

Proof. Necessity. Since $f : (P, T) \times (P, T) \to (P^{*} (P), T^{★})$ is fuzzy continuous, it follows from Lemma 2.4 that the mapping $f : (P, Q) \times (P, Q) \to (P^{*} (P), Q^{★})$ is fuzzy continuous. Thus, for every $x, y \in P, z_{λ} \in x \circ y, W \in I^{P}, λ \in I_{0}, z_{λ} \hat{q} W$ , we have $Q_{(x, y)_{λ}} (f^{\leftarrow} (W)) \geq Q_{z_{λ}} (W)$ . On the other hand, f^← (W) = ⋃ _j∈J {U_j × V_j | U_jV_j ≤ W}. So we have $\begin{matrix} ⋁_{UV \leq W} [Q_{x_{λ}} (U) \land Q_{y_{λ}} (V)] \geq Q_{z_{λ}} (W) . \end{matrix}$

This means that the necessity is proved.

Sufficiency. From the assumption, for each x, y ∈ P, z_λ ∈ x ∘ y, W ∈ I^P, λ ∈ I₀, we have $\begin{matrix} Q_{(x, y)_{λ}} (f^{\leftarrow} (W)) \\ = ⋁_{UV \leq W} [Q_{x_{λ}} (U) \land Q_{y_{λ}} (V)] \geq Q_{z_{λ}} (W) . \end{matrix}$ So, the mapping $f : (P, Q) \times (P, Q) \to (P^{*} (P)$ , $Q^{★})$ is fuzzy continuous. Therefore, $f : (P, T) \times (P, T) \to (P^{*} (P), T^{★})$ is fuzzy continuous. This completes the proof. □

Theorem 3.6. Let (P, ∘) be a polygroup and $T$ be an I-fuzzy topology on P. Then, the mapping g: P → P, x ↦ x⁻¹ is fuzzy continuous if and only if every x ∈ P, U ∈ I^P and λ ∈ I₀, $Q_{x_{λ}} (U^{- 1}) \geq Q_{x_{λ}^{- 1}} (U)$ holds. Where, $Q = {Q_{x_{λ}} : x_{λ} \in pt (I^{X})}$ is an I-fuzzy quasi-coincident neighborhood system induced by $T$ .

Proof. The proof is similar to the proof of Theorem 3.5. □

Theorem 3.7. Let $(P, \circ, T)$ be an I-fuzzy topological polygroup. Then, the mappings $a φ : P \to P^{*} (P), x \mapsto a \circ x$ $φ_{a} : P \to P^{*} (P), x \mapsto x \circ a$ are fuzzy continuous for every a ∈ P.

Proof. It is straightforward. □

Theorem 3.8. Let $(P, \circ, T)$ be an I-fuzzy topological polygroup. Then, the mapping f: P → P, x ↦ x⁻¹ is a fuzzy homeomorphism.

Proof. Note that (f ∘ f) (x) = x for all x ∈ P, that is, f ∘ f = id (identity mapping), it follows that f^→ = f^← is fuzzy continuous. Hence, f is a fuzzy homeomorphism. □

Theorem 3.9. Let P = 〈P, ∘, e, ⁻¹〉 be a polygroup and $T$ an I-fuzzy topology on P. Then, The condition (i) in Definition 3.4 hold if and only if $\begin{matrix} ⋁_{UV \leq W} [T (U) \land T (V)] \geq T (W) for any W \in I^{P} . \end{matrix}$

Proof. If the condition (i) in Definition 3.4 holds, that is, if f is fuzzy continuous, then for any U, V, W ∈ I^P with UV ≤ W, $x_{λ} \hat{q} U$ and $y_{λ} \hat{q} V$ , we have $\begin{matrix} T (W) & = ⋀_{z_{λ} \hat{q} W} Q_{z_{λ}} (W) \leq Q_{z_{λ}} (W) \\ \leq ⋁_{UV \leq W} [Q_{x_{λ}} (U) \land Q_{y_{λ}} (V)], \end{matrix}$ and so $\begin{matrix} T (W) & \leq ⋁_{UV \leq W} [(⋀_{x_{λ} \hat{q} U} Q_{x_{λ}} (U)) \land (⋀_{y_{λ} \hat{q} V} Q_{y_{λ}} (V))] \\ = ⋁_{UV \leq W} [T (U) \land T (V)] . \end{matrix}$ Conversely, for x, y ∈ P, W ∈ I^P and any z_λ ∈ x ∘ y, suppose $z_{λ} \hat{q} W$ , otherwise, $Q_{z_{λ}} (W) = 0$ . $\begin{matrix} Q_{z_{λ}} (W) & = ⋁_{z_{λ} \hat{q} \bar{W} \leq W} T (\bar{W}) \\ \leq ⋁_{z_{λ} \hat{q} \bar{W} \leq W, UV \leq \bar{W}} [T (U) \land T (V)] \\ \leq ⋁_{x_{λ} \hat{q} U, y_{λ} \hat{q} V, UV \leq \bar{W} \leq W} [Q_{x_{λ}} (U) \land Q_{y_{λ}} (V)] \\ \leq ⋁_{UV \leq W} [Q_{x_{λ}} (U) \land Q_{x_{λ}} (V)] . \end{matrix}$ This means that f is fuzzy continuous. □

Theorem 3.10. Let P = 〈P, ∘, e, ⁻¹〉 be a polygroup and $T$ an I-fuzzy topology on P. Then, The condition (ii) in Definition 3.4 hold if and only if $T (U) \leq T (U^{- 1})$ for any U ∈ I^P.

Proof. If the condition (ii) in Definition 3.4 holds, that is, if g is fuzzy continuous, then for all U ∈ I^P, we have $\begin{matrix} T (U^{- 1}) & = ⋀_{x_{λ} \hat{q} U^{- 1}} Q_{x_{λ}} (U^{- 1}) \\ \geq ⋀_{(x^{- 1})_{λ} \hat{q} U} Q_{(x^{- 1})_{λ}} (U) = T (U) . \end{matrix}$ On the contrary, for all x ∈ P, U ∈ I^P, $\begin{matrix} Q_{(x^{- 1})_{λ}} (U) & = ⋁_{(x^{- 1})_{λ} \hat{q} V \leq U} T (V) \\ \leq ⋁_{(x^{- 1})_{λ} \hat{q} V \leq U} T (V^{- 1}) \\ = ⋁_{x_{λ} \hat{q} V^{- 1} \leq U^{- 1}} T (V^{- 1}) \\ = Q_{x_{λ}} (U^{- 1}) . \end{matrix}$ Thus, g is fuzzy continuous. □

Evidently, every I-fuzzy topological group is an I-fuzzy topological polygroup. We give some other examples.

Example 8. Let P be a polygroup. For any A ∈ I^P, define $T (A) = 1$ . Then, $(P, T)$ is an I-fuzzy topological polygroup.

Example 9. Let P be a polygroup and $T : I^{P} \to I$ be defined as follow: $T (A) = {\begin{matrix} 1 & if A = \underline{0} {or \end{matrix} A = \underline{1}, 0.8 & otherwise .$ Then, $(P, T)$ is an I-fuzzy topological polygroup.

Example 10. Let (P, ∘) be any infinite polygroup. Define an I-fuzzy topology $T : I^{P} \to I$ on P by $\begin{matrix} T (A) \\ = {\begin{matrix} 1 & if A = \underline{λ}, λ \in I, \\ 0.5 & if A is satisfing {x; A (x) = 0} is a finite set, \\ 0 & otherwise . \end{matrix} \end{matrix}$ Then, we can easily verify that $(P, \circ, T)$ is an I-fuzzy topological polygroup.

Example 11. Let (P, ∘) be a polygroup, where P = {e} is singleton. Define an I-fuzzy topology $T : I^{P} \to I$ on P by $T (A) = {\begin{matrix} 1 & if A = \underline{0}, \\ 1 - λ & if A = λ, λ \in (0, 1), \\ 1 & if A = \underline{1} . \end{matrix}$ Then, we can easily verify that $(P, \circ, T)$ is an I-fuzzy topological polygroup.

Theorem 3.11. Let $(P, \circ, T)$ be an I-fuzzy topological polygroup and $Q_{x_{λ}} (.)$ be its corresponding I-fuzzy quasi-coincident neighborhood system of x_λ. Then,

$Q_{x_{λ}} (A^{- 1}) = Q_{x_{λ}^{- 1}} (A)$ , for all x_λ ∈ pt (I^P) and A ∈ I^P.

$T (A) = T (A^{- 1})$ , for all A ∈ I^P.

Proof. (1) By (I-FQ3) in Definition 2.2 and Definition 3.4, we have $\begin{matrix} Q_{x_{λ}} (A) = ⋁_{B \leq A} Q_{x_{λ}} (B) . \end{matrix}$ So, $\begin{matrix} Q_{x_{λ}} (A^{- 1}) & = ⋁_{B \leq A^{- 1}} Q_{x_{λ}} (B) \\ = ⋁_{B^{- 1} \leq A} Q_{x_{λ}} (B) \\ \geq Q_{x_{λ}^{- 1}} (A) . \end{matrix}$ Similarly, we can prove that $Q_{x_{λ}^{- 1}} (A) \geq Q_{x_{λ}} (A^{- 1})$ . (2) We have $\begin{matrix} T (A) & = ⋀_{x_{λ}^{- 1} \hat{q} A} Q_{x_{λ}^{- 1}} (A) \\ = ⋀_{x_{λ} \hat{q} A^{- 1}} Q_{x_{λ}} (A^{- 1}) \\ = T (A^{- 1}) . \end{matrix}$ □

Theorem 3.12. Let $(P, \circ, T)$ be an I-fuzzy topological polygroup and e the identity element in P. Then, for every x_λ ∈ pt (I^P), A ∈ I^P, λ ∈ I₀, we have

$Q_{e_{λ}} (A) = Q_{e_{λ}} (A^{- 1})$ .

$Q_{x_{λ}} (x A) = Q_{x_{λ}} (A x)$ .

$T (A) = T (A^{- 1})$ .

$T (x A) = T (A x)$ .

Proof. (1) It is clear.

(2) For every x_λ ∈ pt (I^P), A ∈ I^P, we have $\begin{matrix} Q_{x_{λ}} (x A) & = Q_{x φ^{\to} (e_{λ})} (x A) \\ \leq Q_{e_{λ}} (_{x} φ^{\leftarrow} (x A)) = Q_{e_{λ}} (A) . \end{matrix}$ Now, we show that $Q_{e_{λ}} (A) = Q_{x_{λ}} (x A)$ . $\begin{matrix} Q_{e_{λ}} (A) & = Q_{x^{- 1}} φ^{\to} (x_{λ)} (A) \\ \leq Q_{x_{λ}} (_{x^{- 1}} φ^{\leftarrow} (A)) = Q_{x_{λ}} (x A) . \end{matrix}$ So, $Q_{x_{λ}} (x A) = Q_{x_{λ}} (A x)$ .

(3) It is clear.

(4) $T (A x) = T (φ_{x^{- 1}}^{\leftarrow} (A)) \geq T (A)$ and $T (A) =$ $T (φ_{x^{- 1}}^{\leftarrow} (A x)) \geq T (A x)$ . Hence, $T (A x) = T (A)$ . Similarly, it is easy to prove that $T (A) = T (x A)$ . □

Theorem 3.13. Let $(P, \circ, T)$ be an I-fuzzy topological polygroup and e the identity element in P, Q_{e
_λ}: I^P → I an I-fuzzy quasi-coincident neighborhood system of e_λ induced by $T$ . Then, Q_{e
_λ}, λ ∈ I₀ has the following properties:

Q_{e
_λ} (P) = 1.

For all $U \in I^{P}, Q_{e_{λ}} (U) > 0 \Rightarrow e_{λ} \hat{q} U$ .

For all U, V ∈ I^P, $Q_{e_{λ}} (U \land V) = Q_{e_{λ}} (U) \land Q_{e_{λ}} (P) .$

For each b < Q_{e
_λ} (U), there exist ϸ ∈ (0, λ), V ∈ I^P such that b < Q_{e
_λ-ϸ} (V). Moreover, Q_{e
_λ} (U) ≤ Q_{e
_α} (U) for all λ ≤ α.

For all W ∈ I^P, $Q_{e_{λ}} (W) \leq ⋁_{UV \leq W} [Q_{e_{λ}} (U) \land Q_{e_{λ}} (V)] .$

For all U ∈ I^P, Q_{e
_λ} (U) = Q_{e
_λ} (U⁻¹).

For all x ∈ P, Q_{e
_λ} (xU) = Q_{e
_λ} (Ux).

Conversely, suppose that (P, ∘) be a polygroup and let a mapping Q_{e
λ}: I^P → I satisfy the conditions (1) - (7). Then, there exists an I-fuzzy topology

T

on P such that

(P, \circ, T)

is an I-fuzzy topological polygroup and the mapping Q_{e
_λ} is an I-fuzzy quasi-coincident neighborhood system of e_λ induced by

T

Proof. Necessity. Conditions (1), (2) and (3) follow directly from Definition 2.2 of the I-fuzzy quasi-coincident neighborhood system. Conditions (5), (6) and (7) are easily obtained by Theorems 3.5 and 3.12. We only prove the properties of (4).

(4) For each b < Q_{e
_λ} (U), it follows that $b < ⋁_{e_{λ} \hat{q} V \leq U} T (V)$ . Then, there is a V ∈ I^P with $e_{λ} \hat{q} V \leq U$ such that $T (V) > b$ . Since $e_{λ} \hat{q} V \leq U$ , we have V (e) >1 - λ, it follows that there exists ϸ > 0 such that V (e) >1 - λ + ϸ > 1 - λ. This shows that $e_{λ - ϸ} \hat{q} V$ , by $b < T (V) = ⋀_{x_{ν} \hat{q} V} Q_{e_{ν}} (V)$ , this deduces that $b < Q_{e_{λ - ϸ}} (V)$ . As for the proof of the relation Q_{e
_λ} (U) ≤ Q_{e
_α} (U) for all λ ≤ α, U ∈ I^P, it is indeed easy to check.

Sufficiency. Suppose that Q_{x
_λ}: I^P → I is a mapping which satisfies conditions (1)-(7). Define Q_{x
_λ} (U) = Q_{e
_λ} (x⁻¹U) and $T : I^{P} \to I$ , $T (U) = ⋀_{x_{λ} \hat{q} U} Q_{x_{λ}} (U)$ , for all U ∈ I^P. First, we show that the mapping $T : I^{P} \to I$ satisfies the conditions (1), (2) and (3) in Definition 2.1.

$T (\underline{0}) = 1$ and $\begin{matrix} T (\underline{1}) & = ⋀_{x_{λ} \hat{q} P} Q_{x_{λ}} (P) \\ = ⋀_{x_{λ} \hat{q} P} Q_{e_{λ}} (x^{- 1} P) \\ = ⋀_{x_{λ} \hat{q} P} Q_{e_{λ}} (P) = 1 . \end{matrix}$

We have $\begin{matrix} T (U \land V) & = ⋀_{x_{λ} \hat{q} (U \land V)} Q_{x_{λ}} (U \land V) \\ = ⋀_{x_{λ} \hat{q} (U \land V)} Q_{e_{λ}} (x^{- 1} (U \land V)) \\ = ⋀_{x_{λ} \hat{q} (U \land V)} Q_{e_{λ}} [(x^{- 1} U) \land (x^{- 1} V)] \\ = ⋀_{x_{λ} \hat{q} (U \land V)} [Q_{e_{λ}} (x^{- 1} U) \land Q_{e_{λ}} (x^{- 1} V)] \\ = ⋀_{x_{λ} \hat{q} (U \land V)} [Q_{x_{λ}} (U) \land Q_{x_{λ}} (V)] \\ \geq [⋀_{x_{λ} \hat{q} U} Q_{x_{λ}} (U) \land ⋀_{x_{λ} \hat{q} V} Q_{x_{λ}} (V)] \\ = T (U) \land T (V) . \end{matrix}$

For any $a < ⋀_{j \in J} T (U_{j})$ and for any j ∈ J, we have $a < T (U_{j})$ . Hence, for any $x_{λ} \hat{q} U_{j}$ , we have $a < Q_{x_{λ}} (U_{j})$ . For any $y_{λ} \hat{q} ⋁_{j \in J} U_{j}$ , there exists j₀ ∈ J such that $y_{λ} \hat{q} U_{j_{0}}$ . Then, we have $a < Q_{y_{λ}} (U_{j_{0}}) \leq Q_{y_{λ}} (⋁_{j \in J} U_{j})$ . So, $\begin{matrix} a < ⋀_{x_{λ} \hat{q} ⋁_{j \in J} U_{j}} Q_{x_{λ}} (⋁_{j \in J} U_{j}) = T (⋁_{j \in J} U_{j}) . \end{matrix}$

By the arbitrariness of a, we have

⋀_{j \in J} T (U_{j}) \leq T (⋁_{j \in J} U_{j})

. Therefore,

(P, T)

is an I-fuzzy topological space. Define

Q = {Q_{x_{λ}} | x_{λ} \in pt (I^{P})}

, for all U ∈ I^P,

Q_{x_{λ}} (U) = Q_{e_{λ}} (x^{- 1} U)

. We can easily verify that

Q

is an I-fuzzy quasi-coincident neighborhood system on P. Now, we show that

(P, \circ, T)

is an I-fuzzy topological polygroup. For all x, y ∈ P, z_λ ∈ x ∘ y, W ∈ I^P,

\begin{matrix} Q_{z_{λ}} (W) \\ = Q_{e_{λ}} (z^{- 1} W) \\ \leq ⋁_{AB \leq z^{- 1} W} [Q_{e_{λ}} (A) \land Q_{e_{λ}} (B)] \\ \leq ⋁_{zAB \leq W} [Q_{x_{λ}} (xA) \land Q_{y_{λ}} (yB)] \\ \leq ⋁_{UV \leq W} [Q_{x_{λ}} (U) \land Q_{y_{λ}} (V)], \end{matrix}

and

\begin{matrix} Q_{x_{λ}^{- 1}} (U) & = Q_{e_{λ}} (xU) \\ = Q_{e_{λ}} (U^{- 1} x^{- 1}) \\ = Q_{e_{λ}} (x^{- 1} U^{- 1}) \\ = Q_{x_{λ}} (U^{- 1}) . \end{matrix}

Therefore,

(P, \circ, T)

is an I-fuzzy topological polygroup and

Q_{e_{λ}}

is an I-fuzzy quasi-coincident neighborhood system of e_λ induced by

T

. □

4 Subpolygroups and quotient polygroups of I-fuzzy topological polygroups

Definition 4.1. [21]

If $(X, T)$ is an I-fuzzy topological space, Y ⊆ X, then $T |_{Y}$ , where $T |_{Y} (U) = ⋁ {T (V) | V \in I^{X}, V |_{Y} = U}$ , is called the relative I-fuzzy topology of $T$ with respect to Y.

Let $(X, T)$ and (Y, δ) be two I-fuzzy topological space. If Y ⊆ X and $δ = T |_{Y}$ , then (Y, δ) is called a subspace of $(X, T)$ .

Theorem 4.2. Let $(P, \circ, T)$ be an I-fuzzy topological polygroup, $Q = {Q_{x_{λ}} : x_{λ} \in pt (I^{P})}$ is its corresponding I-fuzzy quasi-ccoincident neighborhood system. K is a subpolygroup of P, $T |_{K}$ is the induced topology by $T$ with respect to K. For all x_λ ∈ pt (I^K), we define a function $Q_{λ}^{K} : ℐ^{K} \to ℐ$ by $\begin{matrix} Q_{x_{λ}}^{K} (A) = ⋁_{B \land K = A} Q_{x_{λ}} (B) . \end{matrix}$ $Q^{K} = {Q_{x_{λ}}^{K} : x_{λ} \in pt (I^{K})}$ is an I- fuzzy quasi-coincident neighborhood system of $(K, T |_{K})$ , which is called the induced neighborhood system by $Q = {Q_{x_{λ}} : x_{λ} \in pt (I^{P})}$ with respect to K.

Proof. We prove that $Q^{K} = {Q_{x_{λ}}^{K} : x_{λ} \in pt (I^{K})}$ is an I-fuzzy quasi-coincident neighborhood system of $(K, T |_{K})$ , it suffices to show that $Q_{x_{λ}}^{K} (A) = ⋁_{x_{λ} \hat{q} D \leq A} T |_{K} (D) .$

In fact, $\begin{matrix} Q_{x_{λ}}^{K} (A) & = ⋁_{B \land K = A} Q_{x_{λ}} (B) \\ = ⋁_{x_{λ} \hat{q} (B \land K) = A} Q_{x_{λ}} (B) \\ = ⋁_{x_{λ} \hat{q} (B \land K) = A} ⋁_{x_{λ} \hat{q} C \leq B} T (C) \\ = ⋁_{x_{λ} \hat{q} (C \land K) \leq A} T (C) \\ = ⋁_{x_{λ} \hat{q} D \leq A} ⋁_{C \land K = D} T (C) \\ = ⋁_{x_{λ} \hat{q} D \leq A} T |_{K} (D) . \end{matrix}$ □

Theorem 4.3. Let $(P, \circ, T)$ be an I-fuzzy topological polygroup, K a subpolygroup of P and $(K, T |_{K})$ be subspace of I-fuzzy topological space $(P, T)$ . Then, $(K, \circ, T |_{K})$ is also an I-fuzzy topological polygroup, called I-fuzzy topological subpolygroup.

Proof. We show that $(K, \circ, T |_{K})$ satisfies conditions (i) and (ii) in Definition 3.4. First, we show that for any A ∈ I^K and z_λ ∈ x ∘ y there exist B, C ∈ I^K such that, $\begin{matrix} ⋁_{BC \leq A} [Q_{x_{λ}}^{K} (B) \land Q_{y_{λ}}^{K} (C)] \geq Q_{x_{λ}}^{K} (A) . \end{matrix}$ $\begin{matrix} Q_{z_{λ}}^{K} (A) \\ = ⋁_{B \land K = A} Q_{z_{λ}} (B) \\ \leq ⋁_{B \land K = A} ⋁_{CD \leq B} [Q_{x_{λ}} (C) \land Q_{y_{λ}} (D)] \\ \leq ⋁_{(CD \land K) \leq A} [Q_{x_{λ}} (C) \land Q_{y_{λ}} (D)] \\ \leq ⋁_{(C \land K) (D \land K) \leq A} [Q_{x_{λ}} (C) \land Q_{y_{λ}} (D)] \\ \leq ⋁_{EF \leq A} ⋁_{C \land K = E, D \land K = F} [Q_{x_{λ}} (C) \land Q_{y_{λ}} (D)] \\ = ⋁_{EF \leq A} [⋁_{C \land K = E} Q_{x_{λ}} (C) \land ⋁_{D \land K = F} Q_{y_{λ}} (D)] \\ = ⋁_{EF \leq A} [Q_{x_{λ}}^{K} (E) \land Q_{y_{λ}}^{K} (F)] . \end{matrix}$ Next, we show that for any A ∈ I^K, $Q_{x_{λ}}^{K} (A) = Q_{x_{λ}}^{K} (A^{- 1})$ . We have $\begin{matrix} Q_{x_{λ}}^{K} (A) = ⋁_{B \land K = A} Q_{x_{λ}} (B), \end{matrix}$ and $\begin{matrix} Q_{x_{λ}}^{K} (A^{- 1}) & = ⋁_{B \land K = A^{- 1}} Q_{x_{λ}} (B) \\ = ⋁_{B^{- 1} \land K = A} Q_{x_{λ}} (B^{- 1}) . \end{matrix}$ So, we have $Q_{x_{λ}}^{K} (A) = Q_{x_{λ}}^{K} (A^{- 1})$ . Therefore, $(K, \circ, T |_{K})$ is an I-fuzzy topological polygroup. □

Definition 4.4. Let $(P, \circ, T)$ be an I - FTP, N a normal subpolygroup of P. P* = P/N = {x ∘ N|x ∈ P}, and let p: P → P*, x ↦ x ∘ N be the natural mapping, for all A* ⊆ P*, we define $T^{*} : I^{P^{*}} \to I$ by $T^{*} (A^{*}) = T (p^{\leftarrow} (A^{*}))$ . Then, $T^{*}$ is an I-fuzzy topology on P*, which is called the quotient topology induced with respect to p and $T$ in P, and $(P^{*}, T^{*})$ is the space of the right accompanying sets of $(P, T)$ . Similarly, we have the definition of the space of the left accompanying sets of $(P, T)$ .

Let $Q^{*} = {Q_{x_{λ}^{*}}^{*} : x_{λ}^{*} \in pt (I^{P^{*}})}$ be an I- fuzzy quasi-coincident neighborhood system on P*, i.e, for any $x_{λ}^{*} \in pt (I^{P^{*}})$ , A* ⊆ P*, $\begin{matrix} Q_{x_{λ}^{*}}^{*} (A^{*}) = ⋁_{x_{λ}^{*} \hat{q} B^{*} \leq A^{*}} T^{*} (B^{*}) . \end{matrix}$

Example 12. Let the polygroup P = {e, a, b} has the following multiplication table: $\begin{matrix} \circ & e & a & b \\ e & e & a & b \\ a & a & e & b \\ b & b & b & {e, a} \end{matrix}$ and $T : I^{P} \to I$ be defined as follow: $T (A) = {\begin{matrix} 1 & if A = \underline{0} {or \end{matrix} A = \underline{1}, 0.7 & otherwise .$ Then, $(P, T)$ is an I-fuzzy topological polygroup. Also let N = {e, a}, then N is a normal subpolygroup of P. Define P* = P/N = {x ∘ N|x ∈ P} and let p: P → P*, x ↦ x ∘ N be the natural mapping, for all A* ⊆ P*. Also we define $T^{*} : I^{P^{*}} \to I$ by $T^{*} (A^{*}) = T (p^{\leftarrow} (A^{*}))$ . Then $T^{*}$ is an I-fuzzy topology on P*.

Theorem 4.5. Let $(P, \circ, T)$ be an I - FTP, N a normal subpolygroup of P, $(P^{*}, T^{*})$ be the space of the right accompanying sets of $(P, T)$ , f: P → P* be the natural mapping and $Q^{*} = {Q_{x_{λ}^{*}}^{*} : x_{λ}^{*} \in pt (I^{P^{*}})}$ is the I- fuzzy quasi-coincident neighborhood system on P*. Then, for all A* ∈ I^P*, $\begin{matrix} Q_{x_{λ}^{*}}^{*} (A^{*}) = Q_{x_{λ}} (f^{\leftarrow} (A^{*})) . \end{matrix}$

Proof. We have $\begin{matrix} Q_{x_{λ}^{*}}^{*} (A^{*}) & = ⋁_{x_{λ}^{*} \hat{q} B^{*} \leq A^{*}} T^{*} (B^{*}) \\ = ⋁_{x_{λ}^{*} \hat{q} B^{*} \leq A^{*}} T (f^{\leftarrow} (B^{*})) \\ \leq ⋁_{x_{λ}^{*} \hat{q} B^{*} \leq A^{*}} Q_{x_{λ}} (f^{\leftarrow} (B^{*})) \\ \leq ⋁_{x_{λ}^{*} \hat{q} B^{*} \leq A^{*}} Q_{x_{λ}} (f^{\leftarrow} (A^{*})) \\ = Q_{x_{λ}} (f^{\leftarrow} (A^{*})), x_{λ} \hat{q} f^{\leftarrow} (A^{*}) . \end{matrix}$ On the other hand, if $x_{λ} \hat{q} f^{\leftarrow} (A^{*})$ , then $\begin{matrix} Q_{x_{λ}} (f^{\leftarrow} (A^{*})) & = ⋁_{x_{λ} \hat{q} B \leq f^{\leftarrow} (A^{*})} T (B) \\ \leq ⋁_{x_{λ}^{*} \hat{q} B^{*} \leq A^{*}} T (B) \\ \leq ⋁_{x_{λ}^{*} \hat{q} B^{*} \leq A^{*}} T (BN) \\ = ⋁_{x_{λ}^{*} \hat{q} B^{*} \leq A^{*}} T (f^{\leftarrow} (B^{*})) \\ = ⋁_{x_{λ}^{*} \hat{q} B^{*} \leq A^{*}} T^{*} (B^{*}) \\ = Q_{x_{λ}^{*}}^{*} (A^{*}) . \end{matrix}$ Thus, for all A* ∈ I^P*, we have $Q_{x_{λ}^{*}}^{*} (A^{*}) = Q_{x_{λ}} (f^{\leftarrow} (A^{*})), x_{λ} \hat{q} f^{\leftarrow} (A^{*})$ . □

Theorem 4.6. Let $(P, \circ, T)$ be an I- fuzzy topological polygroup. Then, $(P / β^{*}, \otimes, T^{*})$ is an I-fuzzy topological group, where β* is the fundamental relation of P and β* (x) ⊗ β* (y) = β* (z), z ∈ x ∘ y for every x, y ∈ P.

Proof. For all β* (x), β* (y) ∈ P/β*, A* ∈ I^P/β*, we have $\begin{matrix} Q_{(β^{*} (x) β^{*} (y)^{- 1})_{λ}}^{*} (A^{*}) \\ = Q_{(x y^{- 1})_{λ}} (f^{\leftarrow} (A^{*})) \\ \leq ⋁_{B C^{- 1} \leq f^{\leftarrow} (A^{*})} [Q_{x_{λ}} (B) \land Q_{x_{λ}} (C)] \\ \leq ⋁_{B^{*} C^{*^{- 1}} \leq A^{*}} [Q_{x_{λ}} (B) \land Q_{x_{λ}} (C)] \\ \leq ⋁_{B^{*} C^{*^{- 1}} \leq A^{*}} [Q_{x_{λ}} (f^{\leftarrow} (B^{*})) \land Q_{x_{λ}} (f^{\leftarrow} (C^{*}))] \\ \leq ⋁_{B^{*} C^{*^{- 1}} \leq A^{*}} [Q_{β^{*} (x)_{λ}}^{*} (B^{*}) \land Q_{β^{*} (y)_{λ}}^{*} (C^{*})] . \end{matrix}$ This completes the proof. □

Lemma 4.7. Let (P, ∘) be a polygroup, N a normal subpolygroup of P and p: P → P* be the natural mapping. Then,

For all A, B ∈ I^P, p^→ (AB) = p^→ (A) p^→ (B).

For each A ∈ I^P, (p^→ (A)) ⁻¹ = p^→ (A⁻¹).

Proof. (i) For each x ∘ N ∈ P*, we have $\begin{matrix} (p^{\to} (A) p^{\to} (B)) (x \circ N) \\ = ⋁_{(x \circ N) \in yz} [p^{\to} (A) (y) \land p^{\to} (B) (z)] \\ = ⋁_{(x \circ N) \in yz} [(⋁_{y_{0} \in p^{\leftarrow} (y)} A (y_{0})) \land (⋁_{z_{0} \in p^{\leftarrow} (z)} B (z_{0}))] \\ = ⋁_{(x \circ N) \in yz} [(⋁_{y \in p^{\to} (y_{0})} A (y_{0})) \land (⋁_{z \in p^{\to} (z_{0})} B (z_{0}))] \\ = ⋁_{(x \circ N) \in yz} ⋁_{yz \subseteq p^{\to} (y_{0}) p^{\to} (z_{0})} [A (y_{0}) \land B (z_{0})] \\ = ⋁_{(x \circ N) \subseteq p^{\to} (y_{0}) p^{\to} (z_{0})} [A (y_{0}) \land B (z_{0})], \end{matrix}$ and $\begin{matrix} p^{\to} (AB) (x \circ N) \\ = ⋁_{z \in p^{\leftarrow} (x \circ N)} AB (z) \\ = ⋁_{z \in p^{\leftarrow} (x \circ N)} ⋁_{z \in y_{0} z_{0}} [A (y_{0}) \land B (z_{0})] \\ = ⋁_{(x \circ N) \in p^{\to} (z)} [A (y_{0}) \land B (z_{0})] \\ = ⋁_{(x \circ N) \in p^{\to} (y_{0}) p^{\to} (z_{0})} [A (y_{0}) \land B (z_{0})] . \end{matrix}$ Hence, p^→ (AB) = p^→ (A) p^→ (B).

(ii) For each x ∘ N ∈ P* $\begin{matrix} (p^{\to} (A))^{- 1} (x \circ N) & = p^{\to} (A) ((x \circ N)^{- 1}) \\ = p^{\to} (A) (x^{- 1} \circ N) \\ = ⋁_{y \in p^{\leftarrow} (x^{- 1} \circ N)} A (y) . \end{matrix}$ Moreover, we obtain $\begin{matrix} p^{\to} (A^{- 1}) (x \circ N) & = ⋁_{z \in p^{\leftarrow} (x \circ N)} A^{- 1} (z) \\ = ⋁_{z \in p^{\leftarrow} (x \circ N)} A (z^{- 1}) \\ = ⋁_{y^{- 1} \in p^{\leftarrow} (x \circ N)} A (y) \\ = ⋁_{y \in p^{\leftarrow} (x^{- 1} \circ N)} A (y) . \end{matrix}$ Therefore, the proof is completed. □

Lemma 4.8. Let $(P, \circ, T)$ be an I - FTP, N a normal subpolygroup of P, $(P^{*}, T^{*})$ be the spaces of the right accompanying sets of $(P, T)$ and p: P → P* be the natural mapping. Then, $p : (P, \circ, T) \to (P^{*}, \circ, T^{*})$ is a fuzzy open mapping.

Proof. We have to prove for every A ∈ I^P, $T (A) \leq T^{*} (p^{\to} (A))$ , where $\begin{matrix} p^{\to} (A) (y) = ⋁_{x \in p^{\leftarrow} (y)} A (x) . \end{matrix}$ First, for each x ∈ P, A ∈ I^P, we have $\begin{matrix} p^{\leftarrow} (p^{\to} (A)) (x) & = p^{\to} (A) (p (x)) \\ = p^{\to} (A) (x \circ N) \\ = ⋁_{y \in p^{\leftarrow} (x \circ N)} A (y), \end{matrix}$ and $\begin{matrix} (A χ_{N}) (x) & = ⋁_{x \in yz} [A (y) \land χ_{N} (z)] \\ = ⋁_{x \in yz, z \in N} A (y) \\ = ⋁_{y \in x z^{- 1}, z \in N} A (y) \\ = ⋁_{y \in p^{\leftarrow} (x \circ N)} A (y) . \end{matrix}$ Hence, p^← (p^→ (A)) = Aχ_N = ⋁ _u∈NA ∘ u. Also, we obtain $\begin{matrix} T^{*} (p^{\to} (A)) & = T (p^{\leftarrow} (p^{\to} (A))) \\ = T (A χ_{N}) \\ = T (⋁_{u \in N} A \circ u) \\ \geq ⋀_{u \in N} T (A \circ u) \\ = ⋀_{u \in N} T (A) \\ = T (A) . \end{matrix}$ So, for every A ∈ I^P, $T (A) \leq T^{*} (p^{\to} (A))$ and p is a fuzzy open mapping. □

Theorem 4.9. Let $(P, \circ, T)$ be an I - FTP, N a normal subpolygroup of P. P* = P/N = {x ∘ N|x ∈ P}, and let p: P → P*, x ↦ x ∘ N be the natural mapping, $(P^{*}, T^{*})$ is the space of the right accompanying sets of $(P, T)$ . Then, $(P^{*}, \circ, T^{*})$ is also an I - FTP.

Proof. We show that $(P^{*}, T^{*})$ satisfies conditions (i) and (ii) in Definition 3.4. (i) For each $x^{*}, y^{*} \in P^{*}, z_{λ}^{*} \in x^{*} y^{*} and A^{*} \in I^{P^{*}}$ , we have $\begin{matrix} Q_{z_{λ}^{*}}^{*} (A^{*}) \\ = Q_{z_{λ}} (p^{\leftarrow} (A^{*})) \\ \leq ⋁_{BC \leq p^{\leftarrow} (A^{*})} [Q_{x_{λ}} (B) \land Q_{y_{λ}} (C)] \\ \leq ⋁_{B^{*} C^{*} \leq A^{*}} [Q_{x_{λ}} (B) \land Q_{y_{λ}} (C)] \\ \leq ⋁_{B^{*} C^{*} \leq A^{*}} [Q_{x_{λ}} (p^{\leftarrow} (B^{*})) \land Q_{y_{λ}} (p^{\leftarrow} (C^{*}))] \\ = ⋁_{B^{*} C^{*} \leq A^{*}} [Q_{x_{λ}^{*}}^{*} (B^{*}) \land Q_{y_{λ}^{*}}^{*} (C^{*})] . \end{matrix}$ (ii) We have $\begin{matrix} Q_{x_{λ}^{*}}^{*} ((A^{*})^{- 1}) \\ = Q_{x_{λ}} (p^{\leftarrow} ((A^{*})^{- 1})) \\ = Q_{x_{λ}} ((p^{\leftarrow} (A^{*}))^{- 1}) \\ \geq Q_{x_{λ}^{- 1}} (p^{\leftarrow} (A^{*})) \\ = Q_{x_{λ}^{*^{- 1}}}^{*} (A^{*}) . \end{matrix}$ Hence, $(P^{*}, \circ, T^{*})$ is an I-fuzzy topological polygroup. □

The category of classical polygroups and homomorphisms is denoted by PGRP, the category of I-fuzzy topological polygroups and fuzzy continuous homomorphisms is denoted by I-FTPP.

Theorem 4.10. cg I - FTPP is topological over PGRP.

Proof. Let ${(P_{j}, T_{j})}_{j \in J}$ be a family of I-fuzzy topological polygroup, $(P, T)$ a polygroup, and let $\begin{matrix} {f_{i} : (P, T) \to (P_{j}, \circ_{j}, T_{j})}_{j \in J} \end{matrix}$ be a family of surjective polygroup homomorphisms. Define $\begin{matrix} Q_{x_{λ}} (A) = ⋁ {⋀_{j \in J} Q_{f_{j} (x_{λ})} (A_{j}) : \\ J \subset varLambda is finite index set, A_{j} \in I^{P_{j}}, ⋀_{j \in J} f_{j}^{\leftarrow} (A_{j}) \leq A}, \end{matrix}$ where A ∈ I^P, x_λ ∈ pt (I^P), A_j ∈ I^{P
_j} and define a mapping $T_{Q} : I^{P} \to I$ by $T_{Q} (A) = ⋀_{x_{λ} \hat{q} A} Q_{x_{λ}} (A) .$ By Theorem 2.8, $Q_{x_{λ}}$ is the weakest I-fuzzy topology on P such that each f_j is continuous for each j ∈ varLambda. In order that to show that $(P, \circ, T_{Q})$ is an I-fuzzy topological polygroup, we need to prove that for all x, y ∈ P, C ∈ I^P, λ ∈ I, z_λ ∈ x ∘ y, $\begin{matrix} ⋁_{AB \leq C} [Q_{x_{λ}} (A) \land Q_{y_{λ}} (B)] \geq Q_{z_{λ}} (C), \end{matrix}$ and for all x ∈ P, A ∈ I^P, λ ∈ I₀, $\begin{matrix} Q_{x_{λ}} (A^{- 1}) \geq Q_{x_{λ}^{- 1}} (A) . \end{matrix}$ If $a < Q_{z_{λ}} (C)$ , then there exists a finite index set J ⊆ varLambda and C_j ∈ I^{P
_j} such that $⋀_{j \in J} f_{j}^{\leftarrow} (A_{j}) \leq A$ and $a < Q_{f_{j} (z_{λ})} (C_{j})$ for each j ∈ J. Since $(P_{j}, \circ_{j}, T_{j})$ is an I-fuzzy topological polygroup, there exist A_j, B_j ∈ I^{P
_j} such that A_jB_j ≤ C_j, $a < Q_{(f_{j} (x))_{λ}} (A_{j})$ and $a < Q_{(f_{j} (y))_{λ}} (B_{j})$ . Denote $A = ⋀_{j \in J} f_{j}^{\leftarrow} (A_{j}), B = ⋀_{j \in J} f_{j}^{\leftarrow} (B_{j}) .$ Then, AB ≤ C and $a < Q_{x_{λ}} (A) \land Q_{y_{λ}} (B)$ . It follows that $\begin{matrix} ⋁_{AB \leq C} [Q_{x_{λ}} (A) \land Q_{y_{λ}} (B)] \geq Q_{z_{λ}} (C) . \end{matrix}$ Similarly, we can show that $\begin{matrix} Q_{x_{λ}} (A^{- 1}) \geq Q_{x_{λ}^{- 1}} (A) . \end{matrix}$ Finally, by Theorem 2.8(3), if $(Z, \circ, T)$ is an I-fuzzy topological polygroup, and $g : (Z, \circ, T) \to (P, \circ, T_{Q})$ is a polygroup homomorphism, then g is continuous if and only if f_j ∘ g (j ∈ varLambda) is continuous. □

By the proof of Theorem 4.10, we can easily obtain the following corollary.

Corollary 4.11. If ${(P_{j}, T_{j})}_{j \in J}$ be a family of I-fuzzy topological polygroups. Then, the product $(\prod_{j \in J} P_{j}, \prod_{j \in J} T_{j})$ is also an I-fuzzy topological polygroup.

Footnotes

Acknowledgments

The authors are highly grateful to referees for their valuable comments and suggestions which were helpful in improving this paper.

References

Aghabozorgi

, Davvaz

and Jafarpour

, Solvable polygroups and drived subpolygroups, Comm Algebra 41(8) (2013), 3098–3107.

Akram

and Davvaz

, Generalized fuzzy ideals of Kalgebras, Journal of Multi-Valued Logic and Soft Computing 11 (2112), 375–311.

Akram

, Davvaz

and Feng

, Fuzzy soft Lie algebras, Journal of Multi-Valued Logic and Soft Computing 23 (2115), 511–521.

Akram

, Davvaz

and Shum

K.P.

, Generalized fuzzy Lie ideals of Lie algebras, Mohu Xitong yu Shuxue 24(4) (2010), 48–55.

Chang

C.L.

, Fuzzy topological spaces, J Math Anal Appl 24 (1968), 182–190.

Chon

, Properties of fuzzy topological groups and semigroups, Kangweon-Kyungki Math 8 (2000), 103–110.

Comer

S.D.

, Polygroups derived from cogroups, J Algebra 89 (1984), 397–405.

Corsini

, Aviani Editore, Prolegomena of Hypergroup Theory, 1993.

Davvaz

, Cristea

, Fuzzy Algebraic Hyperstructures-An Introduction, Springer, 2015.

10.

Davvaz

, Isomorphism theorems of polygroups, Bull Math Sci Soc (2) 33(3) (2010), 11–19.

11.

Davvaz

, Polygroup Theory and Related System, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.

12.

Fang

J.M.

, I — FTOP is isomorphic to I — FQN and I — AITOP, Fuzzy Sets and System 147 (2004), 317–325.

13.

Fang

J.M.

and Yue

Y.I.

, Base and subbase in I-fuzzy topological space, Journal of Mathematical Research and Exposition 26 (2006), 89–95.

14.

Foster

D.H.

, Fuzzy topological groups, J Math Anal Appl 67 (1979), 549–564.

15.

Freni

, Une note sur le coeur dun hypergroupe et sur la cloture transitive β* de β, Riv di Mat Pura Appl 8 (1991), 153–156.

16.

Ganster

, Georgiou

D.N.

and Jafari

, On fuzzy topological groups and fuzzy continuous functions, Hacettepe Journal of Mathematics and Statistics 34 (2005), 35–43.

17.

Hoskova

, Topological hypergroupoids, Comput Math Appl 64 (2012), 2845–2849.

18.

Jafarpour

, Aghabozorgi

and Davvaz

, On nilpotent and solvable polygroups, Bulletin of Iranian Mathematical Society 39(3) (2013), 487–499.

19.

Koskas

, Groupoides, demi-hypergroupes et hypergroupes, J Math Pures Appl 49(9) (1970), 155–192.

20.

Lowen

, Fuzzy topological spaces and fuzzy compacts, J Math Anal Appl 56 (1976), 621–633.

21.

J.L.

and Yu

C.H.

, Fuzzy topological groups, Fuzzy Sets and Systems 12 (1984), 289–299.

22.

J.L.

and Yu

C.H.

, On the direct product of Fuzzy topological groups, Fuzzy Sets and Systems 17 (1985), 91–97.

23.

, Zhan

and Davvaz

, Notes on approximations in hyperrings, Journal of Multi-Valued Logic and Soft Computing 21 (2117), 341–313.

24.

Marty

, Sur une generalization de la notion de groupe, 8ⁱem congres Math Scandinaves, Stockholm, (1934), 45–90.

25.

Ming

P.P.

and Ming

L.Y.

, Fuzzy topology 1, J Math Anal Appl 76 (1980), 571–599.

26.

Ming

P.P.

and Ming

L.Y.

, Fuzzy topology 2, J Math Anal Appl 77 (1980), 20–37.

27.

Rosenfeld

, Fuzzy groups, J Math Anal Appl 35 (1971), 512–517.

28.

Sostak

, On a fuzzy topological structure, Rendiconti Circolo Mathematico Palermo (Suppl, Ser II) 11 (1985), 89–103.

29.

Yan

C.-H.

and Guo

S.-Z.

, I-fuzzy topological groups, Fuzzy Sets and Systems 161 (2010), 2166–2180.

30.

Ying

, A new approach for fuzzy topology (III), Fuzzy Sets and Systems 55 (1993), 193–207.

31.

Zahedi

M.M.

, Bolurian

and Hasankhani

, On polygroups and fuzzy subpolygroups, J Fuzzy Math 3 (1995), 1–15.

32.

Zhan

, Davvaz

and Shum

K.P.

, Probability n-ary hypergroups, Information Sciences 141 (2111), 1151–1166.

33.

Zhan

and Davvaz

, Study of fuzzy algebraic hypersystems from a general viewpoint, International Journal of Fuzzy Systems 12 (2111), 73–71.