Structures of fuzzy truth values based on a type-2 fuzzy set

Abstract

In this paper, based on a type-2 fuzzy set, structures of fuzzy truth values on a linearly ordered set are investigated. Some special type-2 fuzzy sets are first presented. Then, rough approximations of fuzzy truth values based on a type-2 fuzzy set are introduced. Next, topological and rough structures of fuzzy truth values based on a type-2 fuzzy set are obtained. Finally, rough equal relations of fuzzy truth values based on a type-2 fuzzy set are proposed. These results will be helpful for the study of type-2 fuzzy sets.

Keywords

Type-2 fuzzy sets fuzzy truth values structures fuzzy topologies rough approximations rough equal relations

1 Introduction

Type-2 fuzzy sets were introduced by Zadeh [36], extending the notion of type-1 fuzzy sets (ordinary fuzzy sets). They can be seen as fuzzy sets with special fuzzy sets as truth values, i.e, fuzzy sets with fuzzy truth values. Nowadays, type-2 fuzzy sets have already been used in many aspects [4 , 15–18].

Equivalent expressions of type-2 fuzzy sets were completed by Mendel [4], Karnik and Mendel [11, 12], and Mendel and John [18]. Overviews on type-2 fuzzy sets were presented by Mendel [15, 16].

Operations on type-2 fuzzy sets are one of main research topics. But the truth value algebra of type-2 fuzzy sets is its base. Walker et al. [34] studied the algebra of fuzzy truth values. Hu et al. [8] extended operations of fuzzy true values to general binary operation on a linearly ordered set and triangle norms. Hu et al. [9] proposed extended supremum and extended infimum, and introduces new operations on the algebra of fuzzy truth values.

Rough set theory, proposed by Pawlak [23], studies intelligent systems characterized by insufficient and incomplete information, and has been successfully applied to machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, image processing, signal analysis, knowledge discovery, decision analysis, expert systems and many other fields [24 , 26–28]. Pawlak’s rough set model is based on equivalence relations. The core concept of this theory is upper and lower approximation operations, which are the operations induced by an equivalent relation on the universe. They can also be seen as a closure operator and a interior operator of the topology induced by an equivalent relation on the universe.

So far, however, topological and rough structures of fuzzy truth values based on a type-2 fuzzy set have not been reported. The aim of this paper is to address this topic.

The remaining part of this paper is organized as follows: In Section 2, we recall some basic concepts of fuzzy truth values, fuzzy topologies and type-2 fuzzy sets. In Section 3, we consider rough approximations of fuzzy truth values based on a type-2 fuzzy set and give their properties. In Section 4, we obtain topological structures of fuzzy truth values based on a type-2 fuzzy set and reveal the fact that every finite fuzzy topological space is a fuzzy approximating space. In Section 5, we give rough structures of fuzzy truth values based on a type-2 fuzzy set. In Section 6, we discuss f rough equal relations of fuzzy truth values based on a type-2 fuzzy set. Conclusion is in Section 7.

2 Preliminaries

In this paper, I denotes [0, 1] and stipulate sup ∅ =0.

We briefly recall some basic concepts of fuzzy sets and soft sets.

2.1 Fuzzy truth values

Let X just be a set and Y a set with a binary operation * on it. We denote the set of all mappings from X to Y by Map (X, Y). Then, Map (X, Y) automatically inherits a binary operation, which we also denote by *, defined as follows: for f, g ∈ Map (X, Y), $(f * g) (x) = f (x) * g (x) (x \in X) .$

A fuzzy subset A of the set X is a mapping A : X → I. The set X has no operations on it. So operations on the set Map (X, I) of all fuzzy subsets of X come from operations on I. Some operations on I are ∨, ∧ and c, given by $\begin{matrix} x \lor y = \max {x, y}; \\ x \land y = \min {x, y}; \\ x^{c} = 1 - x . \end{matrix}$

Throughout this paper, J denotes either a linearly ordered set with an involution N or the associated algebra (J, ∨ , ∧ , N), where the involution N is a one-to-one order reversing mapping with the property that N (N (u)) = u (∀ u ∈ J). If for A ⊆ J, supA ∈ J and sup infA ∈ J as the linear order, then J is called complete. If J is bounded, then the minimum and maximum in J are written as 0 and 1, respectively, then the associated algebra (J, ∨ , ∧ , N) is written as (J, ∨ , ∧ , N, 0, 1) (see [34]).

In practice, J is usually I itself or a finite subset of I that includes 0 and 1.

If f ∈ Map (J, I), then the fuzzy set f is called a fuzzy truth value on J.

In this paper, $\bar{1}$ represents the fuzzy set which satisfies $\bar{1} (u) = 1$ for each u ∈ J and $\bar{0}$ represents the fuzzy set which satisfies $\bar{0} (u) = 0$ for each u ∈ J.

Obviously, $\bar{1}$ and $\bar{0}$ are two fuzzy truth values.

Some relations and operations of fuzzy truth values are given by Zadeh [35] as follows: for f, g ∈ Map (J, I),

f = g ⇔ f (u) = g (u) for each u ∈ J;

f ⊆ g ⇔ f (u) ≤ g (u) for each u ∈ J; f⊂ g ⇔ f ⊆ g and f ≠ g;

(f ∪ g) (u) = f (u) ∨ g (u) for each u ∈ J;

(f ∩ g) (u) = f (u) ∧ g (u) for each u ∈ J;

f^N (u) = N (f (u)) for each u ∈ J.

Especially, f^c (u) =1 - f (u) for each u ∈ J.

Obviously, f = g ⇔ f ⊆ g and g ⊆ f.

Two operations of fuzzy truth values are defined by Waker et al. [34] as follows: for f ∈ Map (J, I), $\begin{matrix} f^{L} (u) & = & \underset{s \leq u}{V} f (s) f o r e a c h u \in J; \\ f^{R} (u) & = & \underset{s \geq u}{V} f (s) f o r e a c h u \in J . \end{matrix}$

We may also consider the following operations of fuzzy truth values (see [34]): for f, g ∈ Map (J, I), $\begin{matrix} (f ⊔ g) (u) & = & ⋁_{s \lor t = u} (f (s) \land g (t)) for each u \in J; \\ (f ⊓ g) (u) & = & ⋁_{s \land t = u} (f (s) \land g (t)) for each u \in J . \end{matrix}$

Proposition 2.1. [34] The following hold for f, g ∈ Map (J, I).

f ⊆ f^L, f ⊆ f^R;

(f^L) ^L = f^L, (f^R) ^R = f^R;

(f^L) ^R = (f^R) ^L;

(f^c) ^L = f^R, (f^c) ^R = f^L;

(f ∪ g) ^L = f^L ∪ g^L, (f ∪ g) ^R = f^R ∪ g^R;

(f ∩ g) ^L ⊆ f^L ∪ g^L, (f ∩ g) ^R ⊆ f^R ∩ g^R;

f ⊆ g implies f^L ⊆ g^L and f^R ⊆ g^R.

Proposition 2.2. [34] The following hold for f, g ∈ Map (J, I).

f ⊔ g = (f ∩ g^L) ∪ (f^L ∩ g) = (f ∪ g) ∩ (f^L ∩ g^L);

f ⊓ g = (f ∩ g^R) ∪ (f^R ∩ g) = (f ∪ g) ∩ (f^R ∩ g^R);

(f ⊔ g) ^c = f^c ⊓ g^c, (f ⊓ g) ^c = f^c ⊔ g^c;

(f ⊔ g) ^L = f^L ∩ g^L = f^L ⊔ g = f ⊔ g^L = f^L ⊔ g^L, (f ⊓ g) ^L = f^L ⊓ g^L;

(f ⊔ g) ^R = f^R ⊔ g^R, (f ⊓ g) ^R = f^R ∩ g^R = f^R ⊓ g = f ⊓ g^R = f^R ⊓ g^R;

f ⊆ g implies f ⊔ g = f^L ⊓ g and f ⊓ g = f^R ⊓ g.

A fuzzy set is called a fuzzy point in J, if it takes the value 0 for each v ∈ J except one, say, u ∈ J. If its value at u is λ (0 < λ ≤ 1), we denote this fuzzy point by u_λ, where the point u is called its support and λ is called its height (see [13, 25]).

For a fuzzy point u_λ and f ∈ Map (J, I), we defined u_λ ∈ f by u_λ ⊆ f.

Obviously, u_λ ∈ f ⇔ λ ≤ f (u) .

Example 2.3. Let J = {u₁, u₂, u₃, u₄, u₅, u₆}.

(1) Put f (u₁) =0.1, f (u₂) =0.7, f (u₃) =1, f (u₄) =0.2, f (u₅) =0.5, f (u₆) =0.9 . Then f is a fuzzy set in J. We denote it by $\begin{matrix} f & = & {(u_{1}, 0.1), (u_{2}, 0.7), (u_{3}, 1), (u_{4}, 0.2), \\ (u_{5}, 0.5), (u_{6}, 0.9)} . \end{matrix}$

(2) We have $\begin{matrix} f^{c} & = & {(u_{1}, 0.9), (u_{2}, 0.3), (u_{3}, 0), (u_{4}, 0.8), \\ (u_{5}, 0.5), (u_{6}, 0.1)} . \end{matrix}$

(3) Put $\begin{matrix} g & = & {(u_{1}, 0.4), (u_{2}, 0.2), (u_{3}, 0.8), (u_{4}, 0.3), \\ (u_{5}, 0), (u_{6}, 0.7)} . \end{matrix}$

Then $\begin{matrix} f \cup g & = & {(u_{1}, 0.4), (u_{2}, 0.7), (u_{3}, 1), (u_{4}, 0.3), \\ (u_{5}, 0.5), (u_{6}, 0.9)} . \\ f \cap g & = & {(u_{1}, 0.1), (u_{2}, 0.2), (u_{3}, 0.8), (u_{4}, 0.2), \\ (u_{5}, 0), (u_{6}, 0.7)} . \end{matrix}$

(4) Put $\begin{matrix} h & = & {(u_{1}, 0), (u_{2}, 0.6), (u_{3}, 0), (u_{4}, 0), \\ (u_{5}, 0), (u_{6}, 0)} . \end{matrix}$

Then h isafuzzypointin J .

Pick λ = 0.6 and u = u₂. Then u_λ = h.

Obviously, u_λ ∈ f and u_λ ∉ g.

After the introduction of a fuzzy set, it motivated researcher from all the branches of science and technology for investigation its properties and applications. It has been applied for the studies in analysis by Das [6, 7], Tripathy and Das [29]. Fuzzy fixed point theory has been researched by Tripathy et al. [30] and others.

2.2 Fuzzy topologies

In this article we consider the fuzzy topology introduced by Chang [1]. After Chang, the notion of fuzzy topology has been applied and investigated by Tripathy and Ray [31 –33] and many others.

Definition 2.4. [1] Let τ ⊆ Map (J, I). Then τ is called a fuzzy topology on J, if

(i) $\bar{1}, \bar{0} \in τ$ , (ii) f, g ∈ τ implies f ∩ g ∈ τ, (iii) {f_α : α ∈ Γ} ⊆ τ implies ∪ {f_α : α ∈ Γ} ∈ τ.

The pair (J, τ) is called a fuzzy topological space and every member of τ is called a fuzzy open set in J. f ∈ Map (J, I) is called a fuzzy closed set in J, if f^c ∈ τ.

Moreover, τ is called the indiscrete fuzzy topology on J, if $τ = {\bar{1}, \bar{0}}$ .

Let (J, τ) be a fuzzy topological space. Then for f ∈ Map (J, I), interior of A and the closure of f (see [1]), denoted by int_τ (f) and cl_τ (f), respectively, are defined as follows:

int_τ (f) = ∪ {g : g ⊆ f and g ∈ τ} ,

cl_τ (f) = ∩ {g : g ⊇ f and g^c ∈ τ} .

Definition 2.5. [2, 3] Let τ ⊆ Map (J, I). Then τ is called a generalized fuzzy topology on U, if (i) $\bar{0} \in τ$ , (ii) {f_α : α ∈ Γ} ⊆ τ implies ∪ {f_α : α ∈ Γ} ∈ τ.

Moreover, the pair (J, τ) is called a generalized fuzzy topological space and every member of τ is called a generalized fuzzy open set in J. f ∈ Map (J, I) is called a generalized fuzzy closed set in J, if f^c ∈ τ.

2.3 Type-2 fuzzy sets

Below, we recall the concept of type-2 fuzzy sets.

Definition 2.6. Let X be a nonempty set. Then a mapping A : X → Map (J, I) is called a type-2 fuzzy set on X, where A (x) ≐ A_x ∈ Map (J, I), called a fuzzy grade [43], is a fuzzy truth value on J for any x ∈ X with membership function A (x) (u) = A_x (u) for any u ∈ J.

A_x is called the secondary membership function (MFnd); A_x (u) is called the secondary membership grade (MGnd); the support set of the MFnd, i.e., Supp (A) = {u ∈ J : A_x (u) ≠0} ≐ J_x ⊆ J is called the primary membership function (MFst); and Supp (A_x) is the primary membership grade (MGst) of x.

The footprint of uncertainty (FOU) is a subset of X × J related with all primary memberships, i.e., $FOU (A) = {(x, y) : x \in X, y \in Supp (A_{x})} .$

Remark 2.7. In some literature on type-2 fuzzy sets [11 , 18–21], A type-2 fuzzy set is expressed as $A = {((x, u), A (x, u)) : x \in X, u \in J_{x} \subseteq [0, 1]},$ or $\begin{matrix} A & = & \int_{x \in X} \int_{u \in J_{x}} A (x, u) / (x, u) \\ = & \int_{x \in X} [\int_{u \in J_{x}} A_{x} (u) / u] / x, \end{matrix}$ where “∫∫” denotes union over all admissible x and u, J_x ⊆ I and A (x, u) = A (x) (u) = A_x (u). It is ambiguous on the famous “J_x” in the definition, so is FOU (A) = ⋃ _x∈XJ_x [14 , 17–21] or $FOU (A) = ⋃_{x \in X} [\underline{μ} (x), \bar{μ} (x)]$ [17 , 21], which is the FOU of A.

The set of all type-2 fuzzy sets on X is denoted by Map (X, Map (J, I)).

Below, we consider some special type-2 fuzzy sets.

Definition 2.8. Let A ∈ Map (X, Map (J, I)).

A is called full, if $⋃_{x \in X} A_{x} = \bar{1}$ .

A is called keeping intersection, if for any x₁, x₂ ∈ X, there exists x₃ ∈ X such that A_{x
₁} ∩ A_{x
₂} = A_{x
₃}.

A is called keeping union, if for any x₁, x₂ ∈ X, there exists x₃ ∈ X such that A_{x
₁} ∪ A_{x
₂} = A_{x
₃}.

A is called topological, if {A_x : x ∈ X} is a fuzzy topology on J.

Obviously, every topological type-2 fuzzy set is full, keeping intersection and keeping union.

Example 2.9. Let X = {x₁, x₂, x₃} and J = {u₁, u₂, u₃, u₄}. Suppose that A ∈ Map (X, Map (J, I)) is defined as follows $\begin{matrix} A_{x_{1}} & = & {(u_{1}, 0), (u_{2}, 1), (u_{3}, 0), (u_{4}, 1)}, \\ A_{x_{2}} & = & \bar{0}, \\ A_{x_{3}} & = & {(u_{1}, 1), (u_{2}, 0), (u_{3}, 1), (u_{4}, 0)} . \end{matrix}$

Obviously, A is full.

We have A_{x
₁} ∩ A_{x
₂} = A_{x
₁} ∩ A_{x
₃} = A_{x
₂} ∩ A_{x
₃} = A_{x
₂},

$A_{x_{1}} \cup A_{x_{3}} = \bar{1} \neq A_{x} (\forall x \in X) .$

Thus A is keeping intersection. But A is not keeping union.

Example 2.10. Let X = {x₁, x₂, x₃} and J = {u₁, u₂, u₃, u₄}. Suppose that A ∈ Map (X, Map (J, I)) is defined as follows $\begin{matrix} A_{x_{1}} & = & {(u_{1}, 0), (u_{2}, 1), (u_{3}, 0), (u_{4}, 1)}, A_{x_{2}} = \bar{1}, \\ A_{x_{3}} & = & {(u_{1}, 1), (u_{2}, 0), (u_{3}, 1), (u_{4}, 0)} . \end{matrix}$

Obviously, A is full.

We have A_{x
₁} ∪ A_{x
₂} = A_{x
₁} ∪ A_{x
₃} = A_{x
₂} ∪ A_{x
₃} = A_{x
₂},

$A_{x_{1}} \cap A_{x_{3}} = \bar{0} \neq A_{x} (\forall x \in X) .$

Thus A is keeping union. But A is not keeping intersection.

Example 2.11. Let X = {x₁, x₂, x₃} and J = {u₁, u₂, u₃, u₄}. Suppose that A ∈ Map (X, Map (J, I)) is defined as follows $\begin{matrix} A_{x_{1}} & = & {(u_{1}, 0.1), (u_{2}, 0.2), (u_{3}, 0.3), (u_{4}, 0.5)}, \\ A_{x_{2}} & = & {(u_{1}, 0.2), (u_{2}, 0.8), (u_{3}, 0.4), (u_{4}, 0.7)}, \\ A_{x_{3}} & = & {(u_{1}, 0.5), (u_{2}, 0.9), (u_{3}, 0.5), (u_{4}, 1)} . \end{matrix}$

We have $\begin{matrix} A_{x_{1}} \subset A_{x_{2}} \subset A_{x_{3}}, \\ A_{x_{1}} \cup A_{x_{2}} \cup A_{x_{3}} \\ = {(u_{1}, 0.5), (u_{2}, 0.9), (u_{3}, 0.5), (u_{4}, 1)} \neq \bar{1} . \end{matrix}$

Then A is keeping intersection and keeping union. But A is not full.

From Examples 2.13, 2.14 and 2.15, we have the following relationships:

3 Rough approximations of fuzzy truth values based on a type-2 fuzzy set

In this section, we consider a pair of approximations of fuzzy truth values based on a type-2 fuzzy set and give their properties.

Let A ∈ Map (X, Map (J, I)). For x ∈ X, u ∈ J and f ∈ Map (J, I), denote

A_* (f) _u = {λ ∈ I : ∃ x ∈ X s . t . u_λ ∈ A_x ⊆ f} ,

${A^{*} (f)}_{u} = {λ \in I : \exists x \in X s . t . u_{λ} \in A_{x}, A_{x} \cap f \neq \bar{0}};$

${A_{*} (f)}_{u}^{x} = {λ \in I : u_{λ} \in A_{x} \subseteq f},$

${A^{*} (f)}_{u}^{x} = {λ \in I : u_{λ} \in A_{x}, A_{x} \cap f \neq \bar{0}};$

${A_{*} (f)}^{x} (u) = \sup {A_{*} (f)}_{u}^{x} (u \in J),$

${A^{*} (f)}^{x} (u) = \sup {A^{*} (f)}_{u}^{x} (u \in J) .$

Definition 3.1. Let A ∈ Map (X, Map (J, I)). Then based on A, we define the following operations: for f ∈ Map (J, I), $\begin{matrix} A_{*} (f) (u) & = & \sup {A_{*} (f)}_{u} (u \in J), \\ A^{*} (f) (u) & = & \sup {A^{*} (f)}_{u} (u \in J) . \end{matrix}$

Then A_* (f) and A^* (f) are called the A-lower and A-upper approximations of f, respectively; A_* and A^* are called the A-lower and A-upper approximation operations, respectively.

f ∈ Map (J, I) is called A-definable if A_* (f) = A^* (f); f ∈ Map (J, I) is called A-rough if A_* (f) ≠ A^* (f).

In this paper, we denote $\begin{matrix} τ_{A} & = & {f \in Map (J, I) : f = A_{*} (f)}, \\ σ_{A} & = & {f \in Map (J, I) : f = A^{*} (f)} . \end{matrix}$

Lemma 3.2. Let A ∈ Map (X, Map (J, I)), u ∈ J and f, g ∈ Map (J, I). Then the following properties hold.

A_* (f) _u ⊆ A^* (f) _u.

a) ${A_{*} (f)}_{u} = ⋃_{x \in X} {A_{*} (f)}_{u}^{x}$ ;

b) ${A^{*} (f)}_{u} = ⋃_{x \in X} {A^{*} (f)}_{u}^{x}$ .

a) f ⊆ g ⇒ A_* (f) _u ⊆ A_* (g) _u;

b) $f \subseteq g \Rightarrow A * {(f)}_{u} \subseteq A * {(g)}_{u} .$ .

If A is keeping intersection, then A_* (f ∩ g) _u = A_* (f) _u ∩ A_* (g) _u.

A^* (f ∪ g) _u = A^* (f) _u ∪ A^* (g) _u.

If A is ↑, then A_* (f) _u ⊆ A_* (f) _v and A^* (f) _u ⊆ A^* (f) _v whenever u ≤ v.

If A is ↓, then A_* (f) _u ⊇ A_* (f) _v and A^* (f) _u ⊇ A^* (f) _v whenever u ≤ v.

Proof. (1) and (2) are obvious.

(3) a) When A_* (f) _u =∅. Obviously, A_* (f) _u ⊆ A_* (g) _u.

When A_* (f) _u≠ ∅. Let λ ∈ A_* (f) _u. Then u_λ ∈ A_x ⊆ f for some x ∈ X. Since A_x ⊆ f and f ⊆ g, we have A_x ⊆ g. This implies u_λ ∈ A_x ⊆ g. Thus λ ∈ A_* (g) _u.

Hence A_* (f) _u ⊆ A_* (g) (u) .

b) Suppose A^* (f) _u≠ ∅. Let λ ∈ A^* (f) _u. Then $u_{λ} \in A_{x}, A_{x} \cap f \neq \bar{0}$ for some x ∈ X.

$A_{x} \cap f \neq \bar{0}$ implies A_x (v) ∧ f (v) = (A_x ∩ f) (v) >0 for some v ∈ J .

Since f ⊆ g, we have f (v) ≤ g (v).

Then (A_x ∩ g) (v) = A_x (v) ∧ g (v) ≥ A_x (v) ∧ f (v) >0 .

Thus $A_{x} \cap g \neq \bar{0}$ . This implies λ ∈ A^* (g) _u.

Hence A^* (f) _u ⊆ A^* (g) _u.

(4) By (3), A_* (f ∩ g) _u ⊆ A_* (f) _u ∩ A_* (g) _u.

Conversely, we can suppose A_* (f) _u∩ A_* (g) _u ≠ ∅. Let λ ∈ A_* (f) _u ∩ A_* (g) _u.

Then there exist x₁, x₂ ∈ X such that u_λ ∈ A_{x
₁} ⊆ f and u_λ ∈ A_{x
₂} ⊆ g.

Thus λ ≤ A_{x
₁} (u) and λ ≤ A_{x
₂} (u) .

This implies λ ≤ A_{x
₁} (u) ∧ A_{x
₂} (u) = (A_{x
₁} ∩ A_{x
₂}) (u).

Since A is keeping intersection, A_{x
₁} ∩ A_{x
₂} = A_x for some x ∈ X.

Now λ ≤ A_x (u). So u_λ ∈ A_x.

Since A_{x
₁} ⊆ f and A_{x
₂} ⊆ g, A_x ⊆ f ∩ g. Thus λ ∈ A^* (f ∪ g) _u.

So A_* (f ∩ g) _u ⊇ A_* (f) _u ∩ A_* (g) _u.

Hence A_* (f ∩ g) _u = A_* (f) _u ∩ A_* (g) _u .

(5) By (3), A^* (f ∪ g) _u ⊇ A^* (f) _u ∪ A^* (g) _u.

Conversely, we can suppose A^* (f∪ g) _u ≠ ∅. Let λ ∈ A^* (f ∪ g) _u.

Then u_λ ∈ Map (J, I) and there exists x ∈ X such that u_λ ∈ A_x, $A_{x} \cap (f \cup g) \neq \bar{0}$ .

Now $A_{x} \cap (f \cup g) = (A_{x} \cap f) \cup A_{x} \cap g) \neq \bar{0} .$

Thus $A_{x} \cap f \neq \bar{0}$ or $A_{x} \cap g \neq \bar{0}$ . This implies λ ∈ A^* (f) _u ∪ A^* (g) _u.

So A^* (f ∪ g) _u ⊆ A^* (f) _u ∪ A^* (g) _u.

Hence A^* (f ∪ g) _u = A^* (f) _u ∪ A^* (g) _u .

(6) For λ ∈ A_* (f) _u, we have u_λ ∈ A_x ⊆ f for some x ∈ X. Then λ ≤ A_x (u).

Since A is ↑, A_x (u) ≤ A_x (v).

Then λ ≤ A_x (v). So λ ≤ A_x (v) ⊆ f.

This implies λ ∈ A_* (f) _v.

Thus A_* (f) _u ⊆ A_* (f) _v .

Similarly, one can establish A^* (f) _u ⊆ A^* (f) _v.

(7) For λ ∈ A_* (f) _v, we have v_λ ∈ A_x ⊆ f for some x ∈ X. Then λ ≤ A_x (v).

Since A is ↓, A_x (v) ≤ A_x (u). Then λ ≤ A_x (u).

So λ ≤ A_x (u) ⊆ f.

This implies λ ∈ A_* (f) _u.

Thus A_* (f) _v ⊆ A_* (f) _u .

Similarly, one can establish A^* (f) _u ⊇ A^* (f) _v. □

Theorem 3.3. Let A ∈ Map (X, Map (J, I)), let P = (U, A) be a fuzzy soft approximation space and f, g ∈ Map (J, I). Then the following properties hold.

A_* (f) = ⋃ _x∈XA_* (f) ^x

= ⋃ {A_x : x ∈ X and A_x ⊆ f} .

A^* (f) = ⋃ _x∈XA^* (f) ^x

$= ⋃ {A_{x} : x \in X and A_{x} \cap f \neq \bar{0}} .$

A_* (f) ⊆ A^* (f); A_* (f) ⊆ f.

$A_{*} (\bar{0}) = A^{*} (\bar{0}) = \bar{0}$ ; $A_{*} (\bar{1}) = A^{*} (\bar{1})$ .

f ⊆ g ⇒ A_* (f) ⊆ A_* (g); f ⊆ g ⇒ A^* (f) ⊆ A^* (g).

A^* (f ∪ g) = A^* (f) ∪ A^* (g).

If A is ↑, then (A_* (f)) ^L ⊆ A_* (f^L) and [A^* (f)] ^L ⊆ A^* (f^L).

If A is ↓, then (A_* (f)) ^R ⊆ A_* (f^R) and [A^* (f)] ^R ⊆ A^* (f^R).

Proof. (1), (2) and (4) are obvious.

(3) By Lemma 3.2, A_* (f) ⊆ A^* (f).

Let u ∈ J. If A_* (f) _u =∅, then A_* (f) _u = 0 ≤ g (u).

If A_* (f) _u≠ ∅, then for each λ ∈ A_* (f) _u, u_λ ∈ A_x ⊆ f for some x ∈ X. So λ ≤ A_x (u) ≤ f (u). This implies sup A_* (f) _u ≤ f (u).

Thus A_* (f) ⊆ f .

(5) Let f ⊆ g. For u ∈ J, suppose A_* (f) _u≠ ∅. For each λ ∈ A_* (f) _u, by Lemma 3.2, λ ∈ A_* (g) _u.

Then λ ≤ sup A_* (g) (u).

This implies sup A_* (f) _u ≤ A_* (g) (u),

Thus A_* (f) ⊆ A_* (g) .

Similarly, one can establish f ⊆ g ⇒ A^* (f) ⊆ A^* (g) .

(6) By (5), A^* (f ∪ g) ⊇ A^* (f) ∪ A^* (g) .

Conversely, let u ∈ J, we can suppose A^* (f∪ g) _u ≠ ∅. For each λ ∈ A^* (f ∪ g) _u, by Lemma 3.2, λ ∈ A^* (f) _u ∪ A^* (g) (u).

Then λ ∈ A^* (f) _u or λ ∈ A^* (g) (u).

Thus λ ≤ sup A^* (f) _u or λ ≤ sup A^* (g) _u .

So λ ≤ A^* (f) (u) ∨ A^* (g) (u) = (A^* (f) ∪ A^* (g)) (u) .

This implies A^* (f ∪ g) (u) = sup A^* (f ∪ g) _u ≤ (A^* (f) ∪ A^* (g)) (u) .

Thus A^* (f ∪ g) ⊆ A^* (f) ∪ A^* (g) .

Hence A^* (f ∪ g) = A^* (f) ∪ A^* (g) .

(7) Suppose u ∈ J. For s ≤ u, λ ∈ A_* (f) _s, since A is ↑, by Lemma 3.2(6), we have λ ∈ A_* (f) _u. Then u_λ ∈ A_x ⊆ f for some x ∈ X.

By Proposition 2.1(1), f ⊆ f^L.

Then u_λ ∈ A_x ⊆ f^L.

This implies λ ∈ A_* (f^L) _u.

So A_* (f) _s ⊆ A_* (f^L) _u .

Thus, A_* (f) (s) ≤ A_* (f^L) (u).

It follows from (A_* (f)) ^L (u) = ⋁ _s≤uA_* (f) (s) ≤ A_* (f^L) (u) .

Hence (A_* (f)) ^L ⊆ A_* (f^L).

Similarly, one can establish (A^* (f)) ^L ⊆ A^* (f^L).

(8) Suppose u ∈ J. For s ≥ u, λ ∈ A_* (f) _s, since A is ↓, by Lemma 3.2(7), we have λ ∈ A_* (f) _u. Then u_λ ∈ A_x ⊆ f for some x ∈ X. By Proposition 2.1(1), f ⊆ f^R. Then u_λ ∈ A_x ⊆ f^R.

This implies λ ∈ A_* (f^R) _u.

So A_* (f) _s ⊆ A_* (f^R) _u .

Thus, A_* (f) (s) ≤ A_* (f^R) (u).

It follows from (A_* (f)) ^L (u) = ⋁ _s≤uA_* (f) (s) ≤ A_* (f^R) (u) .

Hence (A_* (f)) ^R ⊆ A_* (f^R).

Similarly, one can establish (A^* (f)) ^R ⊆ A^* (f^R). □

To illustrate Theorem 3.3, we give the following examples.

Example 3.4. Let X = {x₁, x₂} and J = {u₁, u₂, u₃, u₄}. Suppose that A ∈ Map (X, Map (J, I)) is defined as follows: $\begin{matrix} A_{x_{1}} & = & {(u_{1}, 0), (u_{2}, 0.2), (u_{3}, 0.3), (u_{4}, 0)}, \\ A_{x_{2}} & = & {(u_{1}, 0.2), (u_{2}, 0.8), (u_{3}, 0.1), (u_{4}, 0.7)} . \end{matrix}$

(1) Put $f = {(u_{1}, 0.5), (u_{2}, 0), (u_{3}, 0), (u_{4}, 0.7)} .$

We have $A_{x_{1}} \cap f = \bar{0}, A_{x_{2}} \cap f \neq \bar{0} .$

Then ${A^{*} (f)}_{u_{1}} = {A^{*} (f)}_{u_{1}}^{x_{2}} = (0, 0.2] .$

So A^* (f) (u₁) = sup A^* (f) _{u
₁} = 0.2 < 0.5 = A_{x
₁} .

Thus A⊈A^* (f) .

(2) Since $A_{x_{1}} \subseteq \bar{1}, A_{x_{2}} \subseteq \bar{1},$ $A^{*} (\bar{1})_{u_{2}} = (0, 0.8],$ we have $A_{*} (\bar{1}) (u_{2}) = \sup A_{*} (\bar{1})_{u_{2}} = 0.8 < 1 = \bar{1} (u_{2}) .$

Thus $A_{*} (\bar{1}) \neq \bar{1} .$

(3) Put $g = {(u_{1}, 0.1), (u_{2}, 0.8), (u_{3}, 0.4), (u_{4}, 0.6)} .$

Then $f \cup g = {(u_{1}, 0.5), (u_{2}, 0.8), (u_{3}, 0.4), (u_{4}, 0.7)} .$

We have $\begin{matrix} A_{x_{1}} f, A_{x_{2}} f; A_{x_{1}} \subseteq g, A_{x_{2}} g; \\ A_{x_{1}} \subseteq f \cup g, A_{x_{2}} \subseteq f \cup g . \end{matrix}$

A_* (f) _{u
₂} =∅ implies A_* (f) (u₂) =0 .

${A_{*} (g)}_{u_{2}} = {A_{*} (g)}_{u_{2}}^{x_{1}} = (0, 0.2]$ implies A_* (g) (u₂) = sup A_* (g) _{u
₂} = 0.2 .

$A_{*} (f \cup g)_{u_{2}} = A_{*} (f \cup g)_{u_{2}}^{x_{1}} \cup A_{*} (f \cup g)_{u_{2}}^{x_{2}} = (0, 0.8]$ implies A_* (f ∪ g) (u₂) = sup A_* (f ∪ g) _{u
₂} = 0.8 .

Note that (A_* (f) ∪ A_* (g)) (u₂) =0.2 ≠ A_* (f ∪ g) (u₂) =0.8 .

Thus A_* (f) ∪ A_* (g) ≠ A_* (f ∪ g) .

(4) Put $\begin{matrix} h & = & {(u_{1}, 0.2), (u_{2}, 0), (u_{3}, 0.4), (u_{4}, 0.7)}, \\ d & = & {(u_{1}, 0.4), (u_{2}, 0.6), (u_{3}, 0), (u_{4}, 0.5)} . \end{matrix}$

Then $h \cap d = {(u_{1}, 0.2), (u_{2}, 0), (u_{3}, 0), (u_{4}, 0.5)} .$

We have $\begin{matrix} A_{x_{1}} \cap h \neq \bar{0}, A_{x_{2}} \cap h \neq \bar{0}; A_{x_{1}} \cap d \neq \bar{0}, \\ A_{x_{2}} \cap d \neq \bar{0}; \\ A_{x_{1}} \cap (h \cap d) = \bar{0}, A_{x_{2}} \cap (h \cap d) \neq \bar{0} . \end{matrix}$

A^* (h) _{u
₃} = A^* (d) _{u
₃} = (0, 0.3] implies A^* (h) (u₃) = A^* (d) (u₃) =0.3 .

$A^{*} {(h \cap d)}_{u_{3}} = A^{*} {(h \cap d)}_{u_{3}}^{x_{2}} = (0, 0.1]$ implies A^* (h ∩ d) (u₃) =0.1 .

Note that (A^* (h) ∩ A^* (d)) (u₃) =0.3 ≠ A^* (h ∩ d) (u₃) =0.1

Then A^* (h) ∩ A^* (d) ≠ A^* (h ∩ d) .

Example 3.5. Let X = {x₁, x₂} and J = {u₁, u₂, u₃, u₄}. Suppose that A ∈ Map (X, Map (J, I)) is defined as follows: $\begin{matrix} A_{x_{1}} & = & {(u_{1}, 0.1), (u_{2}, 0.2), (u_{3}, 0.3), (u_{4}, 0.5)}, \\ A_{x_{2}} & = & {(u_{1}, 0.2), (u_{2}, 0.8), (u_{3}, 0.1), (u_{4}, 0.7)} . \end{matrix}$

Put $\begin{matrix} f & = & {(u_{1}, 0.1), (u_{2}, 0.8), (u_{3}, 0.4), (u_{4}, 0.7)}, \\ g & = & {(u_{1}, 0.3), (u_{2}, 0.8), (u_{3}, 0.1), (u_{4}, 1)} . \end{matrix}$

Then $f \cap g = {(u_{1}, 0.1), (u_{2}, 0.8), (u_{3}, 0.1), (u_{4}, 0.7)} .$

We have

A_{x
₁} ⊆ f, A_{x
₂}⊈f ; A_{x
₁}⊈g, A_{x
₂} ⊆ g ; A_{x
₁}⊈f ∩ g, A_{x
₂}⊈f ∩ g.

${A_{*} (f)}_{u_{2}} = {A_{*} (f)}_{u_{2}}^{x_{1}} = {λ \in (0, 1] : λ \leq A_{x_{1}} (u_{2})} = (0, 0.2]$ implies A_* (f) (u₂) = sup A_* (f) _{u
₂} = 0.2 .

${A_{*} (g)}_{u_{2}} = {A_{*} (g)}_{u_{2}}^{x_{2}} = {λ \in (0, 1] : λ \leq A_{x_{2}} (u_{2})} = (0, 0.8]$ implies A_* (g) (u₂) = sup A_* (g) _{u
₂} = 0.8 .

A_* (f∩ g) _{u
₂} = ∅ implies A_* (f ∩ g) (u₂) =0 .

Note that (A_* (f) ∩ A_* (g)) (u₂) =0.2 ≠ A_* (f ∩ g) (u₂) =0 . Then A_* (f) ∩ A_* (g) ≠ A_* (f ∩ g) .

Example 3.6. Let X = {x₁, x₂} and J = {u₁, u₂, u₃, u₄}. Suppose that A ∈ Map (X, Map (J, I)) is defined as follows: $\begin{matrix} A_{x_{1}} & = & {(u_{1}, 0.1), (u_{2}, 0.2), (u_{3}, 0.3), (u_{4}, 0.5)}, \\ A_{x_{2}} & = & {(u_{1}, 0.2), (u_{2}, 0.8), (u_{3}, 0.1), (u_{4}, 0.7)} . \end{matrix}$

Put f = {(u₁, 0.1) , (u₂, 0.8) , (u₃, 0.4) , (u₄, 0.7)} .

Then f^c = {(u₁, 0.9) , (u₂, 0.2) , (u₃, 0.6) , (u₄, 0.3)} .

We have A_{x
₁} ⊆ f, A_{x
₂}⊈f, A_{x
₁}⊈f^c, A_{x
₂}⊈f^c.

Since $\begin{matrix} {A_{*} (f)}_{u_{1}} = {A_{*} (f)}_{u_{1}}^{x_{1}} = (0, 0.1], \\ {A_{*} (f)}_{u_{2}} = {A_{*} (f)}_{u_{2}}^{x_{1}} = (0, 0.2], \\ {A_{*} (f)}_{u_{3}} = {A_{*} (f)}_{u_{3}}^{x_{1}} = (0, 0.3], \\ {A_{*} (f)}_{u_{4}} = {A_{*} (f)}_{u_{4}}^{x_{1}} = (0, 0.5], \end{matrix}$ we have $\begin{matrix} A_{*} (f) (u_{1}) = \sup {A_{*} (f)}_{u_{1}} = 0.1, \\ A_{*} (f) (u_{2}) = \sup {A_{*} (f)}_{u_{2}} = 0.2, \\ A_{*} (f) (u_{3}) = \sup {A_{*} (f)}_{u_{3}} = 0.3, \\ A_{*} (f) (u_{4}) = \sup {A_{*} (f)}_{u_{4}} = 0.5 . \end{matrix}$

Then $A_{*} (f) = {(u_{1}, 0.1), (u_{2}, 0.2), (u_{3}, 0.3), (u_{4}, 0.5)} .$

So $(A_{*} (f))^{c} = {(u_{1}, 0.9), (u_{2}, 0.8), (u_{3}, 0.7), (u_{4}, 0.5)} .$

Since A_* (f^c) _{u
₁} = A_* (f^c) _{u
₂} = A_* (f^c) _{u
₃} = A_* (f^c) _{u
₄} = ∅ , we have $\begin{matrix} A_{*} (f^{c}) (u_{1}) & = & A_{*} (f^{c}) (u_{2}) = A_{*} (f^{c}) (u_{3}) \\ = & A_{*} (f^{c}) (u_{4}) = 0 . \end{matrix}$

Then $A_{*} (f^{c}) = \bar{0} .$

Thus A_* (f^c) ≠ (A_* (f)) ^c .

Example 3.7. Let X = {x₁, x₂} and J = {u₁, u₂, u₃, u₄}. Suppose that A ∈ Map (X, Map (J, I)) is defined as follows: $\begin{matrix} A_{x_{1}} & = & {(u_{1}, 0.1), (u_{2}, 0.2), (u_{3}, 0.3), (u_{4}, 0.5)}, \\ A_{x_{2}} & = & {(u_{1}, 0), (u_{2}, 0.8), (u_{3}, 0), (u_{4}, 0.7)} . \end{matrix}$

Put $f = {(u_{1}, 0.7), (u_{2}, 0), (u_{3}, 0.4), (u_{4}, 0)} .$

Then $f^{c} = {(u_{1}, 0.3), (u_{2}, 1), (u_{3}, 0.6), (u_{4}, 1)} .$

We have $\begin{matrix} A_{x_{1}} \cap f \neq \bar{0}, A_{x_{2}} \cap f = \bar{0}, A_{x_{1}} \cap f^{c} \neq \bar{0}, \\ A_{x_{2}} \cap f^{c} \neq \bar{0} . \end{matrix}$

Since $\begin{matrix} {A^{*} (f)}_{u_{1}} = {A^{*} (f)}_{u_{1}}^{x_{1}} = (0, 0.1], \\ {A^{*} (f)}_{u_{2}} = {A^{*} (f)}_{u_{2}}^{x_{1}} = (0, 0.2], \\ {A^{*} (f)}_{u_{3}} = {A^{*} (f)}_{u_{3}}^{x_{1}} = (0, 0.3], \\ {A^{*} (f)}_{u_{4}} = {A^{*} (f)}_{u_{4}}^{x_{1}} = (0, 0.5], \end{matrix}$

we have $\begin{matrix} A^{*} (f) (u_{1}) = \sup {A^{*} (f)}_{u_{1}} = 0.1, \\ A^{*} (f) (u_{2}) = \sup {A^{*} (f)}_{u_{2}} = 0.2, \\ A^{*} (f) (u_{3}) = \sup {A^{*} (f)}_{u_{3}} = 0.3, \\ A^{*} (f) (u_{4}) = \sup {A^{*} (f)}_{u_{4}} = 0.5 . \end{matrix}$

Then $A^{*} (f) = {(u_{1}, 0.1), (u_{2}, 0.2), (u_{3}, 0.3), (u_{4}, 0.5)} .$

So $(A^{*} (f))^{c} = {(u_{1}, 0.9), (u_{2}, 0.8), (u_{3}, 0.7), (u_{4}, 0.5)} .$

Since $\begin{matrix} {A^{*} (f^{c})}_{u_{1}} = {A^{*} (f^{c})}_{u_{1}}^{x_{1}} \cup {A^{*} (f^{c})}_{u_{1}}^{x_{2}} = (0, 0.1], \\ {A^{*} (f^{c})}_{u_{2}} = {A^{*} (f^{c})}_{u_{2}}^{x_{1}} \cup {A^{*} (f^{c})}_{u_{2}}^{x_{2}} = (0, 0.8], \\ {A^{*} (f^{c})}_{u_{3}} = {A^{*} (f^{c})}_{u_{3}}^{x_{1}} \cup {A^{*} (f^{c})}_{u_{3}}^{x_{2}} = (0, 0.3], \\ {A^{*} (f^{c})}_{u_{4}} = {A^{*} (f^{c})}_{u_{4}}^{x_{1}} \cup {A^{*} (f^{c})}_{u_{4}}^{x_{2}} = (0, 0.7], \end{matrix}$

we have $\begin{matrix} A^{*} (f^{c}) (u_{1}) = \sup {A^{*} (f^{c})}_{u_{1}} = 0.1, \\ A^{*} (f^{c}) (u_{2}) = \sup {A^{*} (f^{c})}_{u_{2}} = 0.8, \\ A^{*} (f^{c}) (u_{3}) = \sup {A^{*} (f^{c})}_{u_{3}} = 0.3, \\ A^{*} (f^{c}) (u_{4}) = \sup {A^{*} (f^{c})}_{u_{4}} = 0.7 . \end{matrix}$

Then A^* (f^c) = {(u₁, 0.1) , (u₂, 0.8) , (u₃, 0.3) , (u₄, 0.7)} .

Thus A^* (f^c) ≠ (A^* (f)) ^c .

Lemma 3.8. Let A ∈ Map (X, Map (J, I)) and f ∈ Map (J, I). Then the following properties hold.

If A_* (f) = g, then for each u ∈ J, A_* (f) _u = A_* (g) _u ⊆ A^* (g) _u ⊆ A^* (f) _u .

If A^* (f) = g, then for each u ∈ J, A_* (f) _u ⊆ A^* (f) _u ⊆ A_* (g) _u ⊆ A^* (g) _u .

Proof. (1) By Theorem 3.3, A_* (f) ⊆ f. Then g ⊆ f.

By Lemma 3.2, A_* (f) _u ⊇ A_* (g) _u ⊆ A^* (g) _u ⊆ A^* (f) _u .

It suffices to show A_* (f) _u ⊆ A_* (g) _u.

Suppose A_* (f) _u≠ ∅. For each λ ∈ A_* (f) _u, u_λ ∈ A_{x
₀} ⊆ f for some x₀ ∈ X.

By Theorem 3.3, g = ⋃ {A_x : x ∈ X and A_x ⊆ f}. Then A_{x
₀} ⊆ g. This implies λ ∈ A_* (g) _u. Thus A_* (f) _u ⊆ A_* (g) _u.

(2) By Lemma 3.2, A_* (f) _u ⊆ A^* (f) _u andA_* (g) _u ⊆ A^* (g) _u .

It suffices to show A^* (f) _u ⊆ A_* (g) _u .

Suppose A^* (f) _u≠ ∅. For each λ ∈ A^* (f) _u, u_λ ∈ A_{x
₀}, $A_{x_{0}} \cap f \neq \bar{0}$ for some x₀ ∈ X.

By Theorem 3.3, $g = ⋃ {A_{x} : x \in X and A_{x} \cap f \neq \bar{0}}$ . Then A_{x
₀} ⊆ g. This implies λ ∈ A_* (g) _u. Thus A^* (f) _u ⊆ A_* (g) _u. □

Theorem 3.9. Let A ∈ Map (X, Map (J, I)) and f, g ∈ Map (J, I). Then the following properties hold.

A_* (A_* (f)) = A_* (f).

A_* (A^* (f)) = A^* (f).

A^* (A_* (f)) ⊇ A_* (f).

A^* (A^* (f)) ⊇ A^* (f).

Proof. (1) By Theorem 3.3, it suffices to show that

A_* (g) (u) ≥ g (u) for each u ∈ J, where g = A_* (f) .

Suppose A_* (f) _u≠ ∅. For each λ ∈ A_* (f) _u, by Lemma 3.8, A_* (f) _u = A_* (g) _u. Then λ ∈ A_* (g) _u. Thus λ ≤ sup A_* (g) _u = A_* (g) (u). Hence g (u) = A_* (f) (u) = sup A_* (f) _u ≤ A_* (g) (u).

(2) By Theorem 3.3, it suffices to show that $A_{*} (g) (u) \geq g (u) for each u \in J, where g = A^{*} (f) .$

Suppose A^* (f) _u≠ ∅. For each λ ∈ A^* (f) _u, by Lemma 3.8, A^* (f) _u ⊆ A_* (g) _u. Then λ ∈ A_* (g) _u. Thus λ ≤ sup A_* (g) _u = A_* (g) (u). Hence g (u) = A^* (f) (u) = sup A^* (f) _u ≤ A_* (g) (u).

(3) It suffices to show that $A^{*} (g) (u) \geq g (u) for each u \in J, where g = A_{*} (f) .$

Suppose A_* (f) _u≠ ∅. For each λ ∈ A_* (f) _u, by Lemma 3.8, A_* (f) _u = A_* (g) _u. Then λ ∈ A_* (g) _u. By Lemma 3.2, λ ∈ A^* (g) _u. Thus λ ≤ sup A^* (g) _u = A^* (g) (u). Hence g (u) = A_* (f) (u) = sup A_* (f) _u ≤ A^* (g) (u).

(4) It suffices to show that $A^{*} (g) (u) \geq g (u) for each u \in J, where g = A^{*} (f) .$

Suppose A^* (f) _u≠ ∅. For each λ ∈ A^* (f) _u, by Lemma 3.8, A^* (f) _u ⊆ A_* (g) _u. Then λ ∈ A_* (g) _u. By Lemma 3.2, λ ∈ A^* (g) (u). Thus λ ≤ sup A^* (g) _u = A^* (g) (u). Hence g (u) = A^* (f) (u) = sup A_* (f) _u ≤ A^* (g) (u). □

Theorem 3.10. Let A ∈ Map (X, Map (J, I)). If X is finite, then the following are equivalent:

A is full;

f ⊆ A^* (f) for any f ∈ Map (J, I);

$A^{*} (\bar{1}) = \bar{1}$ ;

$A_{*} (\bar{1}) = \bar{1}$ .

Proof. (1) ⇒ (2). By Theorem 3.3, A_* (f) ⊆ f. It suffices to show that $f (u) \leq A^{*} (f) (u) for each u \in J .$

If f (u) =0, then f (u) ≤ A^* (f) (u).

If f (u) ≠0, by A is full, $\bar{1} = ⋃_{x \in X} A_{x}$ , then $f (u) \leq 1 = \bar{1} (u) = ⋁ {A_{x} (u) : x \in X} .$

Thus f (u) ≤ A_{x
₀} (u) for some x₀ ∈ X. Put λ = A_{x
₀} (u). Then u_λ ∈ A_x .

Note that (A_{x
₀} ∩ f) (u) = A_{x
₀} (u) ∧ f (u) = f (u). Then $A_{x_{0}} \cap f \neq \bar{0}$ . This implies λ ∈ A^* (f) _u. Then λ ≤ A^* (f) (u) . So f (u) ≤ A^* (f) (u) .

(2) ⇒ (3). This is obvious.

(3) ⇒ (4). The proof follows from Theorem 3.3.

(4) ⇒ (1). Suppose $A_{*} (\bar{1}) = \bar{1}$ . By Theorem 3.3, $\begin{matrix} A_{*} (\bar{1}) & = & ⋃ {A_{x} : x \in X and A_{x} \subseteq \bar{1}} \\ = & ⋃ {A_{x} : x \in X} . \end{matrix}$

Then $\bar{1} = ⋃ {A_{x} : x \in X}$ . Thus A is full. □

Theorem 3.11. Let A ∈ Map (X, Map (J, I)). Then the following properties hold.

If A is keeping intersection, then A_* (f ∩ g) = A_* (f) ∩ A_* (g) for any f, g ∈ Map (J, I).

If X is finite, and A is full and keeping union, then $A^{*} (f) = \bar{1}$ for any $f \in Map (J, I) \ {\bar{0}}$ .

1) If A is ↑ and keeping intersection, then A_* (f ⊔ g) ⊇ A_* (f) ⊔ A_* (g);

2) If A is ↓ and keeping intersection, then A_* (f ⊓ g) ⊇ A_* (f) ⊓ A_* (g).

Proof. (1) By Theorem 3.3, A_* (f ∩ g) ⊆ A_* (f) ∩ A_* (g) .

It suffices to show A_* (f ∩ g) (u) ≥ (A_* (f) ∩ A_* (g)) (u) for each u ∈ J .

Suppose A_* (f ∩ g) (u) < (A_* (f) ∩ A_* (g)) (u) for some u ∈ J .

Put c = A_* (f ∩ g) (u) , a = A_* (f) (u) and b = A_* (g) (u) . Then c < min {a, b}.

Since c < a and a = sup A_* (f) (u), we have λ₁ > c for some λ₁ ∈ A_* (f) _u.

Since c < b and b = sup A_* (g) (u), we have λ₂ > c for some λ₂ ∈ A_* (g) _u.

λ₁ ∈ A_* (f) _u implies u_{λ
₁} ∈ A_{x
₁} ⊆ f for some x₁ ∈ X.

λ₂ ∈ A_* (g) (u) implies u_{λ
₂} ∈ A_{x
₂} ⊆ g for some x₂ ∈ X.

Since A is keeping intersection, A_{x
₁} ∩ A_{x
₂} = A_{x
₃} for some x₃ ∈ X. This implies A_{x
₃} ⊆ f ∩ g.

Put λ = min {λ₁, λ₁} . Then λ > c.

Note that λ ≤ A_{x
₃} (u). So u_λ ∈ A_{x
₃}. This implies λ ∈ A_* (f ∩ g) _u. Thus λ ≤ c, contradiction.

(2) Since A is full and keeping union, we have $A (x^{*}) = ⋃_{x \in X} A_{x} = \bar{1}$ for some x^* ∈ X.

Thus, for each $f \in Map (J, I) \ {\bar{0}}$ and each u ∈ J.

Pick λ = 1. Then u_λ ∈ A (x^*) and $A (x^{*}) \cap f = f \neq \bar{0}$ .

So λ ∈ A^* (f) _u.

This implies A_* (f) (u) =1.

Hence $A^{*} (f) = \bar{1}$ .

(3)1) By Proposition 2.2(1), f ⊔ g = (f ∩ g^L) ∪ (f^L ∩ g). Then A_* (f ⊔ g) = A_* [(f ∩ g^L) ∪ (f^L ∩ g)]. By Theory 3.3(5), A_* (f ⊔ g) ⊇ A_* (f ∩ g^L) ∪ A_* (f^L ∩ g) .

Since A is keeping intersection, by 1), we have A_* (f ∩ g^L) = A_* (f) ∩ A_* (g^L), A_* (f^L ∩ g) = A_* (f^L) ∩ A_* (g) .

By Theory 3.3(7), (A_* (f)) ^L ⊆ A_* (f^L), (A_* (g)) ^L ⊆ A_* (g^L).

Thus

A_* (f ⊔ g) ⊇ [A_* (f) ∩ (A_* (g)) ^L] ∪ [(A_* (f)) ^L ∩ A_* (g)] = A_* (f) ⊔ A_* (g) .

2) By Proposition 2.2(2), f ⊓ g = (f ∩ g^R) ∪ (f^R ∩ g). Then A_* (f ⊓ g) = A_* [(f ∩ g^R) ∪ (f^R ∩ g)].

By Theory 3.3(5), A_* (f ⊓ g) ⊇ A_* (f ∩ g^R) ∪ A_* (f^R ∩ g) .

Since A is keeping intersection, by 1), we have A_* (f ∩ g^R) = A_* (f) ∩ A_* (g^R), A_* (f^R ∩ g) = A_* (f^R) ∩ A_* (g) .

By Theory 3.3(8), (A_* (f)) ^R ⊆ A_* (f^R), (A_* (g)) ^R ⊆ A_* (g^R).

Thus

A_* (f ⊓ g) ⊇ [A_* (f) ∩ (A_* (g)) ^R)] ∪ [(A_* (f)) ^R ∩ A_* (g)] = A_* (f) ⊓ A_* (g) . □

Example 3.12. Let X = {x₁, x₂, x₃} and J = {u₁, u₂, u₃, u₄}. Suppose that A ∈ Map (X, Map (J, I)) is defined as follows: $\begin{matrix} A_{x_{1}} & = & {(u_{1}, 0), (u_{2}, 1), (u_{3}, 0), (u_{4}, 1)}, A_{x_{2}} = \bar{1}, \\ A_{x_{3}} & = & {(u_{1}, 1), (u_{2}, 0), (u_{3}, 1), (u_{4}, 0)} . \end{matrix}$

By Example 2.14, A is full and keeping union.

For any $f \in Map (J, I) \ {\bar{0}}$ , $A_{x_{2}} \cap f \neq \bar{0}$ . By Theorem 3.3, $A^{*} (f) \supseteq A_{x_{2}} = \bar{1}$ . Thus $A^{*} (f) = \bar{1}$ .

4 Topological structures of fuzzy truth values based on a type-2 fuzzy set

In this section we investigate topological structures of fuzzy truth values based on a type-2 fuzzy set.

4.1 Fuzzy topologies induced by type-2 fuzzy sets

Theorem 4.1 Let A ∈ Map (X, Map (J, I)). Then the following properties hold.

τ_A is a generalized fuzzy topology on J.

If A is full, keeping intersection, then τ_A is a fuzzy topology on J.

If X is finite, A is full and keeping union, then $σ_{A} = {\bar{0}, \bar{1}}$ is the indiscrete fuzzy topology on J.

Proof. (1) By Theorem 3.3, $\bar{0} \in τ_{A}$ .

Let f_α ∈ τ_A for each α ∈ Γ. Denote g = ∪ {f_α : α ∈ Γ}. Since f_α ⊆ g for each α ∈ Γ, by Theorem 3.3, we have f_α = A_* (f_α) ⊆ A_* (g). So g = ∪ {f_α : α ∈ Γ} ⊆ A_* (g).

By Theorem 3.3, A_* (g) ⊆ g. Thus A_* (g) = g. This implies ∪ {f_α : α ∈ Γ} ∈ τ_A. Hence τ_A is a generalized fuzzy topology on J.

(2) By Theorem 3.10, $\bar{1} \in τ$ . By Theorem 3.11, f ∩ g ∈ τ_A whenever f, g ∈ τ_A. By (1), τ_A is a generalized fuzzy topology on J. Thus τ is a fuzzy topology on J.

(3) This holds by Theorems 3.3, 3.10 and 3.11. □

Definition 4.2. Let A ∈ Map (X, Map (J, I)). Then τ_A is called the fuzzy topology induced by A.

Example 4.3. In Example 2.13, we obtain $\begin{matrix} {f \in Map (J, I) : A_{x} \subseteq f for some x \in X} \\ = {f_{i} : 1 \leq i \leq 8}, \end{matrix}$ where $\begin{matrix} f_{1} = A_{x_{1}} = {(u_{1}, 0), (u_{2}, 1), (u_{3}, 0), (u_{4}, 1)}, \\ f_{2} = A_{x_{2}} = \bar{0}, \\ f_{3} = A_{x_{3}} = {(u_{1}, 1), (u_{2}, 0), (u_{3}, 1), (u_{4}, 0)}, \\ f_{4} = {(u_{1}, 1), (u_{2}, 1), (u_{3}, 0), (u_{4}, 1)}, \\ f_{5} = {(u_{1}, 0), (u_{2}, 1), (u_{3}, 1), (u_{4}, 1)}, \\ f_{6} = {(u_{1}, 1), (u_{2}, 1), (u_{3}, 1), (u_{4}, 0)}, \\ f_{7} = {(u_{1}, 1), (u_{2}, 0), (u_{3}, 1), (u_{4}, 1)}, \\ f_{8} = \bar{1} . \end{matrix}$

By Theorem 3.3, A^* (f₁) = A_{x
₁} ∪ A_{x
₂} = f₁. Then f₁ ∈ τ_A.

Similarly, f₂, f₃, f₈ ∈ τ_A, f₄, f₅, f₆, f₇ ∉ τ_A.

Hence τ_A = {f₁, f₂, f₃, f₈} .

Theorem 4.4. Let A ∈ Map (X, Map (J, I)) and τ_A the fuzzy topology induced by A on J. If A is full and keeping intersection, then the following properties hold.

{A^* (f) : f ∈ Map (J, I)} ⊆ τ_A = {A_* (f) : f ∈ Map (J, I)} .

τ_A ⊇ {A_x : x ∈ X} .

If A is topological, then τ_A = {A_x : x ∈ X} .

A_* is an interior operator of τ_A.

Proof. (1) By Theorem 3.9, {A^* (f) : f ∈ Map (J, I)} ⊆ τ_A .

Obviously, τ_A ⊆ {A_* (f) : f ∈ Map (J, I)} .

Let g ∈ {A_* (f) : f ∈ Map (J, I)}. Then g = A_* (f) for some f ∈ Map (J, I).

By Theorem 3.9, A_* (A_* (f)) = A_* (f).

This implies g ∈ τ_A.

Thus τ_A ⊇ {A_* (f) : f ∈ Map (J, I)} .

Hence {A^* (f) : f ∈ Map (J, I)} ⊆ τ_A = {A_* (f) : f ∈ Map (J, I)} .

(2) For each x ∈ X, by Theorem 3.3,

A_* (A_x) = ⋃ {A_x′ : x′ ∈ X and A_x′ ⊆ A_x} ⊆ f .

Then A_x = A_* (A_x). So A_x ∈ τ_A.

Thus {A_x : x ∈ X} ⊆ τ_A .

(3) By (2), τ_A ⊇ {A_x : x ∈ X}.

Let f ∈ τ_A. If $f = \bar{0}$ , by A is topological, then f ∈ {A_x : x ∈ X}.

If $f \neq \bar{0}$ , then f = A_* (f). Put X₁ = {x ∈ X : A_x ⊆ f}, by Theorem 3.3, f = ∪ {A_x : x ∈ X₁}.

We claim that X₁≠ ∅. Otherwise, X₁ =∅. Then for each u ∈ J, f (u) = sup {A_x (u) : x ∈ X₁} = sup ∅ =0. So $f = \bar{0}$ , a contradiction.

Since A is keeping union, there exists x′ ∈ X such that ∪ {A_x : x ∈ X₁} = A_x′, that is, f = A_x′.

This implies τ_A ⊆ {A_x : x ∈ X}.

Hence τ_A = {A_x : x ∈ X} .

(4) It suffices to show that A_* (f) = int_τ (f) for each f ∈ Map (J, I) .

By (1), A_* (f) ∈ τ_A. By Theorem 3.3, A_* (f) ⊆ f. Thus $A_{*} (f) \subseteq {int}_{τ} (f) .$

Conversely, for each g ∈ τ_A with g ⊆ f, we have g = A_*g ⊆ A_* (f) by Theorem 3.3.

Thus int_τ (f) = ⋃ {g : g ∈ τ_A and g ⊆ f} ⊆ A_* (f) .

Hence A_* (f) = int_τ (f) . □

4.2 Type-2 fuzzy sets induced by fuzzy topologies

We recall that a fuzzy topology is finite if this fuzzy topology has only finite elements.

Definition 4.5. Let τ be a finite fuzzy topology on J. Denote τ = {U_x : x ∈ X}, where X is the set of indexes. Define a mapping A_τ : X → Map (J, I) by A_τ (x) = U_x for each x ∈ X. Then the type-2 fuzzy set A_τ is called the type-2 fuzzy set induced by τ.

Example 4.6. Let J = {u₁, u₂, u₃, u₄} and let τ = {U_{x
₁}, U_{x
₂}, U_{x
₃}, U_{x
₄}} be a finite fuzzy topology where $\begin{matrix} U_{x_{1}} & = & {(u_{1}, 0), (u_{2}, 1), (u_{3}, 0), (u_{4}, 1)}, U_{x_{2}} = \bar{0}, \\ U_{x_{3}} & = & {(u_{1}, 1), (u_{2}, 0), (u_{3}, 1), (u_{4}, 0)}, U_{x_{4}} = \bar{1} . \end{matrix}$

Now we define a mapping f_τ : X → Map (J, I) by A_τ (x) = U_x for each x ∈ X.

Thus, (f_τ) _E is the type-2 fuzzy set induced by τ.

Definition 4.7. Let (J, μ) be a finite fuzzy topological space. If there exists A ∈ Map (X, Map (J, I)) such that A is full and keeping intersection, and τ_A = μ, then (J, μ) is called a fuzzy approximating space.

Example 4.8. Let A ∈ Map (X, Map (J, I)) in Example 2.13 and (J, τ) a fuzzy topological space in Example 4.6. Then A is full and keeping intersection.

By Example 4.3, τ_A = τ.

Thus (J, τ) is a fuzzy approximating space.

The following Proposition 4.9 can easily be proved.

Proposition 4.9. (1) Let τ be a finite fuzzy topology on J and A_τ the type-2 fuzzy set induced by τ. Then A_τ is topological.

(2) Let τ₁ and τ₂ be two finite fuzzy topologies on J, and let A_τ₁ and A_τ₁ be two type-2 fuzzy sets induced respectively by τ₁ and τ₂. If τ₁ ⊆ τ₂, then A_τ₁ ⊆ A_τ₂ .

Theorem 4.10. Let τ be a finite fuzzy topology on J, A_τ the type-2 fuzzy set induced by τ and be the fuzzy topology induced by A_τ. Then

Proof. Put τ = {U_x : x ∈ X}. Then A_τ : X → Map (J, I) is a mapping, where A_τ (x) = U_x for each x ∈ X. By Proposition 4.9, A_τ is topological.

By Theorem 4.4, $τ_{A_{τ}} = {A_{τ} (x) : x \in X}$

Hence $τ_{A_{τ}} = τ$ □

Corollary 4.11. Every finite fuzzy topological space is a fuzzy approximating space.

Theorem 4.12. Let (J, τ) be a finite fuzzy topological space. Then there exists A ∈ Map (X, Map (J, I)) such that A_* (f) = int_τ (f) for each f ∈ Map (J, I) .

Proof. Put τ = {U_x : x ∈ X}, where X is the set of indexes. Define a mapping A : X → Map (J, I) by A_x = U_x for each x ∈ X .

By Proposition 4.9, A is topological.

Let f ∈ Map (J, I). It suffices to show A_* (f) (u) = int_τ (f) (u) for each u ∈ J .

Let λ ∈ A_* (f) _u. Then u_λ ∈ A_x = U_x ⊆ f for some x ∈ X. Then λ ≤ U_x (u) and U_x ∈ τ.

Note that int_τ (f) (u) = ⋁ {g (u) : g ∈ τ and g ⊆ f} . Then $λ \leq ⋁ {g (u) : g \in τ and g \subseteq f} = {int}_{τ} (f) (u) .$

This implies A_* (f) (u) = sup A_* (f) _u ≤ int_τ (f) (u) .

On the other hand, put C_u = {g (u) : g (u) ≠0, g ∈ τ and g ⊆ f} .

Let g (u) ∈ C_u. Then g = A_x for some x ∈ X. Pick λ = g (u). Thus u_λ ∈ A_x ⊆ f .

This implies λ ∈ A_* (f) _u. Then g (u) = λ ≤ sup A_* (f) _u = A_* (f) (u). So sup C_u ≤ A_* (f) (u) .

Thus $\begin{matrix} {int}_{τ} (A) (u) & = & ⋁ {g (u) : g \in τ and g \subseteq f} \\ = & \sup C_{u} \leq A_{*} (f) (u) . \end{matrix}$

Hence A_* (f) (u) = int_τ (A) (u) . □

Theorem 4.13. Given A ∈ Map (X, Map (J, I)). Let τ_A be the fuzzy topology induced by A and A_{τ
_A} the type-2 fuzzy set induced by τ_A. If A is full and keeping intersection, then the following properties hold.

A ⊆ A_{τ
_A} .

If A is topological, then A = A_{τ
_A} .

Proof. (1) By Theorem 4.4, τ_A ⊇ {A_x : x ∈ X}. Denote

τ_A = {U_x : x ∈ X′} , where X ⊆ X′ and U_x = A_x for each x ∈ X .

Thus A_{τ
_A} is a mapping given by

A_{τ
_A} : X′ → Map (J, I) , where A_{τ
_A} (x) = U_x for each x ∈ X′ .

Hence A ⊆ A_{τ
_A} .

(2) Since A is topological, by Theorem 4.4, X = X′.

Hence A = A_{τ
_A} . □

5 Rough structures of fuzzy truth values based on a type-2 fuzzy set

In this section we introduce the concept of A-rough sets and give their properties.

Definition 5.1. Let A ∈ Map (X, Map (J, I)). Then for each f ∈ Map (J, I), f is called A-definable, if A_* (f) = A^* (f); f is called A-rough, if A_* (f) ≠ A^* (f).

Moreover, the sets Pos_A (f) = A_* (f) , ${Neg}_{A} (f) = \bar{1} - A^{*} (f),$ Bnd_A (f) = A^* (f) - A_* (f) are called the A-positive region, the A-negative region and the A-boundary region of f, respectively.

Definition 5.3 illustrates the fact that fuzzy truth values can be divided into two categories.

Example 5.2. Let X = {x₁, x₂, x₃} and J = {u₁, u₂, u₃, u₄}. Suppose that A ∈ Map (X, Map (J, I)) is defined as follows: $\begin{matrix} A_{x_{1}} & = & {(u_{1}, 0.3), (u_{2}, 0), (u_{3}, 0), (u_{4}, 0.4)}, \\ A_{x_{2}} & = & {(u_{1}, 0.2), (u_{2}, 0.4), (u_{3}, 0.6), (u_{4}, 0.3)}, \\ A_{x_{3}} & = & {(u_{1}, 0.9), (u_{2}, 0.2), (u_{3}, 0.5), (u_{4}, 0.3)} . \end{matrix}$

Put f = {(u₁, 0.4) , (u₂, 0.1) , (u₃, 0) , (u₄, 0.7)} .

Since A_{x
₁} ⊂ f, A_{x
₂} ⊄ f and A_{x
₃} ⊄ f, by Theorem 3.3, we have A_* (f) = A_{x
₁}.

Since $A (x_{i}) \cap f \neq \bar{0} (i = 1, 2, 3)$ , by Theorem 3.3, we have A^* (f) = A_{x
₁} ∪ A_{x
₂} ∪ A_{x
₃}.

Thus A_* (f) ≠ A^* (f). This show that f is A-rough.

Denote $\begin{matrix} R & = & {f \in Map (J, I) : f is A - rough}, \\ D & = & {f \in Map (J, I) : A is A - definable} . \end{matrix}$

Theorem 5.3. Let A ∈ Map (X, Map (J, I)). Then for each f ∈ Map (J, I), $f \in D \Leftrightarrow A^{*} (f) \subseteq f .$

Proof. If A is A-definable, then A_* (f) = A^* (f). By Theorem 3.3, A_* (f) ⊆ f. Thus A^* (f) ⊆ f.

Conversely, suppose A^* (f) ⊆ f. To prove that A is A-definable, we only need to show $A^{*} (f) (u) = A_{*} (f) (u) for each x \in X .$

Put g = A^* (f). By Lemma 3.8, A^* (f) _u ⊆ A_* (g) _u. Since g ⊆ f, by Lemma 3.2, we have A_* (g) _u ⊆ A_* (f) _u.

This implies A^* (f) _u = A_* (f) _u.

Hence

A^* (f) (u) = sup A^* (f) _u = sup A_* (f) _u = A_* (f) (u) . □

Corollary 5.4. Let f ∈ Map (X, Map (J, I)). Then for each f ∈ Map (J, I), $f \in R \Leftrightarrow A^{*} (f) ⊈ f .$

Proof. This follows from Theorem 5.3. □

For $P, Q \subseteq Map (J, I)$ , denote

$P \ Q = {f \in Map (J, I) : f \in P and f \notin Q} .$

The following Theorem 5.6 gives the structure of A-rough sets.

Theorem 5.5. Let A ∈ Map (X, Map (J, I)).

$ℛ \cup D = ℳ a p (J, ℐ), ℛ \cap D = \emptyset a n d σ_{A} \subseteq D .$

If A is full, then $ℛ = ℳ a p (J, ℐ), \ σ_{A} a n d D = σ_{A} \subseteq τ_{A} .$

If A is full and keeping union, then $ℛ = ℳ a p (J, ℐ), \ (\bar{0}, \bar{1}) a n d D = {\bar{0}, \bar{1}} = σ_{A} \subseteq τ_{A} .$

Proof. This holds by Theorems 3.3, 3.10, 3.11 and 5.3. □

The following result is a consequence of Theorem 3.11.

Theorem 5.6. Let A ∈ Map (X, Map (J, I)). If A is full and keeping union, then for $f \in Map (J, I) \ {\bar{0}}$ , ${Neg}_{A} (A) = \bar{0}, {Bnd}_{A} (A) = \bar{1} - {Pos}_{A} (A) .$

6 Rough equal relations of fuzzy truth values based on a type-2 fuzzy set

In this section we discuss rough equal relations of fuzzy truth values based on a type-2 fuzzy set.

Definition 6.1. Let A ∈ Map (X, Map (J, I)). For f, g ∈ Map (J, I), we define $\begin{matrix} f {\bar{\sim}}_{A} g \Leftrightarrow A^{*} (f) = A^{*} (g), \\ f {\underline{\sim}}_{A} g \Leftrightarrow A_{*} (f) = A_{*} (g) . \end{matrix}$

These binary relations are called the upper and lower rough equal relation, respectively.

It is easy to verify that two relations above are all equivalence relations. Thus, fuzzy truth values can be classified according to these two relations, respectively.

Proposition 6.2. Let A ∈ Map (X, Map (J, I)). Then

$f {\bar{\sim}}_{A} g \Leftrightarrow f \cup g {\bar{\sim}}_{A} f and f \cup g {\bar{\sim}}_{A} g$ ;

$f {\bar{\sim}}_{A} g$ , $f^{'} {\bar{\sim}}_{A} g^{'} \Rightarrow f \cup f^{'} {\bar{\sim}}_{A} g \cup g^{'}$ ;

$f {\bar{\sim}}_{A} g \Rightarrow f \cup (\bar{1} - g) {\bar{\sim}}_{A} \bar{1}$ ;

f ⊆ g, $g {\bar{\sim}}_{A} \bar{0} \Rightarrow f {\bar{\sim}}_{A} \bar{0}$ ;

f ⊆ g, $f {\bar{\sim}}_{A} \bar{1} \Rightarrow g {\bar{\sim}}_{A} \bar{1}$ .

If A is full and keeping union, then $f {\bar{\sim}}_{A} g$ for any $f, g \in Map (J, I) \ {\bar{0}}$ .

Proof. (1) If $f {\bar{\sim}}_{A} g$ , then A^* (f) = A^* (g). By Theorem 3.3, A^* (f ∪ g) = A^* (f) ∪ A^* (g). This implies A^* (f ∪ g) = A^* (f) = A^* (g). Thus $f \cup g {\bar{\sim}}_{A} f andf \cup g {\bar{\sim}}_{A} g$ .

Conversely, if $f \cup g {\bar{\sim}}_{A} fand f \cup g {\bar{\sim}}_{A} g$ , then $f {\bar{\sim}}_{A} g$ because ${\bar{\sim}}_{A}$ is an equivalence relation.

(2) Let $f {\bar{\sim}}_{A} g and f^{'} {\bar{\sim}}_{A} g^{'}$ . Then A^* (f) = A^* (g), A^* (f′) = A^* (g′). By Theorem 3.3, A^* (f ∪ g′) = A^* (f) ∪ A^* (f′), A^* (g ∪ g′) = A^* (g) ∪ A^* (g′). This implies A^* (f ∪ f′) = A^* (g ∪ g′). Thus $f \cup f^{'} {\bar{\sim}}_{A} g \cup g^{'}$ .

(3) Suppose $f {\bar{\sim}}_{A} g$ . Then A^* (f) = A^* (g). By Theorem 3.3, $A^{*} (f \cup (\bar{1} - g)) = A^{*} (f) \cup A^{*} (\bar{1} - g)$ . This implies $A^{*} (f \cup (\bar{1} - g)) = A^{*} (g) \cup A^{*} (\bar{1} - g) = A^{*} (\bar{1})$ . Thus $f \cup (\bar{1} - g) {\bar{\sim}}_{A} \bar{1}$ .

(4) Assume that $f \subseteq g and g {\bar{\sim}}_{A} \bar{0}$ . By Theorem 3.3, $A^{*} (f) \subseteq A^{*} (g) = A^{*} (\bar{0}) = \bar{0}$ . Then $A^{*} (f) = \bar{0} = A^{*} (\bar{0})$ . Thus $f {\bar{\sim}}_{A} \bar{0}$ .

(5) Suppose that $f \subseteq g and f {\bar{\sim}}_{A} \bar{1}$ . By Theorem 3.3, $A^{*} (g) \supseteq A^{*} (f) = A^{*} (\bar{1})$ . Since $g \subseteq \bar{1}$ , $A^{*} (g) \subseteq A^{*} (\bar{1})$ . This implies $A^{*} (g) = \bar{(} \bar{1})$ . Thus $g {\bar{\sim}}_{A} \bar{1}$ .

(6) This holds by Theorem 3.11. □

By Theorems 3.3 and 3.11, we can prove the following Proposition 6.3.

Proposition 6.3. Let A ∈ Map (X, Map (J, I)).

If A is keeping intersection, then

$f {\underline{\sim}}_{A} g \Leftrightarrow f \cap g {\underline{\sim}}_{A} f and f \cap g {\underline{\sim}}_{A} g;$

If A is keeping intersection, then

$f {\underline{\sim}}_{A} g and f^{'} {\underline{\sim}}_{A} g^{'} \Rightarrow f \cap f^{'} {\underline{\sim}}_{A} g \cap g^{'};$

$f \subseteq g and g {\underline{\sim}}_{A} \bar{0} \Rightarrow f {\underline{\sim}}_{A} \bar{0}$ ;

$f \subseteq g and f {\underline{\sim}}_{A} \bar{1} \Rightarrow g {\underline{\sim}}_{A} \bar{1}$ .

Theorem 6.4. Let A ∈ Map (X, Map (J, I)).

$A_{*} (f) = \cap {g \in Map (J, I) : f {\underline{\sim}}_{A} g} .$

If A is full, then $A^{*} (f) \supseteq \cup {g \in Map (J, I) : f {\bar{\sim}}_{A} g} .$

Proof. (1) For any g ∈ Map (J, I) with $f {\underline{\sim}}_{A} g$ , we have A_* (f) = A_* (g).

By Theorem 3.3, A_* (g) ⊆ g.

This implies A_* (f) ⊆ g. Thus $A_{*} (f) \subseteq \cap {g \in Map (J, I) : f {\underline{\sim}}_{A} g} .$

Conversely, by Theorem 3.9, A_* (A_* (f)) = A_* (f).

This implies $f {\underline{\sim}}_{A} A_{*} (f)$ .

Then $A_{*} (f) \in {g \in Map (J, I) : f {\underline{\sim}}_{A} g}$ .

Thus $A_{*} (f) \supseteq \cap {g \in Map (J, I) : f {\underline{\sim}}_{A} g} .$

Hence $A_{*} (f) = \cap {g \in Map (J, I) : f {\underline{\sim}}_{A} g} .$

(2) For any g ∈ Map (J, I) with $f {\bar{\sim}}_{A} g$ , we have A^* (f) = A^* (g).

Since A is full, by Theorem 3.10, we have g ⊆ A^* (g). This implies g ⊆ A^* (f).

Thus $A^{*} (f) \supseteq \cup {g \in Map (J, I) : f {\bar{\sim}}_{A} g} .$ □

7 Conclusions

As type-2 fuzzy sets are fuzzy sets with fuzzy truth values and the operations on type-2 fuzzy subsets will come pointwise from operations on fuzzy truth values, thus researching fuzzy truth values is important. In this paper, a pair of upper and lower of approximations of fuzzy truth values based on a type-2 fuzzy set has been considered. Topological and rough structures of fuzzy truth values based on a type-2 fuzzy set have been obtained. Moreover, rough equal relations of fuzzy truth values based on a type-2 fuzzy set have been proposed. Based on these results, classifications of fuzzy truth values have been derived. It is worth mentioning that structures of fuzzy truth values based on two different type-2 fuzzy sets are somewhat different. In future work, we will consider some concrete applications of the proposed results.

Footnotes

Acknowledgements

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and suggestions which have helped immensely in improving the quality of the paper. This work is supported by National Natural Science Foundation of China (11461005, 11261005, 11161029, 11461002), Natural Science Foundation of Guangxi (2016GXNSFAA380286, 2016GXNSFAA380045, 2016GXNSFAA380282), Guangxi Province Universities and Colleges Excellence Scholar and Innovation Team Funded Scheme, and Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce Intelligent Information Processing.

References

Chang , Fuzzy topological spaces, Journal of Mathematical Analysis and Applications 24 (1968), 182–190.

Csaszár , Generalized topology, generalized continuity, Acta Mathematica Hungarica 96 (2002), 351–357.

Csaszar , Generalized open sets, Acta Mathematica Hungarica 75 (1997), 65–87.

S.M.

Chen ,

L.W.

Lee and V.

R.L.

Shen , Weighted fuzzy interpolative reasoning systems based on interval type-2 fuzzy sets, Information Sciences 248 (2013), 15–30.

S.M.

Chen and

C.Y.

Wang , Fuzzy decision making systems based on interval type-2 fuzzy sets, Information Sciences 242 (2013), 1–21.

P.C.

Das , Statistically convergent fuzzy sequence spaces by fuzzy metric, Kyungpook Mathematical Journal 54(3) (2014), 413–423.

P.C.

Das , P-absolutely summable type fuzzy sequence spaces by fuzzy metric, Boletim da Sociedade Paranaense de Matema 32(2) (2014), 35–43.

B.Q.

Hu and

C.K.

Kwong , On type-2 fuzzy sets and their i-norm operations, Information Sciences 255 (2014), 58–81.

B.Q.

Hu and

C.Y.

Wang , On type-2 fuzzy relations and interval-valued type-2 fuzzy sets, Fuzzy Sets and Systems 236 (2014), 1–32.

10.

R.I.

John , Type-2 fuzzy sets: An appraisal of theory and applications, International Journal of Uncertainty, Fuzzi-ness and Knowledge-Based Systems 6(6) (1998), 563–576.

11.

N.N.

Karnik and

J.M.

Mendel , Operations on type-2 fuzzy sets, Fuzzy Sets and Systems 122 (2001), 327–348.

12.

N.N.

Karnik and

J.M.

Mendel , Centroid of a type-2 fuzzy set, Information Sciences 132 (2001), 195–220.

13.

Y.M.

Liu and

M.K.

Luo , Fuzzy topology, World Scientific Publishing, Singapore, 1998.

14.

J.M.

Mendel , Uncertain rule-based fuzzy logic systems: Introduction and new directions, Prentice-Hall, Upper Saddle River, NJ, 2001.

15.

J.M.

Mendel , Advances in type-2 fuzzy sets and systems, Information Sciences 177 (2007), 84–110.

16.

J.M.

Mendel , Type-2 fuzzy sets and systems: An overview, IEEE Computation Intelligence Magazine 2 (2007), 20–29.

17.

J.M.

Mendel , Computing with words and its relationships with fuzzistics, Information Sciences 177 (2007), 988–1006.

18.

J.M.

Mendel and

R.I.

John , Type-2 fuzzy sets made simple, IEEE Transactions on Fuzzy Systems 10(2) (2002), 117–127.

19.

J.M.

Mendel ,

R.I.

John and

Liu , Interval type-2 fuzzy logic systems made simple, IEEE Transactions on Fuzzy Systems 14(6) (2006), 808–821.

20.

J.M.

Mendel and

Wu , Type-2 fuzzistics for non-symmetric interval type-2 fuzzy sets: Forward problems, IEEE Transactions on Fuzzy Systems 15(5)(2007), 916–930.

21.

J.M.

Mendel and

Wu , New results about the centroid of an interval type-2 fuzzy set, including the centroid of a fuzzy granule, Information Sciences 177 (2007), 360–377.

22.

Nieminen , On the algebraic structure of fuzzy sets of type 2, Kybernetika 13 (1977), 261–273.

23.

Pawlak , Rough sets, International Journal of Computer and Information Sciences 11 (1982), 341–356.

24.

Pawlak , Rough sets: Theoretical aspects of reasoning about data, Kluwer Academic Publishers, Dordrecht, 1991.

25.

B.M.

Pu and

Y.M.

Liu , Fuzzy topology I. Neighborhood structure of a fuzzy point and Moore-Smith convergence, Journal of Mathematical Analysis and Applications 76 (1980), 571–599.

26.

Pawlak and

Skowron , Rudiments of rough sets, Information Sciences 177 (2007), 3–27.

27.

Pawlak and

Skowron , Rough sets: Some extensions, Information Sciences 177 (2007), 28–40.

28.

Pawlak and

Skowron , Rough sets and Boolean reasoning, Information Sciences 177 (2007), 41–73.

29.

B.C.

Tripathy and

P.C.

Das , On convergence of series of fuzzy real numbers, Kuwait Journal of Science & Engineering 39(1A) (2012), 57–70.

30.

B.C.

Tripathy ,

Paul and

N.R.

Das , A fixed point theorem in a generalized fuzzy metric space, Boletim da Sociedade Paranaense de Matematica 32(2) (2014), 221–227.

31.

B.C.

Tripathy and

G.C.

Ray , On Mixed fuzzy topological spaces and countability, Soft Computing 16(10) (2012), 1691–1695.

32.

B.C.

Tripathy and

G.C.

Ray , Weakly continuous functions on mixed fuzzy topological spaces, Acta Scientiarum-Technology 36(2) (2014), 331–335.

33.

B.C.

Tripathy and

G.C.

Ray , Mixed fuzzy ideal topological spaces, Applied Mathematics and Computations 220 (2013), 602–607.

34.

C.L.

Walker and

E.A.

Walker , The algebra of fuzzy truth values, Fuzzy Sets and Systems 149 (2005), 309–347.

35.

L.A.

Zadeh , Fuzzy sets, Information and Control 8 (1965), 338–353.

36.

L.A.

Zadeh , The concept of a linguistic variable and its application to approximate reasoning (I), Information Sciences 8 (1975), 199–249.

37.

W.X.

Zhang ,

W.Z.

Wu ,

J.Y.

Liang and

D.Y.

Li , Rough set theory and methods, Chinese Scientific Publishers, Beijing, 2001.