Yongwei Yang, Kuanyun Zhu and Xiaolong Xin

Rough sets based on fuzzy ideals in distributive lattices

Open Access
De Gruyter Open Access | Published online: March 10, 2020

Abstract

In this paper, we present a rough set model based on fuzzy ideals of distributive lattices. In fact, we consider a distributive lattice as a universal set and we apply the concept of a fuzzy ideal for definitions of the lower and upper approximations in a distributive lattice. A novel congruence relation induced by a fuzzy ideal of a distributive lattice is introduced. Moreover, we study the special properties of rough sets which can be constructed by means of the congruence relations determined by fuzzy ideals in distributive lattices. Finally, the properties of the generalized rough sets with respect to fuzzy ideals in distributive lattices are also investigated.

MSC 2010: 03G10; 06B10; 08A72

1 Introduction

It is well known that the real world problems under consideration are full of indeterminacy and vagueness. In fact, most of the problems that we deal with are vague rather than precise. In the face of so many uncertain data, classical methods are not always successful in dealing with them, because of various types of uncertainties presented in these problems. As far as known, there are several theories to describe uncertainty, for example, fuzzy set theory, rough set theory and other mathematical tools. Over the years, many experts and scholars are looking for some different ways to solve the problem of uncertainty.

Rough set theory was first introduced by Pawlak [1] which is an extension of set theory, as a new mathematical approach to deal with uncertain knowledge and has attracted the interest of researchers and practitioners in various fields of science and technology. In rough set theory, rough sets can be described by a pair of ordinary sets called the lower and upper approximations. However, these equivalence relations in Pawlak rough sets are restrictive in some areas of applications. To solve this issue, some more general models have been proposed, such as quantitative rough sets based on subsethood measure, generalized rough sets based on relations and so on [2, 3]. Nowadays, rough set theory has been applied to many areas, such as knowledge discovery, machine learning, approximate classification and so on [4, 5, 6]. In particular, many researchers applied this theory to algebraic structures. Wang [7] investigated the topological characterizations of generalized fuzzy rough sets. Zhu and Hu [8] introduced the notion of Z-soft rough fuzzy BCI-algebras (ideals), which is an extended notion of soft rough BCI-algebras (ideals) and rough fuzzy BCI-algebras (ideals), and investigated roughness in BCI-algebras with respects to a Z-soft approximation space. Shao et al. introduced the notions of rough filters, multi-granulation rough filters, and rough fuzzy filters in pseudo-BCI algebras [9]. The lower and upper approximations in various hyperstructures were also discussed by many authors in many literatures [10, 11, 12]. Furthermore, some authors considered rough sets in a fuzzy algebraic system, such as [13, 14] studied some types of fuzzy covering rough set models and their generalizations over fuzzy lattices. The generalization of Pawlak rough set was introduced for two universes on general binary relations. Thus, equivalence relations should be extended to two universes for algebraic sets. It follows from this point of view that Davvaz [15] and Yamak et al. [16] put forward the notion of set-valued homomorphism for groups and rings, respectively.

In particular, Davvaz applied the notion of fuzzy ideal of a ring for definitions of the lower and upper approximations in a ring and studied the characterizations of the approximations [17]. In 2014, Xiao et al. [18] studied rough set model on ideals in lattices. In [18], let I be an ideal in a lattice L. Then θI is a joint-congruence on L. θI is a congruence on L if and only if L is distributive. Based on these congruences, they discussed the algebraic properties of rough sets induced by ideals in lattices. Since fuzzy set is an extension of classical set, it is meaningful to use fuzzy set instead of classical set. Be inspired of [17, 18], we focus on discussing the algebraic properties of rough sets induced by fuzzy ideals in distributive lattices. A novel congruence relation U(μ, t) induced by a fuzzy ideal μ of a distributive lattice is introduced. Some properties of this congruence relation are also investigated. Further, we discuss the lower and upper approximations of a subset of a distributive lattice with respect to a fuzzy ideal. Some characterizations of the above approximations are made and some examples are discussed.

This paper is organized as follows. In Section 2, we recall some concepts and results on lattices, fuzzy sets and rough sets. In Section 3, we study the rough sets which are constructed by a novel congruence relation U(μ, t). In particular, in Section 4, we introduce a special class of set-valued homomorphism with respect to a fuzzy ideal and discuss the properties of the generalized rough set.

2 Preliminaries

In this section, we recall some basic notions and results about lattices, fuzzy sets and rough sets. Throughout this paper, L is always a distributive lattice with the minimum element 0.

Definition 2.1

[19] Let L be a lattice and ∅ ≠ IL. Then I is called an ideal of L if it satisfies the following conditions: for any x, yL,

  1. xI and yI imply xyI;

  2. xL and xy imply xI.

Let A, B be subsets of L, we define the join and meet as follows:

A B = { a b | a A , b B }  and  A B = { a b | a A , b B } .

Let I, J be ideals of L, then IJ is an ideal of L [18].

Definition 2.2

[19] Let L be a lattice. A relation R is called an equivalence relation on L if for all a, b, cL,

  1. Reflexive: (a, a) ∈ R;

  2. Symmetry: (a, b) ∈ R implies (b, a) ∈ R;

  3. Transitivity: (a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R.

An equivalence relation R is called a congruence relation on L, if for all a, b, c, dL, (a, b) ∈ R, (c, d) ∈ R, then (ac, bd ) ∈ R and (ac, bd) ∈ R.

Definition 2.3

[20] Let μ be a fuzzy set of a lattice L. Then μ is called a fuzzy sublattice of L if μ(xy) ∧ μ(xy) ≥ μ(x) ∧ μ(y), for all x, yL.

Let μ be a fuzzy sublattice of L. Then μ is a fuzzy ideal of L, if μ(xy) = μ(x) ∧ μ(y) for all x, yL.

Proposition 2.4

[20] Let μ be a fuzzy sublattice of a lattice L. Then μ is a fuzzy ideal of L if and only if xy implies that μ(x) ≥ μ(y), for all x, yL.

Proposition 2.5

[21] Let μ be a fuzzy set of a lattice L. Then μ is a fuzzy ideal of L if and only if any one of the following sets of conditions is satisfied: for all x, yL,

  1. μ(0) = 1 and μ(xy) = μ(x) ∧ μ(y);

  2. μ(0) = 1, μ(xy) ≥ μ(x) ∧ μ(y) and μ(xy) ≥ μ(x) ∨ μ(y).

Let μ be a fuzzy subset of a lattice L and t ∈ [0, 1]. Then the set μt = {xLμ(x) ≥ t} is called a t-level subset of μ.

Remark 2.6

A fuzzy set μ is a fuzzy ideal of a lattice L if and only if every subset μt is an ideal of L for all t ∈ [0, 1].

Definition 2.7

[1] Let R be an equivalence relation on the universe U and (U, R) be a Pawlak approximation space. A subset XU is called definable if R*X = R*X; otherwise, X is said to be a rough set, where two operators are defined as:

R X = { x U | [ x ] R X } , R X = { x U | [ x ] R X } .

Definition 2.8

[1] Let X and Y be two non-empty sets and BY. Let T : X → 𝒫(Y) be a set-valued mapping, where 𝒫(Y) denotes the family of all non-empty subsets of Y. The lower and upper approximations T(B) and T(B) are defined by

T _ ( B ) = { x U | T ( x ) B } , T ¯ ( B ) = { x U | T ( x ) B } ,

respectively. If T(B) ≠ T(B), then the pair (T(B), T(B)) is said to be a generalized rough set.

3 A novel congruence relation induced by a fuzzy ideal in a distributive lattice

In this section, we introduce a novel congruence relation U(μ, t) induced by a fuzzy ideal μ in a distributive lattice. We define the join and meet of two non-empty subsets in a lattice as follows: AB = {abaA, bB}, AB = {abaA, bB}.

Definition 3.1

Let μ be a fuzzy ideal of L. For each t ∈ [0, 1], the set

U ( μ , t ) = ( x , y ) L × L | { μ ( a ) | a x = a y , a L } t

is called a t-level relation of μ.

Example 3.2

Let L = {0, a, b, c, 1}. We define the binary relationin the following Hasse diagram. It is easy to check that L is a distributive lattice. Let μ = 1 0 + 0.8 a + 0.6 b + 0.4 c + 0 1 . Then it is clear that μ is a fuzzy ideal of L. Choose t = 0.9, then we have U(μ, 0.9) = {(0, 0), (a, a), (b, b), (c, c), (1, 1)}. Thus U(μ, 0.9) is called a 0.9-level relation of μ.

Now we prove that U(μ, t) is a congruence relation on L.

Lemma 3.3

Let μ be a fuzzy ideal of L and t ∈ [0, 1]. Then U(μ, t) is a congruence relation on L.

Proof

It is easy to see that μ(0) = 1 and for any xL, a x = a x μ(a) = ⋁ μ(a) ≥ μ(0) = 1 ≥ t. From Definition 3.1, we get that (x, x) ∈ U(μ, t), i.e., U(μ, t) is reflexive. Obviously, U(μ, t) is symmetric. Let (x, y) ∈ U(μ, t) and (y, z) ∈ U(μ, t). Then we have

a x = a y μ ( a ) t , b y = b z μ ( b ) t ,

and so ( a x = a y μ ( a ) ) ( b y = b z μ ( b ) ) t . Since μ is a fuzzy ideal of L, we obtain that

( a x = a y μ ( a ) ) ( b y = b z μ ( b ) ) = a x = a y , b y = b z ( μ ( a ) μ ( b ) ) = a x = a y , b y = b z μ ( a b ) .

For ax = ay, by = bz, we have abx = aby, aby = abz. Thus abx = abz, i.e., cx = cz, where c = abL. It follows that

t a x = a y , b y = b z μ ( a b ) c x = c z μ ( c ) ,

and so c x = c z μ(c) ≥ t. According to Definition 3.1, we get that (x, z) ∈ U(μ, t). Therefore, U(μ, t) is an equivalence relation on L. Now we show that U(μ, t) is a congruence relation on L. Let (x, y) ∈ U(μ, t) and (u, v) ∈ U(μ, t). Then

a x = a y μ ( a ) t , b u = b v μ ( b ) t ,

and so

( a x = a y μ ( a ) ) ( b y = b z μ ( b ) ) t .

Further, we have

( a x = a y μ ( a ) ) ( b u = b v μ ( b ) ) = a x = a y , b u = b v ( μ ( a ) μ ( b ) ) = a x = a y , u y = b v μ ( a b ) .

For ax = ay, bu = bv, we have ab ∨ (xu) = ab ∨ (yv), i.e., c ∨ (xu) = c ∨ (yv), where c = abL. Hence,

t a x = a y , u y = b v μ ( a b ) c ( x u ) = c ( y v ) μ ( c ) .

Consequently, c ( x u ) = c ( y v ) μ(c) ≥ t, which implies that (xu, yv) ∈ U(μ, t).

Further, let (x1, y1) ∈ U(μ, t) and (x2, y2) ∈ U(μ, t). Then

b x 1 = b y 1 μ ( b ) t , c x 2 = c y 2 μ ( c ) t .

So

( b x 1 = b y 1 μ ( b ) ) ( c x 2 = c y 2 μ ( c ) ) t .

For bx1 = by1 and cx2 = cy2, we have

( b x 1 ) ( c x 2 ) = ( b y 1 ) ( c y 2 ) .

On the other hand, since L is a distributive lattice, we have

[ ( b c ) ( x 1 c ) ( x 2 b ) ] ( x 1 x 2 ) = [ ( b c ) ( y 1 c ) ( y 2 b ) ] ( y 1 y 2 ) .

Since (bx1) ∧ c = (cy1) ∧ c and (cx2) ∧ b = (cy2) ∧ b, we have

( b c ) ( x 1 c ) ( x 2 b ) = ( b c ) ( y 1 c ) ( y 2 b ) .

Notice that μ is a fuzzy ideal of L, we get that

μ [ ( b c ) ( x 1 c ) ( x 2 b ) ] = μ ( b c ) μ ( x 1 c ) μ ( x 2 b ) .

It follows from bcb, x1cc, x2bb that

μ ( b c ) μ ( x 1 c ) μ ( x 2 b ) μ ( b ) μ ( c ) .

Thus

t ( b x 1 = b y 1 μ ( b ) ) ( c x 2 = c y 2 μ ( c ) ) = b x 1 = b y 1 , c x 2 = c y 2 ( μ ( b ) μ ( c ) ) b x 1 = b y 1 , c x 2 = c y 2 ( μ ( b c ) μ ( x 1 c ) μ ( x 2 b ) ) [ ( b c ) ( x 1 c ) ( x 2 b ) ] ( x 1 x 2 ) = [ ( b c ) ( y 1 c ) ( y 2 b ) ] ( y 1 y 2 ) ( μ ( b c ) μ ( x 1 c ) μ ( x 2 b ) ) a ( x 1 x 2 ) = a ( y 1 y 2 ) μ ( a ) ,

and therefore (x1x2, y1y2) ∈ U(μ, t). According to the above discussing, we get that U(μ, t) is a congruence relation on L.□

Remark 3.4

In Lemma 3.3, we say x is congruent to y mod μ, written xt y (mod μ) if

a x = a y μ ( a ) t .

It follows from Definition 3.1 and Lemma 3.3 that we can get many useful properties of these congruence relations. We denote by [x](μ,t) the equivalence class of U(μ, t) containing x of L.

Lemma 3.5

Let μ be a fuzzy ideal of L and t ∈ [0, 1]. Then for all x, yL,

  1. [x](μ,t) ∨ [y](μ,t) ⊆ [xy](μ,t);

  2. [x](μ,t) ∧ [y](μ,t) ⊆ [xy](μ,t).

Proof

The proof is easy, and we omit the details.□

Let μ be a fuzzy ideal of L and t ∈ [0, 1]. Then U(μ, t) is a congruence relation on L. Thus, when U = L and R is the above equivalence relation (congruence relation), then we use (L, μ, t) instead of approximation space (U, R).

Definition 3.6

Let μ be a fuzzy ideal of L, t ∈ [0, 1] and ∅ ⊊ XL. Then

U ( μ , t ) _ ( X ) = { x L | [ x ] ( μ , t ) X }

and

U ( μ , t ) ¯ ( X ) = { x L | [ x ] ( μ , t ) X }

are called the lower approximation and the upper approximation of the set X with respect to μ and t, respectively. It is easy to know that U(μ, t)(X) ⊆ XU(μ, t)(X).

Lemma 3.7

Let μ and ν be two fuzzy ideals of L such that μν and t ∈ [0, 1]. Then [x](μ,t) ⊆ [x](ν,t) for all xL.

Proof

Let a ∈ [x](μ,t). Then we have (a, x) ∈ U(μ, t), i.e., b a = b x μ(b) ≥ t. Since μν, we have μ(b) ≤ ν(b). Thus b a = b x ν(b) ≥ b a = b x μ(b) ≥ t, which implies that (a, x) ∈ U(ν, t), i.e., a ∈ [x](ν,t). Therefore, [x](μ,t) ⊆ [x](ν,t).□

From Lemma 3.7, we get the the following conclusion easily.

Lemma 3.8

Let μ and ν be two fuzzy ideals of L such that μν, t ∈ [0, 1] and ∅ ⊊ XL. Then

  1. U(ν, t)(X) ⊆ U(μ, t)(X);

  2. U(μ, t)(X) ⊆ U(ν, t)(X);

  3. U(μ, t)(X) ∪ U(ν, t)(X) ⊆ U(μν, t)(X);

  4. U(μν, t)(X) ⊆ U(μ, t)(X) ∩ U(ν, t)(X).

The following example shows that the containedness in (3) and (4) of Lemma 3.8 need not be an equality.

Example 3.9

Consider the lattice L in Example 3.2, let μ = 1 0 + 0.6 a + 0.8 b + 0.4 c + 0 1 , ν = 1 0 + 0.8 a + 0.5 b + 0.3 c + 0 1 . Then it is clear that μ and ν are fuzzy ideals of L. Choose t = 0.8, then we have

U ( μ , 0.8 ) = { ( 0 , 0 ) , ( a , a ) , ( b , b ) , ( c , c ) , ( 1 , 1 ) , ( 0 , b ) , ( a , c ) } , U ( ν , 0.8 ) = { ( 0 , 0 ) , ( a , a ) , ( b , b ) , ( c , c ) , ( 1 , 1 ) , ( 0 , a ) , ( b , c ) } .

Thus

U ( μ ν , 0.8 ) = { ( 0 , 0 ) , ( a , a ) , ( b , b ) , ( c , c ) , ( 1 , 1 ) } .

If X = {0, c}, then

U ( μ ν ) , t ¯ ( X ) = { 0 , c } , U ( μ , t ) ¯ ( X ) U ( ν , t ) ¯ ( X ) = { 0 , a , b , c } .

Therefore U(μν, t)(X) ⫋ U(μ, t)(X) ∩ U(ν, t)(X). Further, if X = {c, 1}, then

U ( μ , t ) _ ( X ) U ( ν , t ) _ ( X ) = { 1 } , U ( μ ν , t ) _ ( X ) = { 1 , c } .

Hence U(μ, t)(X) ∪ U(ν, t)(X) ⫋U(μν, t)(X).

The following definition is from Zadeh’s expansion principle.

Definition 3.10

Let μ and ν be two fuzzy sets over L. Define μν over L as follows:

( μ ν ) ( x ) = x = a b ( μ ( a ) ν ( b ) )

for all xL.

Now we investigate the operations of lower approximations and upper approximations of the set X with respect to μ and t, respectively.

Proposition 3.11

Let μ and ν be two fuzzy ideals of L, t ∈ [0, 1] and ∅ ⊊ XL. Then

  1. U(μν, t)(X) ⊆ U(μ, t)(X) ∩ U(ν, t)(X);

  2. U(μν, t)(X) ⊇ U(μ, t)(X) ∪ U(ν, t)(X).

Proof

Since L is a distributive lattice, we have that μν is a fuzzy ideal of L. Let xL. Then (μν)(x) = x = a b (μ(a) ∧ ν(b)) ≥ μ(x) ∧ ν(0). Notice that ν is a fuzzy ideal of L, we obtain that ν(0) = 1. It follows that

( μ ν ) ( x ) = x = a b ( μ ( a ) ν ( b ) ) μ ( x ) ν ( 0 ) μ ( x ) ν ( 0 ) = μ ( x )

and so μμν. In a similar way, we have νμν. According to Lemma 3.8, we get that U(μν, t)(X) ⊆ U(μ, t)(X) ∩ U(ν, t)(X) and U(μν, t)(X) ⊇ U(μ, t)(X) ∪ U(ν, t)(X).□

Proposition 3.12

Let μ and ν be two fuzzy ideals of L, t ∈ [0, 1] and ∅ ⊊ XL. Then

  1. U(μ, t) ∩ U(ν, t) is a congruence relation on L;

  2. U(μ, t) ∩ U(ν, t)(X) ⊇ U(μ, t)(X) ∪ U(ν, t)(X);

  3. U(μ, t) ∩ U(ν, t)(X) ⊆ U(μ, t)(X) ∩ U(ν, t)(X).

Proof

It is straightforward.□

Theorem 3.13

Let μ and ν be two fuzzy ideals of L, t ∈ [0, 1] and ∅ ⊊ XL. Then

  1. U(μν, t)(X) = U(μ, t) ∩ U(ν, t)(X);

  2. U(μν, t)(X) = U(μ, t) ∩ U(ν, t)(X).

Proof

  1. We first show that U(μ, t) ∩ U(ν, t)(X) ⊆ U(μν, t)(X). Let xU(μ, t) ∩ U(ν, t) and y ∈ [x](μν,t). Then (x, y) ∈ U(μν, t),

    a y = a x ( μ ν ) ( a ) t , i . e . , a y = a x ( μ ( a ) ν ( a ) ) t .

    Thus,

    a y = a x μ ( a ) t a n d a y = a x ν ( a ) t .

    Hence, y ∈ [x](μ,t) and y ∈ [x](ν,t). So y ∈ [x](μ,t)∩(ν,t), and therefore yX, which implies that xU(μν, t)(X). Therefore, U(μ, t) ∩ U(ν, t)(X) ⊆ U(μν, t)(X).

    Next we show that U(μν, t)(X) ⊆ U(μ, t) ∩ U(ν, t)(X). Let xU(μν, t)(X) and x′ ∈ [x](μ,t)∩(ν,t). Then x′ ∈ [x](μ,t) and x′ ∈ [x](ν,t), i.e.,

    a x = a x μ ( a ) t a n d b x = b x ν ( b ) t .

    For ax′ = ax and bx′ = bx, we have

    ( a x ) ( b x ) = ( a x ) ( b x ) .

    Since L is a distributive lattice and μ and ν are fuzzy ideals of L, we have

    x ( a b ) = x ( a b ) a n d μ ( a b ) μ ( a ) , ν ( a b ) ν ( b ) ,

    i.e.,

    t ( a x = a x μ ( a ) ) ( b x = b x ν ( b ) ) x ( a b ) = x ( a b ) ( μ ( a ) ν ( b ) ) x ( a b ) = x ( a b ) ( μ ( a b ) ν ( a b ) ) = x ( a b ) = x ( a b ) ( μ ν ) ( a b ) .

    Thus x′ ∈ [x](μν,t), then x′ ∈ X, which implies that xU(μ, t) ∩ U(ν, t)(X). Thus

    U ( μ , t ) U ( ν , t ) _ ( X ) U ( μ ν , t ) _ ( X ) .

    Therefore, U(μν, t)(X) = U(μ, t) ∩ U(ν, t)(X).

  2. Let xU(μν, t)(X). Then there exists x′ ∈ [x](μ,t)∩(ν,t)X, i.e., x′ ∈ X and (x, x′) ∈ U(μν, t), so

    a y = a x ( μ ν ) ( a ) t , i . e . , a y = a x ( μ ( a ) ν ( a ) ) t .

    Thus,

    a x = a x μ ( a ) t a n d a x = a x ν ( a ) t .

    Hence, x′ ∈ [x](μ,t) and x′ ∈ [x](ν,t), which implies that xU(μ, t) ∩ U(ν, t)(X). So

    U ( μ ν , t ) ¯ ( X ) U ( μ , t ) U ( ν , t ) ¯ ( X ) .

    In a similar way, we have U(μν, t)(X) ⊇ U(μ, t) ∩ U(ν, t)(X). Therefore, U(μν, t)(X) = U(μ, t) ∩ U(ν, t)(X).□

Theorem 3.14

Let μ be a fuzzy ideal of L and t ∈ [0, 1]. Then

U ( μ , t ) _ ( μ t ) = μ t = U ( μ , t ) ¯ ( μ t ) .

Proof

It is easy to know that U(μ, t)(μt) ⊆ μtU(μ, t)(μt). Now we show that U(μ, t)(μt) ⊆ μtU(μ, t)(μt). Let xU(μ, t)(μt). Then [x](μ,t)μt ≠ ∅, which means that there exists yμt and y ∈ [x](μ,t), i.e., μ(y) ≥ t and a y = a x μ(a) ≥ t. So there exists aL such that μ(a) ≥ t satisfying ay = ax. Then we have aμt. Since μ is a fuzzy ideal of L, we have μt is an ideal of L and ayμt. Thus axμt. Since xax, we have xμt, which implies that U(μ, t)(μt) ⊆ μt. Therefore, U(μ, t)(μt) = μt. Further, let xμt and y ∈ [x](μ,t). Then (x, y) ∈ U(μ, t), i.e., b x = b y μ(b) ≥ t. So there exists bL such that μ(b) ≥ t satisfying by = bx. Then we have bμt and byμt. Since yay, we have yμt. So [x](μ,t)μt, which implies that xU(μ, t)(μt). Hence μtU(μ, t)(μt). From the above, U(μ, t)(μt) = μt = U(μ, t)(μt).□

Theorem 3.15

Let μ and ν be two fuzzy ideals of L and t ∈ [0, 1]. Then μt = U(μ, t)(μν)t.

Proof

It is easy to know that (μν)t = μtνt. Now we show that μt = U(μ, t)(μtνt). Let xμt, yνt. Then μ(x) ≥ t. Since μ is a fuzzy ideal of L, we have μt is an ideal of L. Further, xyx and xyy, we have xyμt and xyνt, i.e., xyμtνt. Since xx = x ∨ (xy), we have a x = a ( x y ) μ(a) ≥ μ(x) ≥ t, which implies that xy ∈ [x](μ,t). Thus [x](μ,t) ∩ (μtνt) ≠ ∅. So xU(μ, t)(μtνt), that is, μtU(μ, t)(μtνt). On the other hand, it is easy to see that U(μ, t)(μtνt) ⊆ U(μ, t)(μt). Moreover, it follows from Theorem 3.14 that U(μ, t)(μt) = μt. So U(μ, t)(μtνt) ⊆ μt. Therefore, μt = U(μ, t)(μtνt), i.e., μt = U(μ, t)(μν)t.□

Corollary 3.16

Let μ and ν be two fuzzy ideals of L and t ∈ [0, 1]. Then νtU(μ, t)(μtνt).

Proof

Since μ and ν are two fuzzy ideals of L, we have μt and νt are ideals of L. Further, since L is a distributive lattice, we have μtνt is an ideal of L. Let xμt and yνt. Then xyμtνt. On the other hand, b x = b ( x y ) μ(b) ≥ μ(x) ≥ t, which implies that xy ∈ [x](μ,t). Thus [x](μ,t) ∩ (μtνt) ≠ ∅. So yU(μ, t)(μtνt). Therefore, νtU(μ, t)(μtνt).□

In the following discussion, we denote by ↓ a = {xLxa} for aL.

Theorem 3.17

Let μ be a fuzzy ideal of L, t ∈ [0, 1]. Then

  1. U(μ, t)(↓ a) = μt for each aμt;

  2. a μ t U(μ, t)(↓ a) ⊆ μt.

Proof

  1. Since μ is a fuzzy ideal of L, we have μt is an ideal of L. It follows from the definition of ↓ a that ↓ a is an ideal and ↓ aμt for each aμt. It follows from the Theorem 3.15 that U(μ, t)(↓ a) = μt.

  2. Let aμt. Then ↓ aμt. It is easy to see that U(μ, t)(↓ a) ⊆ U(μ, t)(μt). Follows from Theorem 3.14, we obtain that U(μ, t)(μt) = μt. Thus U(μ, t)(↓ a) ⊆ μt. Therefore, a μ t U(μ, t)(↓ a) ⊆ μt.□

Theorem 3.18

Let μ and ν be two fuzzy ideals of L and t ∈ [0, 1]. Then the followings are equivalent:

  1. μν;

  2. νt = U(μ, t)(νt);

  3. νt = U(μ, t)(νt).

Proof

(1) ⇒ (2) Let μν and xU(μ, t)(νt). Then [x](μ,t)νt ≠ ∅. This means that there exists aνt such that a ∈ [x](μ,t), i.e.,

b a = b x μ ( b ) t .

Since μν, we have

b a = b x ν ( b ) b a = b x μ ( b ) t .

So there exists bL such that ν(b) ≥ t satisfying ba = bx, i.e., bνt. So ba = bxνt. Since xbx, we have xνt. Hence, U(μ, t)(νt) ⊆ νt. On the other hand, it is easy to see that νtU(μ, t)(νt). Therefore, νt = U(μ, t)(νt).

(2) ⇒ (1) If νt = U(μ, t)(νt), it follows from Theorem 3.14 and Theorem 3.15 that μt = U(μ, t)(μtνt) ⊆ U(μ, t)(νt) = νt. Therefore, μν.

(2) ⇒ (3) Let νt = U(μ, t)(νt), xνt and a ∈ [x](μ,t). Assume that aνt, then aU(μ, t)(νt). Thus, [x](μ,t)νt = ∅, this implies that aU(μ, t)(νt) = νt, which contradicts with xνt. Thus aνt. Hence, [x](μ,t)νt, this means that xU(μ, t)(νt). Thus νtU(μ, t)(νt). On the other hand, it is easy to see that U(μ, t)(νt) ⊆ νt. Therefore, νt = U(μ, t)(νt).

(3) ⇒ (2) Assume that νt = U(μ, t)(νt). Let xU(μ, t)(νt). Then [x](μ,t)νt ≠ ∅, which means that there exists aνt such that a ∈ [x](μ,t). Since νt = U(μ, t)(νt), we have [x](μ,t) = [a](μ,t)νt, so xU(μ, t)(νt) = νt, i.e., U(μ, t)(νt) ⊆ νt. On the other hand, it is easy to see that νtU(μ, t)(νt). Therefore, νt = U(μ, t)(νt).□

Theorem 3.19

Let μ, ν and ω be fuzzy ideals of L such that μω and t ∈ [0, 1]. Then

U ( μ , t ) ¯ ( U ( ν , t ) ¯ ( ω t ) ) = U ( ν , t ) ¯ ( ω t ) = U ( ν , t ) ¯ ( U ( μ , t ) ¯ ( ω t ) ) .

Proof

Since μω, we have μtωt. It follows from Theorem 3.14 that U(μ, t)(ωt) = ωt. So U(ν, t)(ωt) = U(ν, t)(U(μ, t)(ωt)). Next we show that U(μ, t)(U(ν, t)(ωt)) = U(ν, t)(ωt). First of all, we prove that U(ν, t)(ωt) is an ideal of L. Since ω is a fuzzy ideal of L, we have ωt is an ideal of L. On the other hand, it is easy to see that abU(ν, t)(ωt) for all a, bU(ν, t)(ωt). Let cL, dU(ν, t)(ωt) and cd. Then there exists e ∈ [d](ν,t)ωt. Now let f ∈ [c](ν,t). Then ef ∈ [d](ν,t) ∧ [c](ν,t) ⊆ [cd](ν,t) = [c](ν,t). Since efe, we have efωt. Thus [c](ν,t)A ≠ ∅, this means that cU(ν, t)(ωt). Thus U(ν, t)(ωt) is an ideal of L. Further, μtωtU(ν, t)(ωt). It follows from Theorem 3.14 that U(μ, t)(U(ν, t)(ωt)) = U(ν, t)(ωt).□

Theorem 3.20

Let μ, ν and ω be fuzzy ideals of L such that μω and t ∈ [0, 1]. Then

U ( μ , t ) U ( ν , t ) ¯ ( ω t ) = U ( μ , t ) ¯ ( ω t ) U ( ν , t ) ¯ ( ω t ) .

Proof

Let xU(μ, t)(ωt) ∩ U(ν, t)(ωt). Since μ and ω are two fuzzy ideals of L and μω, we have μtωt. It follows from Theorem 3.14 that xωtU(ν, t)(ωt) = ωtU(ν, t) ∩ U(ν, t)(ωt). So U(μ, t)(ωt) ∩ U(ν, t)(ωt) ⊆ U(μ, t) ∩ U(ν, t)(ωt). It follows from Proposition 3.12 that U(μ, t) ∩ U(ν, t)(ωt) = U(μ, t)(ωt) ∩ U(ν, t)(ωt).□

Theorem 3.21

Let μ and ν be two fuzzy ideals of L such that μν and t ∈ [0, 1]. If ∅ ⊊ AL, then

U ( μ , t ) ¯ ( ν t A ) = U ( μ , t ) ¯ ( ν t ) U ( μ , t ) ¯ ( A ) .

Proof

It is easy to see that U(μ, t)(νtA) ⊆ U(μ, t)(νt) ∩ U(μ, t)(A). Now we show that U(μ, t)(νt) ∩ U(μ, t)(A) ⊆ U(μ, t)(νtA). Let xU(μ, t)(νt) ∩ U(μ, t)(A). Since ν is a fuzzy ideal of L, we have νt is an ideal of L. It follows from Theorem 3.14 that xνtU(μ, t)(A). Thus xνt and xU(μ, t)(A), i.e., [x](μ,t)A ≠ ∅. Thus there exists aA such that a ∈ [x](μ,t), which implies that b a = b x μ(b) ≥ t. This means that there exists bL such that μ(b) ≥ t satisfying ba = bx, i.e., bμt. Since μν, we have μtνt. Thus bνt and ba = bxνt. Since aba, we have aνt. So aAνt, it follows that xU(μ, t)(νtA). And therefore U(μ, t)(νtA) = U(μ, t)(νt) ∩ U(μ, t)(A).□

Theorem 3.22

Let μ be a fuzzy ideal of L and t ∈ [0, 1]. If A, B are ideals of L and μtAB, then

  1. U(μ, t)(A) ∨ U(μ, t)(B) = U(μ, t)(AB);

  2. U(μ, t)(A) ∨ U(μ, t)(B) ⊆ U(μ, t)(AB).

Proof

  1. Let xU(μ, t)(A) ∨ U(μ, t)(B). Then there exist yU(μ, t)(A) and zU(μ, t)(B) such that x = yz, i.e., [y](μ,t)A ≠ ∅ and [z](μ,t)B ≠ ∅, which means that there exist aA and bB such that a ∈ [y](μ,t) and b ∈ [z](μ,t), i.e.,

    y a = y y μ ( y ) t , z b = z z μ ( z ) t .

    For y′ ∨ a = y′ ∨ y, z′ ∨ b = z′ ∨ z, we have (y′ ∨ z′) ∨ (ab) = (y′ ∨ z′) ∨ (yz) = (y′ ∨ z′) ∨ x. Thus

    t ( y a = y y μ ( y ) ) ( z b = z z μ ( z ) ) ( y z ) ( a b ) = ( y z ) ( y z ) ( μ ( y ) μ ( z ) ) ( y z ) ( a b ) = ( y z ) x μ ( y z ) .

    So ab ∈ [x](μ,t). Thus [x](μ,t) ∧ (AB) ≠ ∅, i.e., xU(μ, t)(AB). Therefore, U(μ, t)(A) ∨ U(μ, t)(B) ⊆ U(μ, t)(AB). Next we show that U(μ, t)(AB) ⊆ U(μ, t)(B). Since A and B are ideals of L and L is a distributive lattice, we have AB is also an ideal of L. Since μtAB, we have μtABAB. According to Theorem 3.14, we get that U(μ, t)(AB) = ABU(μ, t)(A) ∨ U(μ, t)(B). Therefore, U(μ, t)(A) ∨ U(μ, t)(B) = U(μ, t)(AB).

  2. It follows from Theorem 3.15 that U(μ, t)(AB) = AB. Since U(μ, t)(A) ∨ U(μ, t)(B) ⊆ AB, we have U(μ, t)(A) ∨ U(μ, t)(B) ⊆ U(μ, t)(AB).□

    Let μ and ν be two fuzzy ideals of L and t ∈ [0, 1]. The composition of U(μ, t) and U(ν, t) is defined as follows:

    U ( μ , t ) U ( ν , t ) = { ( x , y ) L × L | z L s u c h t h a t ( x , z ) U ( μ , t ) a n d ( z , y ) U ( ν , t ) }

    It is not difficult to check that U(μ, t)∗ U(ν, t) is a congruence relation on L if and only if U(μ, t)∗ U(ν, t) = U(ν, t)∗ U(μ, t).

Theorem 3.23

Let μ and ν be two fuzzy ideals of L, t ∈ [0, 1] and U(μ, t)∗ U(ν, t) = U(ν, t)∗ U(μ, t).

  1. If A is a non-empty subset of L, then U(μ, t)∗ U(ν, t)(A) ⊆ U(μ, t)(A) ∩ U(μ, t)(A).

  2. If A is a sublattice of L, then U(μ, t)(A) ∩ U(μ, t)(A) ⊆ U(μ, t)∗ U(ν, t)(A).

Proof

  1. Let xU(μ, t)∗ U(ν, t)(A) and a ∈ [x](μ,t). Since x ∈ [x](ν,t), we have a ∈ [x](μ, t)∗(ν, t). Thus aA. So xU(μ, t)(A). In a similar way, we have xU(ν, t)(A). Therefore,

    U ( μ , t ) U ( ν , t ) _ ( A ) U ( μ , t ) _ ( A ) U ( μ , t ) _ ( A ) .

  2. Let xU(μ, t)(A) ∩ U(μ, t)(A). Then there exist y, zA such that y ∈ [x](μ,t) and z ∈ [x](ν,t), i.e.,

    a y = a x μ ( a ) t , b z = b x ν ( a ) t .

    For ay = ax, bz = bx, we have (zy) ∨ a = (zx) ∨ a, (zx) ∨ b = xb. Hence

    ( z y ) a = ( z x ) a μ ( a ) a y = a x μ ( a ) t ,

    and

    ( z x ) b = x b ν ( a ) z b = x b ν ( b ) t .

    Thus (zy) ∈ [zx](μ,t), (zx) ∈ [x](ν,t), i.e., (zy) ∈ [x](μ, t)∗(ν, t). Since A is a sublattice of L, we have zyA. Thus zy ∈ [x](μ, t)∗(ν, t)A, i.e., xU(μ, t)∗ U(ν, t)(A). Therefore, U(μ, t)(A) ∩ U(μ, t)(A) ⊆ U(μ, t)∗ U(ν, t)(A).□

    The following example shows that the containedness in Theorem 3.22 (2) and Theorem 3.23 need not be an equality.

Example 3.24

Consider the lattice in Example 3.2. Let μ = 1 0 + 0.8 a + 0.6 b + 0.4 c + 0 1 a n d ν = 1 0 + 0.7 a + 0.8 b + 0.3 c + 0 1 . Then it is clear that μ and ν are fuzzy ideals of L. Choose t = 0.8, then μt = {0, a} and νt = {0, b}. Now let A = {a, b}, B = {0, b}. Then we have μtAB and AB = {a, b, c}. Thus

U ( μ , t ) _ ( A ) U ( μ , t ) _ ( B ) = a n d U ( μ , t ) _ ( A B ) = { b , c } .

Therefore,

U ( μ , t ) _ ( A ) U ( μ , t ) _ ( B ) U ( μ , t ) _ ( A B ) .

Let A = {a, b, c}. Then U(μ, t)(A) = {b, c}, U(ν, t)(A) = {a, c}, and

U ( μ , t ) U ( ν , t ) _ ( A ) = , U ( μ , t ) _ ( A ) U ( ν , t ) _ ( A ) = { c } .

Therefore, U(μ, t)∗ U(ν, t)(A) ⊆ U(μ, t)(A) ∩ U(μ, t)(A).

Let A = {a, c} be a sublattice of L. Then U(μ, t)(A) = {0, a, b, c}, U(ν, t)(A) = {a, c}, and

U ( μ , t ) ¯ ( A ) U ( μ , t ) ¯ ( A ) = { b , c } a n d U ( μ , t ) U ( ν , t ) ¯ ( A ) = { 0 , a , b , c } .

Therefore, U(μ, t)(A) ∩ U(μ, t)(A) ⊆ U(μ, t)∗ U(ν, t)(A).

Theorem 3.25

Let μ and ν be two fuzzy ideals of L, t ∈ [0, 1], U(μ, t)∗ U(ν, t) = U(ν, t)∗ U(μ, t) and A be an ideal of L.

  1. If μtA, then U(μ, t)∗ U(ν, t)(A) = U(μ, t)(A) ∩ U(ν, t)(A).

  2. If μt, νtA, then U(μ, t)∗ U(ν, t)(A) = U(μ, t)(A) ∩ U(ν, t)(A).

Proof

  1. Let xU(μ, t)(A) ∩ U(ν, t)(A) and x′ ∈ [x](μ, t)∗ (ν, t). Then there exists yL such that x′ ∈ [y](μ,t) and y ∈ [x](ν,t). So x d = y d μ(d) ≥ t and yA, which means that there exists dL such that μ(d) ≥ t satisfying x′ ∨ d = yd. Thus dμt. Since A is an ideal of L and μtA, we get that ydA. Further, since x′ ∨ d = ydx′, we have x′ ∈ A. So xU(μ, t)∗ U(ν, t)(A). Therefore, U(μ, t)(A) ∩ U(ν, t)(A) ⊆ U(μ, t)∗ U(ν, t)(A). On the other hand, it follows from Theorem 3.23 that U(μ, t)∗ U(ν, t)(A) = U(μ, t)(A) ∩ U(ν, t)(A).

  2. Let xU(μ, t)∗ U(ν, t)(A). Then there exist x′ ∈ A and yL such that x′ ∈ [y](μ,t) and y ∈ [x](ν,t). So yU(μ, t)(A). Since A is an ideal of L and μtA, it follows from Theorem 3.15 that U(μ, t)(A) = A. So yA. Thus xU(ν, t)(A). Since U(μ, t)∗ U(ν, t) = U(ν, t)∗ U(μ, t), we have xU(μ, t)(A). Therefore, U(μ, t)∗ U(ν, t)(A) ⊆ U(μ, t)(A) ∩ U(ν, t)(A). From Theorem 3.23, we get that U(μ, t)∗ U(ν, t)(A) = U(μ, t)(A) ∩ U(ν, t)(A).□

Proposition 3.26

Let μ be a fuzzy ideal of L and t ∈ [0, 1]. Then

  1. [0](μ,t) is an ideal of L;

  2. [0](μ,t) = μt.

Proof

  1. Let x, y ∈ [0](μ,t). Then xy ∈ [0](μ,t) ∨ [0](μ,t) ⊆ [0 ∨ 0](μ,t) = [0](μ,t). Thus, xy ∈ [0](μ,t). Now let xL, a ∈ [0](μ,t) and xa. Then (a, 0) ∈ U(μ, t), i.e., a c = 0 c μ(c) ≥ t. For ac = 0 ∨ c, we have xac. Thus x d = 0 d μ(d) ≥ μ(c) ≥ t, i.e., x ∈ [0](μ,t). Therefore, [0](μ,t) is an ideal of L.

  2. We first show that μt ⊆ [0](μ,t). Let xμt. Then μ(x) ≥ t. Thus a x = a 0 μ(a) ≥ μ(x) ≥ t. It follows from Definition 3.1 that (0, x) ∈ U(μ, t), i.e., x ∈ [0](μ,t). Therefore, [0](μ,t)μt. Now we prove that [0](μ,t)μt. Let y ∈ [0](μ,t). Then (y, 0) ∈ U(μ, t), i.e., a y = a 0 μ(a) ≥ t. For ay = a ∨ 0, we know that ya. Since μ is a fuzzy ideal of L, we have μ(y) ≥ μ(a). Thus μ(y) ≥ a y = a 0 μ(a) ≥ t, i.e., yμt. Therefore, [0](μ,t)μt.□

4 Generalized roughness in distributive lattices with respect to fuzzy ideals

In this section, we investigate generalized roughness in a distributive lattice L with respect to a fuzzy ideal μ and t, where t ∈ [0, 1]. Let J be a distributive lattice and η: L → 𝒫(J) be a set-valued mapping, where 𝒫(J) denotes the family of all non-empty subsets of J. Let μ be a fuzzy ideal of J, t ∈ [0, 1] and X be a non-empty subset of J. We denote η μ t (x) = {b ∈ [a](μ,t)∣a ∈ η(x)} for all xL. Obviously, η μ t is a set-valued mapping from L to 𝒫(J). Further, η(x) ⊆ η μ t (x) for all xL. Thus, η μ t _ (X) = {xL η μ t (x) ⊆ X} and η μ t ¯ ( X ) = { x L | η μ t ( x ) X } are called generalized lower and upper approximations of X with respect to μ and t, respectively. In this section, J is always a distributive lattice and 𝒫(J) denotes the set of all non-empty subsets of J.

Definition 4.1

Let η : L → 𝒫(J) be a mapping. Then

  1. η is called a ∨-homomorphic set-valued mapping if η(x) ∨ η(y) ⊆ η(xy) for all x, yL.

  2. η is called a ∧-homomorphic set-valued mapping if η(x) ∧ η(y) ⊆ η(xy) for all x, yL.

η is called a homomorphic set-valued mapping if it is both a ∨-homomorphic set-valued mapping and a ∧-homomorphic set-valued mapping.

Theorem 4.2

Let μ and ν be fuzzy ideals of J, t ∈ [0, 1] and η: L → 𝒫(J) be a homomorphic set-valued mapping. Then

  1. η μ t is a homomorphic set-valued mapping.

  2. η μ t η ν t is a homomorphic set-valued mapping.

Proof

  1. Let x, yL and z η μ t (x) ∨ η μ t (y). Then there exist x′ ∈ η μ t (x) and y′ ∈ η μ t (y) such that z = x′ ∨ y′. It follows from the definition of η μ t that there exist aη(x), bη(y) such that x′ ∈ [a](μ,t) and y′ ∈ [b](μ,t), i.e.,

    x c = a c μ ( c ) t , y d = b d μ ( d ) t .

    For x′ ∨ c = ac, y′ ∨ d = bd, we have (x′ ∨ y′) ∨ (cd) = (ab) ∨ (cd). Since μ is a fuzzy ideal of J, we get that μ(cd) = μ(c) ∧ μ(d). Thus,

    t ( y d = b d μ ( c ) ) ( y d = b d μ ( d ) ) ( x y ) ( c d ) = ( a b ) ( c d ) ( μ ( c ) μ ( d ) ) = ( x y ) ( c d ) = ( a b ) ( c d ) μ ( c d ) ,

    and so z = x′ ∨ y′ ∈ [ab](μ,t). Since η is a homomorphic set-valued mapping, we have abη(x) ∨ η(y) ⊆ η(xy). Thus z = x′ ∨ y′ ∈ η μ t (xy). Therefore, η μ t (x) ∨ η μ t (y) ⊆ η μ t (xy). In a similar way, we have η μ t (x) ∧ η μ t (y) ⊆ η μ t (xy). Hence, η μ t is a homomorphic set-valued mapping.

  2. Let x, yL and z ( η μ t η ν t ) (x) ∨ ( η μ t η ν t ) (y). Then there exist x′ ∈ ( η μ t η ν t ) (x) and y′ ∈ ( η μ t η ν t ) (y) such that z = x′ ∨ y′, which means that there exist a, bη(x) and c, dη(y) such that x′ ∈ [a](μ,t) ∩ [b](μ,t) and y′ ∈ [c](μ,t) ∩ [d](μ,t). Thus

    x y ( [ a ] ( μ , t ) [ c ] ( μ , t ) ) ( [ b ] ( ν , t ) [ d ] ( ν , t ) ) [ a c ] ( μ , t ) [ b d ] ( ν , t ) .

    Since η is a homomorphic set-valued mapping, we have ac, bdη(x) ∨ η(y) ⊆ η(xy). It follows that z ( η μ t η ν t ) (xy), and so ( η μ t η ν t ) (x) ∨ ( η μ t η ν t ) (y) ⊆ ( η μ t η ν t ) (xy). In a similar way, we have ( η μ t η ν t ) (x) ∧ ( η μ t η ν t ) (y) ⊆ ( η μ t η ν t ) (xy). Therefore, η μ t η ν t is a homomorphic set-valued mapping.□

Theorem 4.3

Let μ be a fuzzy ideal of J, t ∈ [0, 1] and η : L → 𝒫(J) be a homomorphic set-valued mapping. If ∅ ⊊ X, YJ, then

  1. η μ t (X) ∨ η μ t (Y) ⊆ η μ t (XY);

  2. η μ t (X) ∧ η μ t (Y) ⊆ η μ t (XY).

Proof

Let c η μ t (X) ∨ η μ t (Y). Then there exist x η μ t (X) and y η μ t (Y) such that c = xy. Thus there exist x′ ∈ X, y′ ∈ Y and aη(x), bη(y) such that x′ ∈ [a](μ,t), y′ ∈ [b](μ,t). So x′ ∨ y′ ∈ [ab](μ,t) ∩ (AB) and abη(x) ∨ η(y) ⊆ η(xy). Hence, η μ t (xy) ∩ (AB) ≠ ∅, i.e., c η μ t (XY). Therefore, η μ t (X) ∨ η μ t (Y) ⊆ η μ t (XY).

(2) The proof is similar to that of (1).□

Proposition 4.4

Let μ and ν be fuzzy ideals of J, t ∈ [0, 1] and η: L → 𝒫(J) be a homomorphic set-valued mapping. If ∅ ⊊ XJ and μν, then

  1. η ν t _ ( X ) η μ t _ ( X ) .

  2. η μ t ¯ ( X ) η ν t ¯ ( X ) .

Proof

It is straightforward.□

According to Proposition 4.4, we can get the following result easily.

Corollary 4.5

Let μ and ν be fuzzy ideals of J, t ∈ [0, 1] and η : L → 𝒫(J) be a homomorphic set-valued mapping. If ∅ ⊊ XJ, then

  1. η μ t _ ( X ) η ν t _ ( X ) η μ ν t _ ( X ) .

  2. η μ ν t ¯ ( X ) η μ t ¯ ( X ) η ν t ¯ ( X ) .

Lemma 4.6

Let μ and ν be fuzzy ideals of J, t ∈ [0, 1] and η: L → 𝒫(J) be a homomorphic set-valued mapping. Then

η μ ν t ( x ) η μ t ( x ) η ν t ( x )

for all xL.

Proof

Let xL and a η μ ν t ( x ) . Then there exists bη(x) such that a ∈ [b](μν,t), i.e., a c = b c (μν)(c) ≥ t. On the other hand,

t a c = b c ( μ ν ) ( c ) = a c = b c ( μ ( c ) ν ( c ) ) = ( a c = b c μ ( c ) ) ( a c = b c ν ( c ) ) ,

that is,

a c = b c μ ( c ) t a n d a c = b c ν ( c ) t ,

which means that a ∈ [b](μ,t) and a ∈ [b](ν,t). And so, a η μ t ( x ) η ν t ( x ) . Therefore, η μ ν t ( x ) η μ t ( x ) η ν t ( x ) .

From Lemma 4.6, we get the following result.

Theorem 4.7

Let μ and ν be fuzzy ideals of J, t ∈ [0, 1] and η: L → 𝒫(J) be a homomorphic set-valued mapping. If ∅ ⊊ XJ, then

  1. η μ ν t _ ( X ) η μ t η μ t _ ( X ) .

  2. η μ ν t ¯ ( X ) η μ t η μ t ¯ ( X ) .

Lemma 4.8

Let μ be a fuzzy ideal of J, t ∈ [0, 1] and η : L → 𝒫(J) be a homomorphic set-valued mapping. Let xL. Then the following statements are equivalent:

  1. η(x) ⊆ μt;

  2. η μ t (x) = μt.

Proof

(1) ⇒ (2) Let a η μ t (x). Then there exists bη(x) ⊆ μt such that a ∈ [b](μ,t), that is, a c = b c μ(c) ≥ t, which means that there exists cJ such that μ(c) ≥ t satisfying ac = bc. Thus cμt and ac = bcμt. Since aac, we have aμt. Therefore, η μ t (x) ⊆ μt. Next we show that μt η μ t (x). Let fμt. Since η(x) ≠ ∅, we have there exists dη(x) ⊆ μt, i.e., μ(d) ≥ t. On the other hand, since μ be a fuzzy ideal of J, we have μ(fd) = μ(f) ∧ μ(d) ≥ t. Thus f e = d e μ(e) ≥ μ(fd) ≥ t. So f ∈ [d](μ,t). Hence, f η μ t (x), i.e., μt η μ t (x). Therefore, η μ t (x) = μt.

(2) ⇒ (1) Let gη(x). Since g ∈ [g](μ,t), we have g η μ t (g) ⊆ μt. Therefore, η(x) ⊆ μt.□

Theorem 4.9

Let μ and ν be fuzzy ideals of J, t ∈ [0, 1] and η: L → 𝒫(J) be a homomorphic set-valued mapping. If μtXJ and η(x) ⊆ μt for all xL, then η μ t _ ( x ) = η μ t ¯ ( x ) = L .

Proof

According to Lemma 4.8, we get the conclusion easily.□

Theorem 4.10

Let μ and ν be fuzzy ideals of J, t ∈ [0, 1], μν and η: L → 𝒫(J) be a homomorphic set-valued mapping. If xη(x) for all xL, then the following are equivalent:

  1. η(x) ⊆ νt for all xνt;

  2. η μ t _ (νt) = νt.

Proof

(1) ⇒ (2) Let x η μ t _ (νt). Then η μ t (x) ⊆ νt. Since xη(x) ⊆ η μ t (x), we have xνt. Now let a′ ∈ νt. Then for any y η μ t (x), there exists a′ ∈ η(x) such that y ∈ [a′](μ,t), i.e., y c = a c (μ)(c) ≥ t, which means that there exists cJ such that μ(c) ≥ t satisfying yc = a′ ∨ c. Thus cμt. Since μν, we have μtνt. On the other hand, since η(x) ⊆ νt, we have acνt. So yνt. Thus, η μ t _ (νt) ⊆ νt. Therefore, η μ t _ (νt) = νt.

(2) ⇒ (1) Let xνt and yη(x). Since η(x) ⊆ η μ t (x), we have y η μ t (x). On the other hand, η μ t _ (νt) = νt, we have η μ t (x) ⊆ νt. Thus yνt. Therefore, η(x) ⊆ νt for all xνt.□

5 Conclusion

The study of rough sets in the distributive lattice theory is an interesting topic of rough set theory. In this paper, we introduce the special class of rough sets and generalized rough sets with respect to a fuzzy ideal in a distributive lattice, that is the universe of objects is endowed with a distributive lattice and a congruence relation is defined with respect to a fuzzy ideal. The main conclusions in this paper and the further work to do are listed as follows.

  1. A novel congruence relation U(μ, t) induced by a fuzzy ideal μ of a distributive lattice is introduced.

  2. Roughness in distributive lattices with respect to fuzzy ideals are investigated,

  3. Generalized roughness in distributive lattices with respect to fuzzy ideals are investigated.

Acknowledgements

The authors are very grateful to the editor and the anonymous reviewers for their constructive comments and suggestions that have led to an improved version of this paper. The work was supported partially by National Natural Science Foundation of China (No. 11971384), Higher Education Key Scientific Research Program Funded by Henan Province (No. 20A110011, 20B630002) and Research and Cultivation Fund Project of Anyang Normal University (No. AYNUKP-2018-B26).

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Received: 2019-05-15
Accepted: 2020-01-18
Published Online: 2020-03-10

© 2020 Yongwei Yang et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.