Upper rough approximation operators of quantale-valued similarities related to fuzzy orderings

Abstract

Let L be a commutative unital quantale. For every L-fuzzy relation E on a nonempty set X, we define an upper rough approximation operator on L^X, which is a fuzzy extension of the classical Pawlak upper rough approximation operator. We show that this operator has close relation with the subsethood operator on X. Conversely, by an L-fuzzy closure operator on X, we can easily get an L-fuzzy relation. We show that this relation can be characterized by more smooth ways. Without the help of the lower approximation operator, L-fuzzy rough sets can still be studied by means of constructive and axiomatic approaches, and L-fuzzy similarities and L-fuzzy closure operators are one-to-one corresponding. We also show that, the L-topology induced by the upper rough approximation operator is stratified and Alexandrov.

Keywords

commutative unital quantale upper rough approximation operator

1 Introduction

Since Pawlak [16, 17] proposed the concept of rough sets in the last eighties, rough set theory has received wide attention in both of theoretical research and practical applications. There are usually two approaches for the development of this theory: the constructive approach and the axiomatic approach. In the constructive approach, the upper and lower approximation operators are constructed from some basic mathematical structures, such as binary relations [24 , 46], neighborhood systems [6, 11, 45] and algebras [14, 21]. Based on constructive method, extensive research has also been carried out to compare the theory of rough sets with others [1 , 37]. In contrast with the constructive approach, the axiomatic approach takes set-theoretic operators as basic notions. Under some axioms on a pair of set-theoretic operators, there exists a binary relation such that the upper and lower rough approximation operators coincide with the set-theoretic operators [3 , 33].

In the framework of fuzzy rough set theory, various fuzzy generalizations of approximation operators have been proposed and investigated. Considering the truth value table [0, 1] with a triangular norm, a triangular conorm and an implication operator, many researchers proposed different types of fuzzy rough sets and provided axiomatic characterizations of fuzzy rough approximation operators, such as Morsi and Yakout [13], Thiele [25, 26], Radzikowska and Kerre [18], Wu et al. [34, 36], Mi et al. [12] and Liu [8]. With the development of fuzzy set theory, it seems that taking [0, 1] as the lattice background is relatively strong. For this trend, Radzikowska et al. [19] and She et al. [22] chose a complete residuated lattice (L, ∧ , ∨ , * , → , 0, 1) to investigate L-fuzzy rough approximation operators. Certainly, it still requires that the upper and lower fuzzy rough approximation operators are dual to each other. If the upper and lower fuzzy rough approximation operators can not be dual in a natural way, then some conditions of the lattice are required to guarantee this duality. For example, in [22], the upper and lower L-fuzzy rough approximation operators are not dual unless L is regular, i.e., (a → 0) →0 = a for each a ∈ L, and most of conclusions with respect to the lower L-fuzzy rough approximation operators needs this condition.

In the axiomatic approach, the duality between the upper and lower rough approximation operators is like the case of the closure operators and the interior operators in topological spaces, while each one of them can already completely determine the binary relation by itself. Thus from viewpoint of pure mathematics, we just need use one kind of operators to study rough set theory, the other operator can be obtained by the duality principle. This principle is sometimes used by researchers as a technique of constructing one operator from another, especially when the truth value table is an abstract lattice without the law of double negation. The aim of this paper is to study L-fuzzy rough set by using the L-fuzzy upper rough approximation operator. The lattice L in this paper will be a commutative unital quantale [20], which is a complete lattice with a monoid operation being distributive over arbitrary joins. The kind of lattices is more general than the unit interval, complete residuated lattices and some other lattices in the literature.

Now let us focus on fuzzy versions of the upper and lower rough approximation operators. For a set X, both of these two set-theoretic operators are firstly monotone with respect to pointwise order on the L-powerset L^X. While considering L^X as a complete lattice, it seems have no big difference from 2^X, which means that from the order we can not extract much more fuzzy information from L^X than 2^X. But in fact, in fuzzy topology theory, a viewpoint arises in recently years that we should use the concept of fuzzy order in order to make a difference between a fuzzy topology and a crisp topology. How to make the difference? A fuzzy order maybe is a good choice [39, 40]. The fuzzy order on L^X is called subsethood degree between L-fuzzy subsets of X, which already has a history for 60 years [5] and recently reaches a hot point in fuzzy topology theory and fuzzy domain theory [38 , 41]. For more details, for all A, B ∈ L^X, the subset degree of A being contained in B is defined as the value of ⋀_x∈XA (x) → B (x). That is to say, every L-fuzzy subset has a subset degree in another one, no longer the cases of strict containedness or discontainedness. It also indicates that a fuzzy order on L^X is more smooth than a traditional pointwise order.

As has been stated above, we will select a commutative unital quantale as the truth value table. For an L-fuzzy similarity, we define the upper rough approximation operator. Bao et al. [2] described L-fuzzy rough approximation operators with serial, reflexive, symmetric, transitive L-fuzzy relations as well as their compositions based on a complete residuated lattice L by a single axiom, respectively. Recently, Wang et al. [28, 29] further characterized Euclidean L-fuzzy rough approximation operators by a single axiom. Considering L being a complete Heyting algebra or a frame with an order-reversing involution, Pang et al. [15] systematically investigated axiomatic characterizations of L-rough sets with respect to serial, reflexive, symmetric, transitive, adjoint, Euclidean and mediate L-fuzzy relation as well as their compositions by a single axiom, respectively. Consequently for a GL-quantale as the truth value table, Wei et al. [31] studied single axiomatic characterizations of fuzzy rough sets based on the theory of L-valued rough sets with an L-set as the basic universe. For more information on single axiomatic characterization of fuzzy rough sets, we can refer to [9, 35].

In this paper, we will select a commutative unital quantale L as the truth value table and use the upper rough approximation operators to study single axiomatic characterizations of fuzzy rough sets with respect to fuzzy relations with fuzzy ordering feature. We show that, beyond some usual conditions, this operator is monotone with respect to the fuzzy order of subsethood degree and is tensored with respect to monoid operation. We call such an operator an L-fuzzy closure operator. And conversely, when we have such a kind of L-fuzzy closure operators at hand and want to use it to induce an L-fuzzy similarity, one will see that the fuzzy monotonicity is crucial in order to get all desired results. comparing to the related papers [2 , 35], a commutative unital quantale is most general truth value table for the study of fuzzy rough set theory, especially for L-fuzzy rough approximation operators by a single axiom. Thus the results obtained in this paper have more wider applicability in the theoretical level.

This paper is organized as follows: In Section 2, we will list some basic concepts and results related to commutative unital quantale quantales and L-fuzzy relations. In Section 3, we will study the upper rough approximation operators induced by L-fuzzy relations by means of constructive and axiomatic approaches. In Section 4, we will study the properties of the L-topology induced by the upper rough approximation operators. And we will make a conclusion in the last section.

2 Preliminaries

Let L be a poset. For a subset S ⊆ L, we use ⋁S and ⋀S to denote its supremum and infimum respectively, if they exist. If S = {a_i| i ∈ I} is indexed, then we write ⋁S = ⋁ _ia_i and ⋀S = ⋀ _ia_i. A complete lattice is a poset such that every subset (including the empty set) has supremum and infimum, with 1, 0 as the top and the bottom elements respectively.

A monoid is a semigroup (L, *) with a special element e ∈ G satisfying a * e = a = e * a for every a ∈ G, here e is called the unit of L. If the operation * is commutative, then G is called commutative.

Definition 2.1. Let (L ; ∧ , ∨ , * , e) be an algebra of type (2, 2, 2, 2, 0). If

(Q1) (L ; ∨ , ∧) is a complete lattice;

(Q2) (L ; * , e) is a commutative monoid;

(Q3) a * (⋁ _ib_i) = ⋁ _ia * b_i holds for all a ∈ L and {b_i| i ∈ I} ⊆ L, then L is called a commutative unital quantale.

Condition (Q3) is equivalently to that, there is an implication operation → : L × L ⟶ L satisfying that

(Q3’) (* , →) is an adjoint pair, i.e., a * b ≤ c iff a ≤ b → c for all a, b, c ∈ L. The unit e and the operation * are always logically treated as the truth value and the conjunction respectively when we deal with fuzziness and uncertainty. Notice that e is not always the same as the top element 1, and also * is not always as ∧, for example:

Example 2.2. Let L = {0, e, a, 1} be the four-element diamond lattice, that is, it is ordered by 0 < e, a < 1 and a ∨ e = 1, a ∧ e = 0. Define * : L × L ⟶ L by

*	e	a	1
0	0	0	0
e	e	a	1
a	a	a	1
1	1	1	1

Then L is a commutative unital quantale with e as the unit. In fact,

(Q1) Obviously, L is a complete lattice and furthermore * is idempotent.

(Q2) Clearly e is the unit of *, we only need to show that (G, *) is a semigroup, that is, (x * y) * z = x * (y * z) for all x, y, z ∈ L. If one of x, y, z is 0, then both sides are 0, the same; if one of x, y, z is e, then the equality holds; if one of x, y, z is 1 but none of them is 0, then both sides are 1, the same; and lastly if x = y = z = a, then both sides are a, the same.

(Q3) We need to show x * (y ∨ z) = (x * y) ∨ (x * z) for all x, y, z ∈ L. In fact, if y, z are compatible, then the equality holds since * is monotone; otherwise {y, z} = {e, a}, then the left side is x * 1, and the right side is x * (x * 1) = (x * x) *1 = x * 1, the equality holds.

If we want to make the lattice L = {0, e, a, 1} to be a commutative unital quantale, then only a * 1 should be determined. Since the operation * should be monotone firstly, there are two options a and 1 for a * 1. Example 2.2 has chosen 1, and of course we can also choose a.

Example 2.3. Let L = {0, e, a, 1} be the complete lattice in Example 2.2. Define * : L × L ⟶ L by

*	e	a	1
0	0	0	0
e	e	a	1
a	a	a	a
1	1	a	1

Then L is a commutative unital quantale with e as the unit. We leave it as an exercise for the readers.

Proposition 2.4. Let L be a commutative unital quantale. Then for all a, b, c ∈ L and all {a_i| i ∈ I} ⊆ L,

(1) a * 0 =0;

(2) a ≤ b iff e ≤ a → b;

(3) e → a = a;

(4) b ≤ c implies a → b ≤ a → c;

(5) b ≤ c implies c → a ≤ b → a;

(6) b → (⋀ _ia_i) = ⋀ _i (b → a_i);

(7) (⋁ _ia_i) → b = ⋀ _i (a_i → b);

(8) (a → b) * (b → c) ≤ a → c;

(9) b → c ≤ (a * b) → (a * c).

Let X be a nonempty set. Every mapping A : X ⟶ L is called an L-fuzzy subset of X. Let L^X denote the set of all L-fuzzy subset of X. Then L^X, under the pointwise order, also is a complete lattice. For a ∈ L and x ∈ X, we use the symbol a_x denote the L-fuzzy subset of X which sends x to a and others to 0. This kind of an L-fuzzy subset is called a fuzzy point of X. For A, B ∈ L^X and ⊗ ∈ {∨ , ∧ , * , →}, the notation A ⊗ B denotes the L-fuzzy subset sending x to A (x) ⊗ B (x); for a ∈ L, a ⊗ A denotes the L-fuzzy subset sending x to a ⊗ A (x).

Proposition 2.5. For every A ∈ L^X, A = ⋁ _y∈XA (y) * e_y.

Proof. For every x ∈ X, (⋁ _y∈XA (y) * e_y) (x) = ⋁ _y∈XA (y) * e_y (x) = A (x) * e_x (x) = A (x) * e = A (x) . Therefore, A = ⋁ _y∈XA (y) * e_y.

Every mapping E : X × X ⟶ L is called an L-fuzzy (binary) relation on X.

Definition 2.6. Let E : X × X ⟶ L be an L-fuzzy relation. We call E an L-fuzzy similarity on X if

(e-Ref) e-reflectivity: ∀x ∈ X, E (x, x) ≥ e;

(*-Tran) *-transitivity: ∀x, y, z ∈ X, E (x, y) * E (y, z) ≤ E (x, z).

Condition (e-Ref) can be interpreted as the reflectivity of the relation since e is the truth value; and Condition (*-Tran) can be interpreted as the transitivity of the relation since * is the conjunction. Notice that an L-fuzzy similarity need not be symmetric, otherwise it will be an L-fuzzy equivalence.

Example 2.7. By Proposition 2.4, the equality I (a, b) = a → b defines a nonsymmetric L-fuzzy similarity on L.

Define Sub_X : L^X × L^X ⟶ L by ${Sub}_{X} (A, B) = ⋀_{x \in X} A (x) \to B (x) (\forall A, B \in L^{X}) .$ Then it is easy to investigate that Sub_X is an L-fuzzy similarity on L^X, that is we have, for all A, B, C ∈ L^X,

(1) Sub_X (A, A) ≥ e;

(2) Sub_X (A, B) * Sub_X (B, C) ≤ Sub_X (A, C).

Indeed, Sub_X is rather than an L-fuzzy similarity, since besides the conditions of (e-Ref) and (*-Tran), it also satisfies

(e-ASym) (e-antisymmetry):

Sub_X (A, B) ∧ Sub_X (B, A) ≥ e imply A = B (∀A, B ∈ L^X).

If an L-fuzzy relation satisfies (e-Ref), (*-Tran) and (e-ASym), then it is called an L-fuzzy order. For example I (a, b) in Example 2.7 is an L-fuzzy order on L. And here, Sub_X is an L-fuzzy order on L^X. The value Sub_X (A, B) is called the subsethood degree of A in B, which is an extenstion of the classical subset relation in set theory.

Proposition 2.8. For every x ∈ X and every A ∈ L^X, it holds that ${Sub}_{X} (e_{x}, A) = A (x) .$

Proof. Straightforward.

3 Constructive and axiomatic approaches to L-fuzzy rough sets

3.1 The constructive approach

In this section, we will define study the axioms of upper rough approximation operators induced by L-fuzzy similarities.

Let X be a set and E : X × X ⟶ L be an L-fuzzy relation. Define c_E : L^X ⟶ L^X by $c_{E} (A) (x) = ⋁_{y \in X} E (x, y) * A (y) (\forall x \in X),$ called the upper rough approximation operator induced by the L-fuzzy similarity E.

Firstly, we study some basic properties of the upper rough approximation operator c_E.

Theorem 3.1. Let E be an L-fuzzy relation on a set X. Then

(FM) (Fuzzy monotone): ∀A, B ∈ L^X,

Sub_X (A, B) ≤ Sub_X (c_E (A) , c_E (B));

(JP) (Join-preserving):∀ {A_i| i ∈ I} ⊆ L^X,

c_E (⋁ _iA_i) = ⋁ _ic_E (A_i);

(TN) (Tensored)

c_E (a * A) = a * c_E (A) (∀ a ∈ L, ∀ A ∈ L^X) .

Proof. (FM) Let A, B ∈ L^X. $\begin{matrix} {Sub}_{X} (c_{E} (A), c_{E} (B)) \\ = & ⋀_{x \in X} c_{E} (A) (x) \to c_{E} (B) (x) \\ = & ⋀_{x \in X} (⋁_{y \in X} E (x, y) * A (y)) \to (⋁_{z \in X} E (x, z) * B (z)) \\ = & ⋀_{x \in X} ⋀_{y \in X} (E (x, y) * A (y)) \to (⋁_{z \in X} E (x, z) * B (z)) \\ \geq & ⋀_{x, y \in X} (E (x, y) * A (y)) \to (E (x, y) * B (y)) \\ \geq & ⋀_{y \in X} A (y) \to B (y) \\ = & {Sub}_{X} (A, B) . \end{matrix}$ (JP) Let {A_i| i ∈ I} ⊆ L^X. $\begin{matrix} c_{E} (⋁_{i} A_{i}) & = & ⋁_{y \in X} E (x, y) * (⋁_{i} A_{i}) (y) \\ = & ⋁_{y \in X} ⋁_{i} E (x, y) * A_{i} (y) \\ = & ⋁_{i} ⋁_{y \in X} E (x, y) * A_{i} (y) \\ = & ⋁_{i} c_{E} (A_{i}) . \end{matrix}$ (TN) Let a ∈ L and A ∈ L^X. For every x ∈ X, $\begin{matrix} c_{E} (a * A) (x) & = & ⋁_{y \in X} E (x, y) * (a * A) (y) \\ = & a * ⋁_{y \in X} E (x, y) * A (y) \\ = & a * c_{E} (A) (x) . \end{matrix}$ Then c_E (a * A) = a * c_E (A) .

By Proposition 2.4, Condition (FM) can imply the classical monotonicity:

(CM) A ≤ B implies c_E (A) ≤ c_E (B) (∀A, B ∈ L^X).

Condition (FM) is a very natural condition for self-operators on L-powerset. It means that upper rough approximation operator is closely related to the L-fuzzy order Sub_X on L^X, for which we consider it as a crucial property of the upper rough approximation operators in fuzzy rough set theory.

We know that the upper rough approximation operator in various fuzzy rough set model is always a special type of lattice-theoretic closure operators, and thus it should be always join-preserving (cf. Theorem 3.1), enlarging and idempotent. By Theorem 3.2, Theorem 3.3 and Corollary 3.4, we will see that these properties of the upper rough approximation operator c_E will also reflect the properties of the L-fuzzy relation E.

Theorem 3.2. Let E be an L-fuzzy relation on X. Then E is e-reflective iff

(EL) (Enlarging) c_E (A) ≥ A (∀ A ∈ L^X) .

Proof. The necessity: For every x ∈ X, $\begin{matrix} c_{E} (A) (x) & = & ⋁_{y \in X} E (x, y) * A (y) \\ \geq & E (x, x) * A (x) \\ = & e * A (x) = A (x) . \end{matrix}$ Then c_E (A) ≥ A.

The sufficiency: For x ∈ X, we have $\begin{matrix} e & = & e_{x} (x) \leq c_{E} (e_{x}) (x) \\ = & ⋁_{y \in X} E (x, y) * e_{x} (y) \\ = & E (x, x) * e \\ = & E (x, x) . \end{matrix}$

Theorem 3.3. Let E be an L-fuzzy relation on X. Then E is *-transitive iff

(SI) (Semi-idempotent) c_E (c_E (A)) ≤ c_E (A) (∀ A ∈ L^X).

Proof. The necessity: For every x ∈ X, $\begin{matrix} c_{E} (c_{E} (A)) (x) & = & ⋁_{y \in X} E (x, y) * c_{E} (A) (y) \\ = & ⋁_{y \in X} E (x, y) * ⋁_{z \in X} E (y, z) * A (z) \\ = & ⋁_{y, z \in X} E (x, y) * E (y, z) * A (z) \\ \leq & ⋁_{z \in X} E (x, z) * A (z) \\ = & c_{E} (A) (x) . \end{matrix}$ Hence c_E (c_E (A)) ≤ c_E (A).

The sufficiency: For all x, z ∈ X, it holds that $\begin{matrix} c_{E} (c_{E} (e_{z})) (x) & = & ⋁_{y \in X} E (x, y) * c_{E} (e_{z}) (y) \\ = & ⋁_{y, u \in X} E (x, y) * E (y, u) * e_{z} (u) \\ = & ⋁_{y \in X} E (x, y) * E (y, z) . \end{matrix}$ While c_E (e_z) (x) = ⋁ _u∈XE (x, u) * e_z (u) = E (x, z) . By (SI), E is *-transitive.

The conditions (EL) and (SI) can imply the idempotency, that is we have

Corollary 3.4. Let E be an L-relation on X. Then E is an L-fuzzy similarity iff

(ID) (Idempotent) ∀A ∈ L^X, c_E (c_E (A)) = c_E (A).

Definition 3.5. Let c : L^X ⟶ L^X be a mapping. If is satisfies (FM), (EL) and (ID), then we call c a basic L-fuzzy closure operator on X, which is a standard fuzzy closure operator of the fuzzy poset (L^X, Sub_X) [42]. Together with (JP) and (TN), we call c an L-fuzzy closure operator on X.

3.2 The axiomatic approach

Now we suppose there is an L-fuzzy closure operator. Let us see that how to defined the related L-fuzzy relation.

Let c : L^X ⟶ L^X be a mapping. Define E_c : X × X ⟶ L by $E_{c} (x, y) = c (e_{y}) (x) .$

Notice that we usually define the L-fuzzy relation as c (1_y) (x). But here we replace 1_y by e_y, which means in a commutative unital quantale, the unit e is more important than the top element 1, this is the big difference of this model from those in [2 , 35].

Theorem 3.6. For every L-fuzzy relation E : X × X ⟶ L, it always holds that E_{c
_E} = E.

Proof. For all x, y ∈ X, it holds that E_{c
_E} (x, y) = c_E (e_y) (x) = ⋁ _z∈ZE (x, z) * e_y (z) = E (x, y) * e = E (x, y) . Then, E_{c
_E} = E.

Theorem 3.6 means that in order to get the L-fuzzy relation back, we need no further condition. This is not the case in order to get a L-fuzzy closure operator back and Lemma 3.7 and Theorem 3.8 make a preparation for that purpose.

Lemma 3.7. Let c₁ and c₂ be two selfmappings on L^X. If they satisfy (TN) and (JP), then c₁ = c₂ iff c₁ (e_y) = c₂ (e_y) holds for all y ∈ X.

Proof. We only need to show the sufficiency. In fact, for every A ∈ L^X, $\begin{matrix} c_{1} (A) & = & c_{1} (⋁_{y \in X} A (y) * e_{y}) \\ = & ⋁_{y \in X} c_{1} (A (y) * e_{y}) \\ = & ⋁_{y \in X} A (y) * c_{1} (e_{y}) \\ = & ⋁_{y \in X} A (y) * c_{2} (e_{y}) \end{matrix}$ $\begin{matrix} = & ⋁_{y \in X} c_{2} (A (y) * e_{y}) \\ = & c_{2} (⋁_{y \in X} A (y) * e_{y}) = c_{2} (A) . \end{matrix}$ Then c₁ (A) = c₂ (A) for every A ∈ L^X.

Theorem 3.8. Let c : L^X ⟶ L^X satisfy (TN) and (JP). Then c_{E
_c} = c.

Proof. Since E_c is an L-fuzzy relation on X. By Theorem 1, c_{E
_c} satisfies (TN) and (JP). For all x, y ∈ X, it holds that c_{E
_c} (e_y) (x) = ⋁ _z∈XE_c (x, z) * e_y (z) = E_c (x, y) = c (e_y) (x) . Then c_{E
_c} (e_y) = c (e_y) for all y ∈ X. By Lemma 5, c_{E
_c} = c.

Theorem 3.9. If c satisfies (EL), then E_c is e-reflective.

Proof. For every x ∈ X, E_c (x, x) = c (e_x) (x) ≥ e_x (x) = e.

Theorem 3.10. 3If c satisfies (FM) and (SI), then E_c is *-transitive.

Proof. For all x, y, z ∈ X,

$\begin{matrix} \begin{matrix} E_{c} (x, y) * E_{c} (y, z) \\ = & {Sub}_{X} (e_{x}, c (e_{y})) * {Sub}_{X} (e_{y}, c (e_{z})) \\ \leq & {Sub}_{X} (e_{x}, c (e_{y})) * {Sub}_{X} (c (e_{y}), c (c (e_{z}))) \\ \leq & {Sub}_{X} (e_{x}, c (e_{y})) * {Sub}_{X} (c (e_{y}), c (e_{z})) \\ \leq & {Sub}_{X} (e_{x}, c (e_{z})) \\ = & E_{c} (x, z) . \end{matrix} \end{matrix}$

By Theorems 3.9 and 3.10, we have

Corollary 3.11. (1) If c satisfies (TN), (JP) and (EL), then there exists an e-reflective L-fuzzy relation E on X such that c = c_E.

(2) If c satisfies (TN), (JP), (FM) and (SI), then there exists a *-transitive L-fuzzy relation E on X such that c = c_E.

(3) If c is an L-fuzzy closure operator, then there exists an L-fuzzy similarity E on X such that c = c_E.

Theorem 3.12. 6 If c is a basic L-fuzzy closure operator, then $E_{c} (x, y) = ⋀_{A \in L^{X}} c (A) (y) \to c (A) (x) (\forall x, y \in X) .$

Proof. For all x, y ∈ X, $\begin{matrix} c (e_{y}) (x) * c (A) (y) \\ = & {Sub}_{X} (e_{x}, c (e_{y})) * {Sub}_{X} (e_{y}, c (A)) \\ \leq & {Sub}_{X} (e_{x}, c (e_{y})) * {Sub}_{X} (c (e_{y}), c (c (A))) \\ = & {Sub}_{X} (e_{x}, c (e_{y})) * {Sub}_{X} (c (e_{y}), c (A)) \\ \leq & {Sub}_{X} (e_{x}, c (A)) \\ = & c (A) (x) . \end{matrix}$ By the arbitrariness of A ∈ L^X, $c (e_{y}) (x) \leq ⋀_{A \in L^{X}} c (A) (y) \to c (A) (x) .$ Conversely, $\begin{matrix} ⋀_{A \in L^{X}} c (A) (y) \to c (A) (x) \\ \leq & c (e_{y}) (y) \to c (e_{y}) (x) \\ \leq & e_{y} (y) \to c (e_{y}) (x) \\ = & e \to c (e_{y}) (x) \\ = & c (e_{y}) (x) . \end{matrix}$ Hence, $E_{c} (x, y) = ⋀_{A \in L^{X}} c (A) (y) \to c (A) (x) .$

Remark 3.13. The equality in Theorem 3.12 gives a common way to induce the L-fuzzy relation from a basic L-fuzzy operator, which is more smooth than the direct way of E_c (x, y) = c (e_y) (x). This way is more effective than the direct way, since we need not to judge which one of c (1_y) (x) and c (e_y) (x) is the proper definition for the induced similarity, even if the unit e is not the same as the top element 1 in a quantale.

Remark 3.14. Let E be an L-fuzzy similarity. For the lower rough approximation operator, we can define it as $i_{E} (A) (x) = ⋀_{y \in X} E (y, x) \to A (y) .$ In order to make c_E and i_E dual to each other., the lattice L still should be assumed to be regular, that is (a → 0) →0 = a (∀ a ∈ L), thus L is regular. Otherwise we should suppose there exists an abstract order-reversing involution ^′, then define i_E (A) = (c_E (A′)) ′. The former of course is a special case of the latter since the operator () →0 is an order-reserving involution on L.

4 Properties of the L-topology induced by upper rough approximation operators

In this section, we will study the properties of the L-topology induced by upper rough approximation operators. Similar work has been done in [6 , 27], while therein the truth value tables are the unit interval or a complete residuated lattice, which are special cases of commutative unital quantales.

Theorem 4.1.Let E be an L-fuzzy relation on X and let c be the induced upper rough approximation operator. Then δ_c = {A ∈ L^X| c (A) = A} has the following properties:

(TCJ) ⋁_iA_i ∈ δ_c (∀ {A_i| i ∈ I} ⊆ δ_c);

(TNS) a * A ∈ δ_c (∀a ∈ L, A ∈ δ_c).

Proof. We know that c satisfies the conditions (FM), (JP) and (TN).

(TCJ) Let {A_i| i ∈ I} ⊆ δ_c. By (JP), we have c (⋁ _iA_i) = ⋁ _ic (A_i) = ⋁ _iA_i and so ⋁_iA_i ∈ δ_c.

(TNS) Let A ∈ δ_c and a ∈ L. By (TN), we have c (a * A) = a * c (A) = a * A and so a * A ∈ δ_c.

Theorem 4.2. Let E be an e-reflective L-fuzzy relation on X and let c be the induced upper rough approximation operator. Then δ_c has the following properties:

(TCM) ⋀_iA_i ∈ δ_c (∀ {A_i| i ∈ I} ⊆ δ_c);

(TNA) a → A ∈ δ_c (∀a ∈ L, A ∈ δ_c).

Proof. We know that c satisfies the conditions (FM), (JP), (TN), (EL) and (ID); the family δ_c satisfies (TCJ) and (TNS).

(TCM) Let {A_i| i ∈ I} ⊆ δ_c. By (FM), we have c (⋀ _iA_i) ≤ ⋀ _ic (A_i) = ⋀ _iA_i. Together with (EL), we have c (⋀ _iA_i) = ⋀ _iA_i and ⋀_iA_i ∈ δ_c.

(TNA) Let A ∈ δ_c and a ∈ L. By (TNS) and (JP), we have a * c (a → A) = c (a * (a → A)) ≤ c (A) = A, we have c (a → A) ≤ a → A. Together with (EL), we have c (a → A) = a → A and so a → A ∈ δ_c.

If δ ⊆ L^X satisfies (TCJ) and (TCM), then we called it an Alexandrov L-topology on X. If an Alexandrov L-topology satisfies (TNS) and (TNA), then we called it a stratified Alexandrov L-topology on X. If E is an e-reflective L-fuzzy relation on X, then we can get a stratified Alexandrov L-topology on X

Suppose that δ is a stratified Alexandrov L-topology on X. Define c_δ : L^X ⟶ L^X by $c_{δ} (A) = ⋀ {B \in δ | A \leq B} .$ Then A ∈ δ_c iff c_δ (A) = A.

Theorem 4.3. Suppose that δ is stratified Alexandrov L-topology on X. Then c_δ is an L-fuzzy closure operator on X.

Proof. (EL) is automatically holds; (ID) holds directly by (TCM).

(FM) Let A, B ∈ L^X. If C ∈ δ and B ≤ C, then by (TNS), we have Sub_X (A, B) * C ∈ δ and ${Sub}_{X} (A, B) * C \geq {Sub}_{X} (A, B) * B \geq A .$ Therefore, $\begin{matrix} {Sub}_{X} (c_{δ} (A), c_{δ} (B)) \\ = & ⋀_{C \in δ, C \leq B} {Sub}_{X} (c_{δ} (A), C) \\ = & ⋀_{C \in δ, C \leq B} ⋀_{x \in X} c_{δ} (A) (x) \to C (x) \\ = & ⋀_{C \in δ, C \leq B} ⋀_{x \in X} ⋀ {D (x) | D \in δ | A \leq D} \to C (x) \\ \geq & ⋀_{C \in δ, C \leq B} ⋀_{x \in X} ({Sub}_{X} (A, B) * C (x)) \to C (x) \\ \geq & {Sub}_{X} (A, B) . \end{matrix}$ (JP) Let {A_i| i ∈ I} ⊆ L^X. If C ∈ δ and C ≥ ⋁ _ic_δ (A_i), by (EL) and (FM), we have C ≥ c_δ (A_i) ≥ A_i for every i ∈ I. Then C ≥ ⋁ _iA_i and C = c_δ (C) ≥ c_δ (⋁ _iA_i). By the arbitrariness of C ∈ δ, we have c_δ (⋁ _iA_i) ≤ ⋁ _ic_δ (A_i). Together with (FM), c_δ (⋁ _iA_i) = ⋁ _ic_δ (A_i).

(TN) Since c_δ (A) ∈ δ and a * c_δ (A) ≥ a * A, we have c_δ (a * A) ≤ a * c_δ (A). Let B ∈ δ and a * A ≤ B. Then A ≤ a → B ∈ δ and then a * c_δ (A) ≤ a * (a → B) ≤ B. Then a * c_δ (A) ≤ c_δ (a * A).

Theorem 4.4. If δ satisfies (TCM), then δ_{c
_δ} = δ.

Proof. A ∈ δ_{c
_δ} iff A = c_δ (A) = ⋀ {B ∈ δ| A ≤ B} iff A ∈ δ.

Theorem 4.5. If c satisfies (FM) and (ID), then c_{δ
_c} = c.

Proof. Clearly, c_{δ
_c} (A) = ⋀ {B ∈ δ_c| A ≤ B} = ⋀ {B ∈ L^X| c (B) = B, A ≤ B} . On one hand, if A ≤ B and c (B) = B, then c (A) ≤ c (B) = B. Therefore, c (A) ≤ c_{δ
_c} (A). On the other hand, since A ≤ c (A) and c (c (A)) = c (A), we have c_{δ
_c} (A) ≤ c (A). Hence, c_{δ
_c} (A) = c (A) for every A ∈ L^X.

Results in this section indicate that all definable subsets induced by an e-reflective L-fuzzy relation form a stratified Alexandrov L-topology. These kind of topology are not only closed under arbitrary joins and meets, but also closely related to the operation * and → on the lattice L. It is also easy to see that L-fuzzy similarities can be characterized by stratified Alexandrov L-topologies completely.

5 Conclusion

Let a commutative unital quantale L be the truth value table. We define and study the upper rough approximation operators for an L-fuzzy similarity. Results showed that the upper rough approximation operator is a kind of L-fuzzy version of closure operators. It is monotone with respect to the natural L-fuzzy order on the L-powerset, which is a distinguish feature beyond others. By using this feature, every L-fuzzy similarity can be completely determined by the so-called L-fuzzy closure operator. The related results are exactly the constructive and axiomatic approaches to L-fuzzy rough set theory. Meanwhlie, we also got a natural and smooth way to induce an L-fuzzy relation by an L-fuzzy closure operator without the usage of fuzzy points.

In Pawlak theory of rough sets, one always use a pair of closure operator and interior operator to determine the binary relation. Since these two operators are dual to each other, each of them can determine the binary relation by itself. For the quantale-valued case in this paper, we have not defined and studied the lower rough approximation operators. In order to do so, we need more requirements for the lattice L, since the lower one is always assumed to be dual to the upper one. It will make the lattice too special: a commutative unital quantale with the double negation or a bi-commutative unital quantale. A key point of this paper is to state that (fuzzy) rough set theory, from viewpoint of pure mathematics, can be only studied by one operator and then get the other one by the dual principle.

We also investigated the properties of the related L-fuzzy topology. In order to establish a strictly close relation, we need an Alexandrov one which is stratified.

Footnotes

Acknowledgment

This paper is supported by National Natural Science Foundation of China (11871189, 12111540250, 12231007), Natural Science Foundation of Hebei Province (A2020208008), Jiangsu Provincial Innovative and Entrepreneurial Talent Support Plan (JSSCRC2021521).

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