M -fuzzifying derived spaces

Abstract

In this paper, the notions of M-fuzzifying derived sets and M-fuzzifying derived operators are introduced, when M is a fuzzy lattice. Then their characterizations are given. What’s more, it is shown that the category of M-fuzzifying topological spaces, the category of M-fuzzifying closure spaces and the category of M-fuzzifying neighborhood spaces are all isomorphic to the category of M-fuzzifying derived spaces. Besides, we prove that the M-fuzzifying derived operators induced by M-fuzzifying neighborhood systems are equivalent to those in the sense of Ying.

Keywords

Isomorphism of categories MSC (2010): 54A40

1 Introduction

The concept of derived sets was first introduced by Georg Cantor in 1872. Later, derived sets became a very useful tool in several areas, involving topology, algebra, modal logic, etc. In mathematics, more specifically in point-set topology, derived set is an important and interesting topic. It is well known that neighborhood systems, interior operators and closure operators play an important role in general topology and they are good ways to characterize topology. In addition, the concepts of accumulation points, derived sets and derived operators are also very important and there is a close relationship between derived sets and topologies in general topology [4, 8]. F.G. Shi showed that there exists a one-to-one correspondence between derived operators and topologies in [12].

With the development of fuzzy mathematics [22], the notion of derived sets has been extended to fuzzy set theory. Many papers about accumulation points, derived sets, derived operators in L-topological spaces appeared, such as [2 , 18]. Nevertheless, studies on fuzzifying derived sets and fuzzifying derived operators are not so many and incomplete.

In 1991, M.S. Ying introduced the notion of [0, 1]-fuzzifying (or fuzzifying, in short) derived set by using fuzzifying neighborhood systems in [20]. But he did not give the axiomatic conditions of fuzzifying derived operators and did not discuss the relation between fuzzifying topologies and fuzzifying derived operators. In [1], although the definition of fuzzifying derived operators was given, they emphasized the fuzzifying derived operator is a family of operators on some point. Namely, D = {d_x | x ∈ X} is called a fuzzifying derived operator on X, where d_x : 2^X ⟶ [0, 1] is a map satisfying some conditions. However, this kind definition of fuzzifying derived operators looks a bit complex and can not give the definition of fuzzifying derived set of an arbitrary subset of X.

The aims of this paper is to introduce the notions of M-fuzzifying derived sets and M-fuzzifying derived operators. In addition, the relationships among M-fuzzifying topological spaces, M-fuzzifying closure spaces, M-fuzzifying neighborhood spaces and M-fuzzifying derived spaces are all discussed from the perspective of category theory.

This paper is organized as follows. In Section 2, some preliminary notions and results are recalled. In Section 3, M-fuzzifying derived sets and M-fuzzifying derived operators are introduced and their characterizations are given. In Section 4, it is shown that the category of M-fuzzifying topological spaces, the category of M-fuzzifying closure spaces and the category of M-fuzzifying neighborhood spaces are all isomorphic to the category of M-fuzzifying derived spaces. Besides, we prove that the M-fuzzifying derived operators induced by M-fuzzifying neighborhood systems are equivalent to those in the sense of Ying.

2 Preliminaries

Throughout this paper, M denotes a fuzzy lattice, i.e., a completely distributive lattice with an order-reserving involution ^′. The smallest element and the largest element in M are denoted by ⊥ and ⊤, respectively.

Let X be a non-empty set. We denote the set of all subsets on X by 2^X and denote the set of all M-subsets on X by M^X. M^X is also a fuzzy lattice when it inherits the structure of the lattice M in a natural way, by defining ∧, ∨ , ≤ and ^′ pointwisely. For all A ∈ M^X, A′ denotes the complementary set of A. We often do not distinguish a crisp subset A ∈ 2^X from its characteristic function χ_A. Also, we adopt the convention that ⋀∅ = ⊤ and ⋁∅ = ⊥.

An element a in M is called a co-prime element if b ∨ c ⩾ a implies b ⩾ a or c ⩾ a [6]. The set of non-zero co-prime elements in M is denoted by J (M). From [6], we know that in a completely distributive lattice, each element is the sup of co-prime elements.

We say that a is wedge below b, in symbols, a ≺ b, if for every subset D ⊆ M, ⋁D ≥ b implies a ≤ d for some d ∈ D [3]. We denote β (a) = {x ∈ M ∣ x ≺ a} and β^* (a) = β (a) ∩ J (M). a ≺ ^opb means that if for every subset D ⊆ M, ⋀D ≤ b implies d ≤ a for some d ∈ D. Therefore a ≺ ^opb ⇔ a′ ≺ b′ ⇔ b ≺ a. We denote α (a) = {x ∈ M ∣ x ≺ ^opa}. A complete lattice M is completely distributive if and only if a = ⋁ β (a) = ⋁ β^* (a) = ⋀ α (a) for each a ∈ M [19]. The wedge below relation in a completely distributive lattice has the interpolation property, i.e., if a ≺ b, then there exists c ∈ M such that a ≺ c ≺ b. Moreover, it is easy to see that a ≺ ⋀ _i∈Ib_i implies a ≺ b_i for every i ∈ I, whereas a ≺ ⋁ _i∈Ib_i implies a ≺ b_i for some i ∈ I [9].

Lemma 2.1. ([19]) Let M be a completely distributive lattice and let {a_i ∣ i ∈ I} ⊆ M. Then

α (⋀ _i∈Ia_i) = ⋃ _i∈Iα (a_i), i.e., α is a ⋀-⋃ mapping.

β (⋁ _i∈Ia_i) = ⋃ _i∈Iβ (a_i), i.e., β is a union-preserving mapping.

From Lemma 2.1, we know that α is an order-reversing map and β is an order-preserving map.

For any A ∈ M^X and for each a ∈ M, we use the following notations [14], A_[a] = {x ∈ X ∣ A (x) ≥ a}, A^[a] = {x ∈ X ∣ a ∉ α (A (x))}.For all a ∈ M and A ∈ 2^X, define two M-fuzzy sets a ∧ A and a ∨ A as follows [14]: $\begin{matrix} (a \land A) (x) = {\begin{matrix} a, & x \in A; \\ ⊥, & x \notin A . \end{matrix} \end{matrix}$ $\begin{matrix} (a \lor A) (x) = {\begin{matrix} ⊤, & x \in A; \\ a, & x \notin A . \end{matrix} \end{matrix}$

Theorem 2.2. ([14]) If M is a completely distributive lattice, then for each A ∈ M^X,

A = ⋁ _a∈M (a ∧ A_[a]) = ⋁ _a∈J(M) (a ∧ A_[a]).

A = ⋀ _a∈M (a ∨ A^[a]) = ⋀ _a∈α(⊥) (a ∨ A^[a]).

∀a ∈ M, A_[a] = ⋂ _b∈β(a)A_[b] = ⋂ _b∈β^*(a)A_[b].

∀a ∈ M, A^[a] = ⋃ _b∈α(a)A_[b].

In what follows, the notions of M-fuzzifying topologies, M-fuzzifying closure operators and M-fuzzifying neighborhood systems are recalled.

Definition 2.3. ([7 , 20]) An M-fuzzifying topology on X is a mapping $T : 2^{X} ⟶ M$ satisfying the following axioms:

$T (\emptyset) = T (X) = ⊤$ ;

∀A₁, A₂ ∈ 2^X, $T (A_{1}) \land T (A_{2}) \leq T (A_{1} \cap A_{2})$ ;

∀ {A_j ∣ j ∈ J} ⊆2^X, $⋀_{j \in J} T (A_{j}) \leq T (⋃_{j \in J} A_{j})$ .

$T (A)$ is interpreted as the degree of openness of A. The pair $(X, T)$ is called an M-fuzzifying topological space. A continuous map between two M-fuzzifying topological spaces $(X, T_{X})$ and $(Y, T_{Y})$ is a mapping f : X ⟶ Y such that $T_{Y} (B) \leq T_{X} (f^{\leftarrow} (B))$ for all B ∈ 2^Y, where f^← (B) = {x ∈ X ∣ f (x) ∈ B}.

The category of M-fuzzifying topological spaces and their continuous maps is denoted by MY-FTop.

Definition 2.4. ([16]) An M-fuzzifying closure operator on X is a mapping cl : 2^X ⟶ M^X satisfying the following conditions:

∀x∈ X, cl (∅) (x) = ⊥;

∀x∈ A, cl (A) (x) = ⊤;

cl (A ∪ B) = cl (A) ∨ cl (B);

cl (A) (x) = ⋀ _x∉B⊇A ⋁ _y∉Bcl (B) (y).

cl (A) (x) is called the degree to which x belongs to the closure of A. The pair (X, cl) is called an M-fuzzifying closure space.

A continuous map between two M-fuzzifying closure spaces (X, cl_X) and (Y, cl_Y) is a mapping f : X ⟶ Y such that cl_X (A) (x) ≤ cl_Y (f^→ (A)) (f (x)) for all x ∈ X and A ∈ 2^X, where f^→ (A) = {f (x) | x ∈ A}.

The category of M-fuzzifying closure spaces and their continuous maps is denoted by MY-FCS.

Definition 2.5. ([15 , 21]) An M-fuzzifying neighborhood system on X is defined to be a set $N = {N_{x} ∣ x \in X}$ of maps ${N_{x} : 2^{X} ⟶ M}$ satisfying the following conditions:

$N_{x} (\emptyset) = ⊥$ , $N_{x} (X) = ⊤$ ;

∀x ∉ A, $N_{x} (A) = ⊥$ ;

$N_{x} (A \cap B) = N_{x} (A) \land N_{x} (B)$ ;

$N_{x} (A) = ⋁_{x \in B \subseteq A} ⋀_{y \in B} N_{y} (B)$ .

$N_{x} (A)$ is called the degree to which A is a neighborhood of x. The pair $(X, N)$ is called an M-fuzzifying neighborhood space.

A continuous map between two M-fuzzifying neighborhood spaces $(X, N_{X})$ and $(Y, N_{Y})$ is a mapping f : X ⟶ Y such that $(N_{Y})_{f (x)} (B) \leq (N_{X})_{x} (f^{\leftarrow} (B))$ for all x ∈ X and B ∈ 2^Y.

The category of M-fuzzifying neighborhood spaces and their continuous maps is denoted by MY-FNS.

In general topology, the fundamental importance of accumulation points and derived sets are well known. Let $(X, T)$ be a topological space and A ∈ 2^X. Then x ∈ X is called an accumulation point of A provided that U∩ (A - {x}) ≠ ∅ for any $U \in U_{x}$ , where $U_{x}$ is the set of all neighborhoods of x. The set of all accumulation points of A is called the derived set of A, denoted by $d^{T} (A)$ . Meanwhile, we can obtain a mapping $d^{T} : 2^{X} ⟶ 2^{X}$ defined by $A \mapsto d^{T} (A)$ . The following theorem reveals the properties of derived sets.

Theorem 2.6. ([12]) Let $(X, T)$ be a topological space. For each A ∈ 2^X, $d^{T} (A)$ denotes the derived set of A. Then

$d^{T} (\emptyset) = \emptyset$ ;

∀x ∈ X, $x \notin d^{T} ({x})$ ;

$d^{T} (A \cup B) = d^{T} (A) \cup d^{T} (B)$ ;

$d^{T} (d^{T} (A)) \subseteq A \cup d^{T} (A)$ .

If a mapping d : 2^X ⟶ 2^X satisfying (D1)-(D4), then d is called a derived operator on X. The pair (X, d) is called a derived space. There exists a one-to-one correspondence between derived operators and topologies [12].

Theorem 2.7. ([12]) Let d : 2^X ⟶ 2^X be a derived operator on X. Then $T^{d} = {A \in 2^{X} | d (A^{'}) \subseteq A^{'}}$ is a topology on X and $d^{T^{d}} = d$ , $T^{d^{T}} = T$ .

A mapping f : X ⟶ Y between two derived spaces (X, d_X) and (Y, d_Y) is called continuous if f^→ (d_X (A)) ⊆ f^→ (A) ∪ d_Y (f^→ (A)) for all A ∈ 2^X.

3 M-fuzzifying derived spaces

In this section, we will introduce the notions of M-fuzzifying derived sets and M-fuzzifying derived operators. Then we will give their characterizations.

Now, we shall generalize the concept of accumulation points to M-fuzzifying settings as follows.

Definition 3.1. Let $T : 2^{X} ⟶ M$ be an M-fuzzifying topology. Define a mapping $d^{T} : 2^{X} ⟶ M^{X}$ by ∀A ∈ 2^X, ∀ x ∈ X, $d^{T} (A) (x) = ⋀_{x \notin B \supseteq A - {x}} {(T (B^{'}))}^{'},$

$d^{T} (A) (x)$ is called the degree to which x is an accumulation point of A with respect to $T$ . The mapping $d^{T} (A) : X ⟶ M$ is called the M-fuzzifying derived set of A with respect to $T$ .

Remark 3.2. If M = 2 and $d^{T} (A) (x) = ⊤$ , then $T$ is a topology and $⋀_{x \notin B \supseteq A - {x}} {(T (B^{'}))}^{'} = ⊤$ . For all B ∈ 2^X with x ∉ B, B ⊇ A - {x} implies $B^{'} \notin T$ . That is, $B^{'} \in T$ implies A - {x} ⊈ B (i.e., (A - {x})∩ B′ ≠ ∅). It is exactly the definition of accumulation points in general topology.

Example 3.3. Let X = {x, y, z} and M = [0, 1]. Define $T : 2^{X} ⟶ M$ $\begin{matrix} T (\emptyset) = T (X) = 1; \\ T ({x}) = T ({y}) = 0.8; T ({z}) = 0.6; \\ T ({x, y}) = T ({y, z}) = 0.8; T ({x, z}) = 0.6 . \end{matrix}$

It is easy to check that $T$ satisfies the axioms of (MYT1)-(MYT3). Hence $T$ is a [0, 1]-fuzzifying topology. By Definition 3.1, we obtain $\begin{matrix} d^{T} (\emptyset) (x) = d^{T} (\emptyset) (y) = d^{T} (\emptyset) (z) = 0; \\ d^{T} ({x}) (x) = 0; \\ d^{T} ({x}) (y) = 0.2; d^{T} ({x}) (z) = 0.2; \\ d^{T} ({y}) (x) = 0.2; \\ d^{T} ({y}) (y) = 0; d^{T} ({y}) (z) = 0.4; \\ d^{T} ({z}) (x) = 0.2; \\ d^{T} ({z}) (y) = 0.2; d^{T} ({z}) (z) = 0; \\ d^{T} ({x, y}) (x) = 0.2; \\ d^{T} ({x, y}) (y) = 0.2; d^{T} ({x, y}) (z) = 0.4; \\ d^{T} ({y, z}) (x) = 0.2; \\ d^{T} ({y, z}) (y) = 0.2; d^{T} ({y, z}) (z) = 0.4; \\ d^{T} ({x, z}) (x) = 0.2; \\ d^{T} ({x, z}) (y) = 0.2; d^{T} ({x, z}) (z) = 0.2; \\ d^{T} (X) (x) = 0.2; \\ d^{T} (X) (y) = 0.2; d^{T} (X) (z) = 0.4; \end{matrix}$

Before studying the properties of M-fuzzifying derived sets, we need the following lemma.

Lemma 3.4. If $T : 2^{X} ⟶ M$ is an M-fuzzifying topology, then $\begin{matrix} ⋁_{y \notin B} ⋀_{y \notin D \supseteq B - {y}} {(T (D^{'}))}^{'} = T (B^{'})^{'} . \end{matrix}$

Proof. Let LHS denote the left hand side of equality. Then $LHS \leq T (B^{'})^{'}$ is trivial. The key of proofs is that $(LHS)^{'} \leq T (B^{'})$ . In fact, we have $\begin{matrix} (LHS)^{'} & = & ⋀_{y \notin B} ⋁_{y \notin D \supseteq B - {y}} T (D^{'}) \\ = & ⋁_{f \in \prod_{y \notin B} B_{y}} ⋀_{y \notin B} T (f (y)^{'}) \\ \leq & ⋁_{f \in \prod_{y \notin B} B_{y}} T (⋁_{y \notin B} f (y)^{'}) \\ = & ⋁_{f \in \prod_{y \notin B} B_{y}} T ({(⋀_{y \notin B} f (y))}^{'}) \\ = & T (B^{'}) \end{matrix}$ where B_y = {D ∣ y ∉ D ⊇ B - {y}}, ⋀_y∉Bf (y) = B - {y} = B for all y ∉ B and for any $f \in \prod_{y \notin B} B_{y}$ .□

Theorem 3.5. Let $T : 2^{X} ⟶ M$ be an M-fuzzifying topology. Then for any A ∈ 2^X,

∀x ∈ X, $d^{T} (\emptyset) (x) = ⊥;$

∀x ∈ X, $d^{T} ({x}) (x) = ⊥;$

$d^{T} (A \cup B) = d^{T} (A) \lor d^{T} (B);$

$d^{T} (A) (x) = ⋀_{x \notin B \supseteq A - {x}} ⋁_{y \notin B} d^{T} (B) (y) .$

Proof. (1) and (2) are obvious.□

(3) By Definition 3.1, it is easily seen that $d^{T} (\cdot)$ is an order-preserving mapping. Then $d^{T} (A) \lor d^{T} (B) \leq d^{T} (A \cup B)$ . We need to prove that $d^{T} (A) \lor d^{T} (B) \geq d^{T} (A \cup B) .$ By (MYT2) and (A ∪ B) - {x} = (A - {x}) ∪ (B - {x}) , we have $\begin{matrix} d^{T} (A) (x) \lor d^{T} (B) (x) \\ = ⋀_{x \notin D_{1} \supseteq A - {x}} {(T (D_{1}^{'}))}^{'} \lor ⋀_{x \notin D_{2} \supseteq B - {x}} {(T (D_{2}^{'}))}^{'} \\ = ⋀_{x \notin D_{1} \supseteq A - {x}, x \notin D_{2} \supseteq B - {x}} {(T (D_{1}^{'}))}^{'} \lor {(T (D_{2}^{'}))}^{'} \\ = ⋀_{x \notin D_{1} \cup D_{2} \supseteq (A \cup B) - {x}} {(T (D_{1}^{'}) \land T (D_{2}^{'}))}^{'} \\ \geq ⋀_{x \notin D_{1} \cup D_{2} \supseteq (A \cup B) - {x}} {(T (D_{1} \cup D_{2})^{'})}^{'} \\ \geq d^{T} (A \cup B) (x) . \end{matrix}$

(4) By Lemma 3.4, we know $\begin{matrix} d^{T} (A) (x) & = & ⋀_{x \notin B \supseteq A - {x}} {(T (B^{'}))}^{'} \\ = & ⋀_{x \notin B \supseteq A - {x}} ⋁_{y \notin B} ⋀_{y \notin D \supseteq B - {y}} {(T (D^{'}))}^{'} \\ = & ⋀_{x \notin B \supseteq A - {x}} ⋁_{y \notin B} d^{T} (B) (y) . \end{matrix}$

□

By Definition 3.1, we get a mapping $d^{T} : 2^{X} ⟶ M^{X}$ defined by $A \mapsto d^{T} (A)$ . The mapping $d^{T}$ has properties (1)-(4). If an arbitrary mapping d : 2^X ⟶ M^X has these properties, then d is called an M-fuzzifying derived operator. That is,

Definition 3.6. An M-fuzzifying derived operator on X is a mapping d : 2^X ⟶ M^X satisfying the following conditions:

∀x ∈ X, d (∅) (x) = ⊥;

∀x ∈ X, d ({x}) (x) =⊥;

d (A ∪ B) = d (A) ∨ d (B);

d (A) (x) = ⋀ _{x∉B⊇A-{x}} ⋁ _y∉Bd (B) (y).

The pair (X, d) is called an M-fuzzifying derived space.

A mapping f : X ⟶ Y between two M-fuzzifying derived spaces (X, d_X) and (Y, d_Y) is called continuous if $\begin{matrix} d_{X} (A) (x) & \leq & f^{\to} (A) (f (x)) \lor d_{Y} (f^{\to} (A)) (f (x)) \\ = & {\begin{matrix} ⊤, & f (x) \in f^{\to} (A); \\ d_{Y} (f^{\to} (A)) (f (x)), & f (x) \notin f^{\to} (A) . \end{matrix} \end{matrix}$ for all x ∈ X and for any A ∈ 2^X.

It is not difficult to check that M-fuzzifying derived spaces and their continuous mappings form a category, denoted by MY-FDS.

Theorem 3.7. If a mapping d : 2^X ⟶ M^X satisfies (MYD1)-(MYD3), then $d (A) (x) = d (A - {x}) (x)$ for any A ∈ 2^X and for all x ∈ X.

Proof. d (A) (x) = d ((A - {x}) ∪ {x}) (x) = d (A - {x}) (x) ∨ d ({x}) (x) = d (A - {x}) (x).□

The following theorems give equivalent characterizations of (MYD4).

Theorem 3.8. M-fuzzifying derived operators Let d : 2^X ⟶ M^X be a mapping satisfying (MYD1)-(MYD3). Then (MYD4) is equivalent to the following condition: (MYD4^*) $\forall a \in J (M), d (d (A)_{[a]})_{[a]} \subseteq A \cup d (A)_{[a]} .$

Proof. Sufficiency. Let RHS denote the right hand side of (MYD4). It follows from (MYD3) that d is an order-preserving map. On one hand, for any x ∉ B ⊇ A - {x}, we know d (A) (x) = d (A - {x}) (x) ≤ d (B) (x) ≤ ⋁ _y∉Bd (B) (y). Then d (A) (x) ≤ RHS. On the other hand, take any a ∈ J (M) with anleqd (A) (x). There exists some b ∈ β^* (a) such that bnleqd (A) (x). By Theorem 3.7, we have d (A - {x}) (x) = d (A) (x) ngeqb. This shows x ∉ d (A - {x}) _[b]. Then x ∉ (A - {x}) ∪ d (A - {x}) _[b]. Let B = (A - {x}) ∪ d (A - {x}) _[b]. Then x ∉ B ⊇ A - {x}. By (MYD4^*), we get y ∉ d (d (A - {x}) _[b]) _[b] for all y ∉ B. Since b ∈ J (M), we have d (B) (y) = d (A - {x}) (y) ∨ d (d (A - {x}) _[b]) (y) ngeqb. Hence ⋁_y∉Bd (B) (y) ngeqa. Therefore anleqRHS. By the arbitrariness of a, we obtain RHS ≤ d (A) (x).

Necessity. For each a ∈ J (M), take any x ∉ A ∪ d (A) _[a], we have x ∉ A and x ∉ d (A) _[a]. Then d (A) (x) ngeqa. By (MYD4), there exists some B such that x ∉ B ⊇ A - {x} and ⋁_y∉Bd (B) (y) ngeqa. This implies d (B) (y) ngeqa for all y ∉ B. So B ⊇ d (B) _[a]. Note that x ∉ A. Then A - {x} = A. Under these facts, we have x ∉ B ⊇ d (B) _[a] ⊇ d (A - {x}) _[a] = d (A) _[a] ⊇ d (A) _[a] - {x}. By (MYD4), we have d (d (A) _[a]) (x) ngeqa, i.e., x ∉ d (d (A) _[a]) _[a]. Therefore d (d (A) _[a]) _[a] ⊆ A ∪ d (A) _[a].□

Theorem 3.9. M-fuzzifying derived operators Let d : 2^X ⟶ M^X be a mapping satisfying (MYD1)-(MYD3). If α (a ∨ b) = α (a) ∩ α (b) for each a, b ∈ M, then (MYD4) is also equivalent to the following condition: (MYD4^**) $\forall a \in α (⊥), d (d (A)^{[a]})^{[a]} \subseteq A \cup d (A)^{[a]} .$

Proof. Sufficiency. Let RHS denote the right hand side of (MYD4). d (A) (x) ≤ RHS is obvious. We only need to prove RHS ≤ d (A) (x). Take any a ∈ α (⊥) with a ∈ α (d (A) (x)). There exists some b ∈ β (a) such that b ∈ α (d (A) (x)). By Theorem 3.7, we have b ∈ α (d (A - {x}) (x)), i.e., x ∉ d (A - {x}) ^[b]. Let B = (A - {x}) ∪ d (A - {x}) ^[b]. Then x ∉ B ⊇ A - {x}. By (MYD4^**), we get y ∉ d (d (A - {x}) ^[b]) ^[b] for all y ∉ B. It follows that b ∈ α (d (A - {x}) (y)) ∩ α (d (d (A - {x}) ^[b]) (y)) = α (d (A - {x}) (y) ∨ d (d (A - {x}) ^[b]) (y)) = α (d (B) (y)) for all y ∉ B. So a ∈ α (⋁ _y∉Bd (B) (y)). This shows a ∈ ⋃ _{x∉B⊇A-{x}}α (⋁ _y∉Bd (B) (y)) = α (RHS). By the arbitrariness of a, we obtain RHS ≤ d (A) (x).

Necessity. For each a ∈ α (⊥), take any x ∉ A ∪ d (A) ^[a], we have x ∉ A and x ∉ d (A) ^[a]. Then a ∈ α (d (A) (x)) = ⋃ _{x∉B⊇A-{x}}α (⋁ _y∈Bd (B) (y)), which means there exists some B such that x ∉ B ⊇ A - {x} and a ∈ α (⋁ _y∉Bd (B) (y)). This implies a ∈ α (d (B) (y)) for all y ∉ B. So B ⊇ d (B) ^[a]. Hence x ∉ B ⊇ d (B) ^[a] ⊇ d (A - {x}) ^[a] = d (A) ^[a] ⊇ d (A) ^[a] - {x}. By (MYD4), we get a ∈ α (d (d (A) ^[a]) (x)), i.e., x ∉ d (d (A) ^[a]) ^[a]. Therefore d (d (A) ^[a]) ^[a] ⊆ A ∪ d (A) ^[a].□

Given an M-fuzzifying derived operators d : 2^X ⟶ M^X, we define two mappings d_[a] : 2^X ⟶ 2^X (a ∈ J (M)) and d^[a] : 2^X ⟶ 2^X (a ∈ α (⊥)) as follows: $\begin{matrix} d_{[a]} (A) = d (A)_{[a]} and d^{[a]} (A) = d (A)^{[a]} \end{matrix}$ for each A ∈ 2^X, respectively.

Theorem 3.10. If d : 2^X ⟶ M^X is an M-fuzzifying derived operator, then for each A ∈ 2^X,

d_[a] (A) ≤ d_[b] (A) for any a, b ∈ J (M) with b ∈ β^* (a) and d_[a] (A) = ⋂ _b∈β^*(a)d_[b] (A).

d^[b] (A) ≤ d^[a] (A) for any a, b ∈ α (⊥ _M) with b ∈ α (a) and d^[a] (A) = ⋃ _b∈α(a)d^[b] (A).

Proof. By Theorem 2.2, we know that d (A) _[a] = ⋂ _b∈β^*(a)d (A) _[b] and d (A) ^[a] = ⋃ _b∈α(a)d (A) ^[b]. Since d_[a] (A) = d (A) _[a] and d^[a] (A) = d (A) ^[a], (1) and (2) are true trivially.

The following theorem shows the relations among d, d_[a] and d^[a].

Theorem 3.11. Let d : 2^X ⟶ M^X be a mapping. Then the following statements are equivalent.

d is an M-fuzzifying derived operator on X.

For each a ∈ J (M), d_[a] is a crisp derived operator on X.

If α (a ∨ b) = α (a) ∩ α (b) for any a, b ∈ M, then for each a ∈ α (⊥), d^[a] is a crisp derived operator on X.

Proof. We only prove (1)⇔(2). (1)⇔(3) is similar to (1)⇔(2) and omitted here.

(1)⇒(2) For each a ∈ J (M), we need to check that d_[a] satisfies (D1)-(D4).

Since d (∅) (x) = ⊥ and d ({x}) (x) =⊥, we know d_[a] (∅) = ∅ and x ∉ d_[a] ({x}). So (D1) and (D2) hold. By the definition of co-prime element, we have a ≤ d (A ∪ B) (x) = d (A) (x) ∨ d (B) (x) if and only if a ≤ d (A) (x) or a ≤ d (B) (x). Then (D3) holds. By Theorem 3.8, (D4) holds.

(2)⇒(1) Define d : 2^X ⟶ M^X by $d (A) (x) = ⋁ {a \in J (M) ∣ x \in d_{[a]} (A)} .$ We need to check d satisfies (MYD1), (MYD2), (MYD3) and (MYD4). (MYD1) ∀x ∈ X, d (∅) (x) = ⋁ ∅ = ⊥.

(MYD2) ∀x ∈ X, d ({x}) (x) =⋁ ∅ = ⊥.

(MYD3) By the definition of d, we easily know that d (A) ≤ d (B) whenever A ⊆ B. Then d (A) ∨ d (B) ≤ d (A ∪ B). It suffices to prove d (A ∪ B) ≤ d (A) ∨ d (B). Suppose that d (A ∨ B) (x) = ⋁ {a ∈ J (M) ∣ x ∈ d_[a] (A ∪ B)}. Take any a ∈ J (M) with x ∈ d_[a] (A ∪ B). Then x ∈ d_[a] (A) or x ∈ d_[a] (B). So a ≤ d (A) (x) or a ≤ d (B) (x), which implies a ≤ d (A) (x) ∨ d (B) (x). Hence d (A ∪ B) ≤ d (A) ∨ d (B).

(MYD4) Since d satisfies (MYD1)-(MYD3), it follows from Theorem 3.8 that (MYD4) holds.□

Now we consider that a family of crisp derived operators forms an M-fuzzifying derived operator.

Theorem 3.12. Let d (a) :2^X ⟶ 2^X ∣ a ∈ J (M)} be a family of crisp derived operators on X satisfying d (a) (A) = ⋂ _b∈β^*(a)d (b) (A) for each a ∈ J (M) and for any A ∈ 2^X. Then there exists an M-fuzzifying derived operator d : 2^X ⟶ M^X such that ∀A ∈ 2^X, ∀x ∈ X, $\begin{matrix} d (A) (x) = ⋁ {a \in J (M) ∣ x \in d (a) (A)} \end{matrix}$ and d_[a] (A) = d (a) (A) for any A ∈ 2^X .

Proof. By Theorem 3.11, in order to prove d is an M-fuzzifying derived operator, it suffices to prove that d_[a] (A) = d (a) (A) for any A ∈ 2^X.

On one hand, suppose that x ∈ d (a) (A). Then d (A) ≥ a, which means x ∈ d (A) _[a] = d_[a] (A). Hence d (a) (A) ⊆ d_[a] (A). On the other hand, assume that x ∈ d_[a] (A), i.e., d (A) (x) ≥ a. Then β^* (d (A) (x)) ⊇ β^* (a). Since d (a) (A) = ⋂ _b∈β^*(a)d (b) (A) and β^* is order-preserving, it is easily seen that d (a₂) (A) ⊆ d (a₁) (A) whenever a₁ ≤ a₂. For any b ∈ β^* (a), we have b ∈ β^* (d (A) (x)) = ⋃ _c∈J(M)β (c ∧ d (c) (A) (x)). There exists some c ∈ J (M) such that c ∧ d (c) (A) (x) ≥ b. So c ≥ b and x ∈ d (c) (A) ⊆ d (b) (A). Hence x ∈ ⋂ _b∈β^*(a)d (b) (A) = d (a) (A). Therefore d_[a] (A) ⊆ d (a) (A).□

Theorem 3.13. Let {d (a) :2^X ⟶ 2^X ∣ a ∈ α (⊥)} be a family of crisp derived operators on X satisfying d (a) (A) = ⋃ _b∈α(a)d (b) (A) for each a ∈ α (⊥) and for any A ∈ 2^X. Then there exists an M-fuzzifying derived operator d : 2^X ⟶ M^X such that ∀A ∈ 2^X, ∀x ∈ X, $\begin{matrix} d (A) = ⋀ {a \in α (⊥) ∣ x \in d (a) (A)} \end{matrix}$ and d^[a] (A) = d (a) (A) for any A ∈ 2^X.

Proof. The proof is analogous to that of Theorem 3.12 and omitted here.□

4 Categories isomorphic to MY-FDS

In this section,we will show that the category MY-FTop, the category MY-FCS and the category MY-FNS are all isomorphic to the category MY-FDS. Besides, the M-fuzzifying derived operators induced by M-fuzzifying neighborhood systems in this section are equivalent to those in the sense of Ying.

4.1 Category MY-FTop isomorphic to MY-FDS

By Definition 3.1 and Theorem 3.5, we have known that an M-fuzzifying topology can induce an M-fuzzifying derived operator. Conversely, an M-fuzzifying derived operator also can induce an M-fuzzifying topology as follows.

Theorem 4.1. Let (X, d) be an M-fuzzifying derived space. Define $T^{d} : 2^{X} ⟶ M$ by ∀A ∈ 2^X, $\begin{matrix} T^{d} (A) = ⋀_{x \notin A^{'}} {(d (A^{'}) (x))}^{'} . \end{matrix}$ Then $T^{d}$ is an M-fuzzifying topology on X.

Proof. We need to check (MYT1)-(MYT3).

(MYT1) is trivial.

(MYT2) For all A, B ∈ 2^X, we have $\begin{matrix} T^{d} (A) \land T^{d} (B) \\ = (⋀_{x \notin A^{'}} {(d (A^{'}) (x))}^{'}) \land (⋀_{y \notin B^{'}} {(d (B^{'}) (y))}^{'}) \\ = ⋀_{x \notin A^{'}, y \notin B^{'}} ({(d (A^{'}) (x))}^{'} \land {(d (B^{'}) (y))}^{'}) \\ \leq ⋀_{x \notin A^{'} \cup B^{'}} {(d (A^{'}) (x) \lor d (B^{'}) (x))}^{'} \\ = ⋀_{x \notin A^{'} \cup B^{'}} {(d (A^{'} \cup B^{'}) (x))}^{'} (by MYD 3) \\ = ⋀_{x \notin (A \cap B)^{'}} {(d ((A \cap B)^{'}) (x))}^{'} \\ = T^{d} (A \cap B) . \end{matrix}$

(MYT3) By (MYD3), it is not difficult to see that d (·) is order-preserving. For all {A_j ∣ j ∈ J} ⊆2^X, we have $\begin{matrix} T^{d} (⋁_{j \in J} A_{j}) \\ = ⋀_{x \notin {(⋁_{j \in J} A_{j})}^{'}} d {({(⋁_{j \in J} A_{j})}^{'} (x))}^{'} \\ = ⋀_{j \in J} ⋀_{x \notin A_{j}^{'}} d {(⋀_{j \in J} A_{j}^{'} (x))}^{'} \\ \geq ⋀_{j \in J} ⋀_{x \notin A_{j}^{'}} d {(A_{j}^{'} (x))}^{'} = ⋀_{j \in J} T^{d} (A_{j}) . \end{matrix}$ □

In what follows, the relationships between the continuous mappings of M-fuzzifying derived spaces and the continuous mappings of M-fuzzifying topological spaces are discussed.

Theorem 4.2. Let (X, d_X) and (Y, d_Y) be M-fuzzifying derived spaces. If f : (X, d_X) ⟶ (Y, d_Y) is continuous, then $f : (X, T^{d_{X}}) ⟶ (Y, T^{d_{Y}})$ is continuous.

Proof. We need to show $T^{d_{Y}} (B) \leq T^{d_{X}} (f^{\leftarrow} (B))$ for all B ∈ 2^Y. By Theorem 4.1, it suffices to prove ⋁_{x∉f^←(B′)}d_X (f^← (B′)) (x) ≤ ⋁ _y∉B′d_Y (B′) (y). Let a ∈ M with a ≺ ⋁ _{x∉f^←(B′)}d_X (f^← (B′)) (x). Then there exists some x ∉ f^← (B′) such that a ≤ d_X (f^← (B′)) (x). Note that x ∉ f^← (B′) if and only if f (x) ∉ B′ and f : (X, d_X) ⟶ (Y, d_Y) is continuous. It follows that $\begin{matrix} a & \leq & d_{X} (f^{\leftarrow} (B^{'})) (x) \\ \leq & f^{\to} (f^{\leftarrow} (B^{'})) (f (x)) \lor d_{Y} \\ \times (f^{\to} (f^{\leftarrow} (B^{'})) (f (x))) \\ \leq & B^{'} (f (x)) \lor d_{Y} (B^{'}) (f (x)) = d_{Y} (B^{'}) (f (x)) . \end{matrix}$ This implies a ≤ ⋁ _y∉B′d_Y (B′) (y). By the arbitrariness of a, we get ⋁_{x∉f^←(B′)}d_X (f^← (B′)) (x) ≤ ⋁ _y∉B′d_Y (B′) (y). Therefore $f : (X, T^{d_{X}}) ⟶ (Y, T^{d_{Y}})$ is continuous.□

Theorem 4.3. Let $(X, T_{X})$ and $(Y, T_{Y})$ be M-fuzzifying topological spaces. If $f : (X, T_{X}) ⟶ (Y, T_{Y})$ is continuous, then $f : (X, d^{T_{X}}) ⟶ (Y, d^{T_{Y}})$ is continuous.

Proof. We only need to check that $d^{T_{X}} (A) (x) \leq d^{T_{Y}} (f^{\to} (A)) (f (x))$ whenever f (x) ∉ f^→ (A). Note that f (x) ∉ f^→ (A) implies x ∉ A. It suffices to prove that $⋁_{f (x) \notin D \supseteq f^{\to} (A)} T_{Y} (D^{'}) \leq ⋁_{x \notin B \supseteq A} T_{X} (B^{'})$ whenever f (x) ∉ f^→ (A). Let a ∈ M with $a ≺ ⋁_{f (x) \notin D \supseteq f^{\to} (A)} T_{Y} (D^{'})$ . Then there exists some D such that f (x) ∉ D ⊇ f^→ (A) and $a ≺ T_{Y} (D^{'})$ . This implies x ∉ f^← (D) ⊇ f^← (f^→ (A)) ⊇ A. By the continuity of $f : (X, T_{X}) ⟶ (Y, T_{Y})$ , we have $a ≺ T_{Y} (D^{'}) \leq T_{X} (f^{\leftarrow} (D^{'})) = T_{X} (f^{\leftarrow} (D)^{'}) .$ This shows $a \leq ⋁_{x \notin B \supseteq A} T_{X} (B^{'})$ . From the arbitrariness of a, we obtain $⋁_{f (x) \notin D \supseteq f^{\to} (A)} T_{Y} (D^{'}) \leq ⋁_{x \notin B \supseteq A} T_{X} (B^{'})$ . Therefore $f : (X, d^{T_{X}}) ⟶ (Y, d^{T_{Y}})$ is continuous.□

The following theorem shows that there exists a one-to-one correspondence between M-fuzzifying derived operators and M-fuzzifying topologies.

Theorem 4.4. If $T : 2^{X} ⟶ M$ is an M-fuzzifying topology on X and d : 2^X ⟶ M^X is an M-fuzzifying derived operator on X, then $T^{d^{T}} = T$ and $d^{T^{d}} = d$ .

Proof. Firstly, by Lemma 3.4, $T^{d^{T}} = T$ follows from $\begin{matrix} T^{d^{T}} (A) & = & ⋀_{x \notin A^{'}} {(d^{T} (A^{'}) (x))}^{'} \\ = & ⋀_{x \notin A^{'}} ⋁_{x \notin B \supseteq A^{'} - {x}} T (B^{'}) \\ = & T (A^{″}) = T (A) . \end{matrix}$

Secondly, by (MFD4), $d^{T^{d}} = d$ follows from $\begin{matrix} d^{T^{d}} (A) (x) & = & ⋀_{x \notin B \supseteq A - {x}} {(T^{d} (B^{'}))}^{'} \\ = & ⋀_{x \notin B \supseteq A - {x}} ⋁_{y \notin B} d (B) (y) \\ = & d (A) (x) . \end{matrix}$ □

By Definition 3.1, Theorem 3.5 and Theorem 4.1–4.3, we obtain a functor mathdsF: MY - FTop ⟶ MY - FDS such that

$\begin{matrix} {\begin{matrix} F ((X, T)) = (X, d^{T}), & (X, T) \in ob (MY - FTop); \\ F (f) = f, & f \in mor (MY - FTop) \end{matrix} . \end{matrix}$

and a functor mathdsG: MY-FDS ⟶ MY-FTop such that $\begin{matrix} {\begin{matrix} G ((X, d)) = (X, T^{d}), & (X, d) \in ob (MY - FDS); \\ G (f) = f, & f \in mor (MY - FDS) . \end{matrix} \end{matrix}$

By Theorem 4.4, we get the following theorem.

Theorem 4.5. The category MY-FTop is isomorphic to the category MY-FDS.

4.2 Category MY-FCS isomorphic to MY-FDS

In [15], F.G. Shi proved that there is a one-to-one correspondence between M-fuzzifying closure spaces and M-fuzzifying topologies. That is,

Theorem 4.6. ([15]) Let $(X, T)$ be an M-fuzzifying topological space and let (X, cl) be an M-fuzzifying closure space.

(1) Define a mapping ${cl}^{T} : 2^{X} ⟶ M^{X}$ by ∀A ∈ 2^X, ∀ x ∈ X, $\begin{matrix} {cl}^{T} (A) (x) = ⋀_{x \notin B \supseteq A} {(T (B^{'}))}^{'} . \end{matrix}$

Then ${cl}^{T}$ is an M-fuzzifying closure operator on X.

(2) Define a mapping $T^{cl} : 2^{X} ⟶ M$ by ∀A ∈ 2^X, $\begin{matrix} T^{cl} (A) = ⋀_{x \notin A^{'}} {(cl (A^{'}) (x))}^{'} . \end{matrix}$ Then $T^{cl}$ is an M-fuzzifying topology on X.

(3) ${cl}^{T^{cl}} = cl$ and $T^{{cl}^{T}} = T$ .

The following lemma is trivial.

Lemma 4.7. Let d : 2^X ⟶ M^X be an M-fuzzifying derived operator. Then ∀x ∉ A, $\begin{matrix} ⋀_{x \notin B \supseteq A - {x}} ⋁_{y \notin B} d (B) (y) = ⋀_{x \notin B \supseteq A} ⋁_{y \notin B} d (B) (y) . \end{matrix}$

In [15], F.G. Shi also proved that the category MY-FTop is isomorphic to the category MY-FCS. By Theorem 4.5, we know that the category MY-FTop is isomorphic to the category MY-FDS. Therefore, we get the following theorem.

Theorem 4.8. The category MY-FCS is isomorphic to the category MY-FDS.

There is a functor mathdsF: MY-FCS ⟶ MY-FDS such that $\begin{matrix} {\begin{matrix} F ((X, cl)) = (X, d^{cl}), & (X, cl) \in ob (MY - FCS); \\ F (f) = f, & f \in mor (MY - FCS), \end{matrix} \end{matrix}$ where d^cl : 2^X ⟶ M^X defined by ∀A ∈ 2^X, ∀ x ∈ X, $\begin{matrix} d^{cl} (A) (x) = cl (A - {x}) (x) \end{matrix}$ and a functor mathdsG: MY-FDS ⟶ MY-FCS such that $\begin{matrix} {\begin{matrix} G ((X, d)) = (X, {cl}^{d}), & (X, d) \in ob (MY - FDS); \\ G (f) = f, & f \in mor (MY - FDS), \end{matrix} \end{matrix}$ where cl^d : 2^X ⟶ M^X defined by ∀A ∈ 2^X, ∀ x ∈ X, $\begin{matrix} {cl}^{d} (A) (x) & = & {\begin{matrix} ⊤, & x \in A; \\ d (A) (x), & x \notin A . \end{matrix} \end{matrix}$

Remark 4.9. (1) By Definition 3.1, Theorem 4.1, Theorem 4.6, Lemma 4.7 and (MYC4), (MYD4) we have

i) ∀x ∈ X, $\begin{matrix} d^{T^{cl}} (A) (x) & = & ⋀_{x \notin B \supseteq A - {x}} {(T^{cl} (B^{'}))}^{'} \\ = & ⋀_{x \notin B \supseteq A - {x}} ⋁_{y \notin B} cl (B) (y) \\ = & cl (A - {x}) (x) = d^{cl} (A) (x) . \end{matrix}$

ii) ∀x ∉ A, $\begin{matrix} {cl}^{T^{d}} (A) (x) & = & ⋀_{x \notin B \supseteq A} {(T^{d} (B^{'}))}^{'} \\ = & ⋀_{x \notin B \supseteq A} ⋁_{y \notin B} d (B) (y) \\ = & ⋀_{x \notin B \supseteq A - {x}} ⋁_{y \notin B} d (B) (y) \\ = & d (A) (x) = {cl}^{d} (A) (x) . \end{matrix}$

(2) Let M = 2 (namely, $T$ is a topology). If d^cl (A) (x) =⊤, then x ∈ d^cl (A) if and only if x ∈ cl (A - {x}). If cl^d (A) (x) =⊤, then cl^d (A) = A ∪ d (A). These are exactly the conclusions of general topological spaces.

4.3 Category MY-FNS isomorphic to MY-FDS

In [20], M.S. Ying showed that there is a one-to-one correspondence between fuzzifying neighborhood systems and fuzzifying topologies from the view of multi-valued logic. Besides, he also gave the definition of fuzzifying derived sets by logical language. These results can easily be generalized to M-fuzzifying cases and be translated by analysis language in the following definition and theorem.

Definition 4.10. ([20]) Let $N = {N_{x} : 2^{X} ⟶ M ∣ x \in X}$ be an M-fuzzifying neighborhood system on X. For each A ∈ 2^X, the M-fuzzifying derived set of A is a mapping ${\underline{d}}^{N} (A) : X ⟶ M$ defined by ∀x ∈ X, $\begin{matrix} {\underline{d}}^{N} (A) (x) = ⋀_{B \cap (A - {x}) = \emptyset} (N_{x} (B))^{'} . \end{matrix}$

Theorem 4.11. ([20]) Let $(X, T)$ be an M-fuzzifying topological space and $N = {N_{x} : 2^{X} ⟶ M ∣ x \in X}$ be an M-fuzzifying neighborhood system on X.

(1) Define a set $N^{T} = {N_{x}^{T} ∣ x \in X}$ of maps $N_{x}^{T} : 2^{X} ⟶ M$ by ∀A ∈ 2^X, $\begin{matrix} N_{x}^{T} (A) = ⋁_{x \in B \subseteq A} T (B) . \end{matrix}$ Then $N^{T} = {N_{x}^{T} ∣ x \in X}$ is an M-fuzzifying neighborhood system on X.

(2) Define a mapping $T^{N} : 2^{X} ⟶ M$ by ∀A ∈ 2^X, $\begin{matrix} T^{N} (A) = ⋀_{x \in A} N_{x} (A) . \end{matrix}$ Then $T^{N}$ is an M-fuzzifying topology on X.

(3) $N^{T^{N}} = N$ and $T^{N^{T}} = T$ .

Since we prove that the category MY-FTop is isomorphic to the category MY-FDS and F.G. Shi also show that the category MY-FTop is isomorphic to the category MY-FNS in [15]. Therefore, we obtain the following theorem.

Theorem 4.12. The category MY-FNS is isomorphic to the category MY-FDS.

There is a functor mathdsF: MY-FNS ⟶ MY-FDS such that $\begin{matrix} {\begin{matrix} F ((X, N)) = (X, d^{N}), & (X, N) \in ob (MY - FNS); \\ F (f) = f, & f \in mor (MY - FNS), \end{matrix} \end{matrix}$ where $d^{N} : 2^{X} ⟶ M^{X}$ defined by ∀A ∈ 2^X, ∀ x ∈ X, $\begin{matrix} d^{N} (A) (x) = {(N_{x} (A - {x})^{'})}^{'} \end{matrix}$ and a functor mathdsG: MY-FDS ⟶ MY-FNS such that $\begin{matrix} {\begin{matrix} G ((X, d)) = (X, N^{d}), & (X, d) \in ob (MY - FDS); \\ G (f) = f, & f \in mor (MY - FDS), \end{matrix} \end{matrix}$ where the set $N^{d} = {N_{x}^{d} ∣ x \in X}$ of maps $N_{x}^{d} : 2^{X} ⟶ M$ defined by ∀A ∈ 2^X, $\begin{matrix} N_{x}^{d} (A) & = & {\begin{matrix} {(d (A^{'}) (x))}^{'}, & x \in A; \\ ⊥, & x \notin A . \end{matrix} \end{matrix}$

Remark 4.13. (1) By Definition 3.1, Theorem 4.1, Theorem 4.11, Lemma 4.7 and (MYN4), (MYD4) we have

i) ∀x ∈ X, $\begin{matrix} d^{T^{N}} (A) (x) & = & ⋀_{x \notin B \supseteq A - {x}} {(T^{N} (B^{'}))}^{'} \\ = & ⋀_{x \notin B \supseteq A - {x}} ⋁_{y \in B^{'}} {(N_{y} (B^{'}))}^{'} \\ = & {(⋁_{x \in B^{'} \subseteq (A - {x})^{'}} ⋀_{y \in B^{'}} N_{y} (B^{'}))}^{'} \\ = & {(N_{x} (A - {x})^{'})}^{'} = d^{N} (A) (x) \end{matrix}$

ii) ∀x ∈ A, (i.e., ∀x ∉ A′) $\begin{matrix} N_{x}^{T^{d}} (A) & = & ⋁_{x \in B \subseteq A} T^{d} (B) \\ = & ⋁_{x \in B \subseteq A} ⋀_{y \notin B^{'}} {(d (B^{'}) (y))}^{'} \\ = & {(⋀_{x \notin B^{'} \supseteq A^{'}} ⋁_{y \notin B^{'}} d (B^{'}) (y))}^{'} \\ = & {(⋀_{x \notin B^{'} \supseteq A^{'} - {x}} ⋁_{y \notin B^{'}} d (B^{'}) (y))}^{'} \\ = & {(d (A^{'}) (x))}^{'} = N_{x}^{d} (A) \end{matrix}$

(2) Let M = 2 (namely, $T$ is a topology). If $d^{N} (A) (x) = ⊥$ , then $x \notin d^{N} (A)$ if and only if $(A - {x})^{'} \in N_{x}$ . If $N_{x}^{d} (A) = ⊤$ and x ∈ A, then $A \in N_{x}^{d}$ if and only if x ∉ d (A′). These conclusions can be obtained by the definition of accumulation points in general topology.

In what follows, we show that the M-fuzzifying derived operators induced by M-fuzzifying neighborhood systems in Theorem 4.12 are equivalent to those in Definition 4.10.

Theorem 4.14. Let $N = {N_{x} : 2^{X} ⟶ M ∣ x \in X}$ be an M-fuzzifying neighborhood system on X. Then $d^{N} = {\underline{d}}^{N}$ .

Proof. We need to check $d^{N} (A) (x) = {\underline{d}}^{N} (A) (x)$ , i.e., $\begin{matrix} {(N_{x} (A - {x})^{'})}^{'} = ⋀_{B \cap (A - {x}) = \emptyset} {(N_{x} (B))}^{'} \end{matrix}$ for each A ∈ 2^X and for all x ∈ X.

On one hand, it is clear that ${(N_{x} (A - {x})^{'})}^{'} \geq ⋀_{B \cap (A - {x}) = \emptyset} {(N_{x} (B))}^{'}$ . On the other hand, take any a ∈ M with $a ⋀_{B \cap (A - {x}) = \emptyset} {(N_{x} (B))}^{'}$ . There exists some B ∈ 2^X such that B∩ (A - {x}) = ∅ and $a {(N_{x} (B))}^{'}$ . This shows B ⊆ (A - {x}) ′. Since $N_{x} (\cdot)$ is order-preserving, it follows that ${(N_{x} (B))}^{'} \geq {(N_{x} (A - {x})^{'})}^{'}$ . Hence $a {(N_{x} (A - {x})^{'})}^{'}$ . From the arbitrariness of a, we obtain ${(N_{x} (A - {x})^{'})}^{'} \leq ⋀_{B \cap (A - {x}) = \emptyset} {(N_{x} (B))}^{'}$ .□

Footnotes

Acknowledgement

The authors would like to express their sincere thanks to the anonymous reviewers for their careful reading and constructive comments. This work is supported by Langfang municipal science and technology project (2017029040) and the National Natural Science Foundation of China (11371002) and Specialized Research Fund for the Doctoral Program of Higher Education (20131101110048).

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