# Rough sets based on fuzzy ideals in distributive lattices

Yongwei Yang
• School of Mathematics and Statistics, Anyang Normal University, Anyang, 455000, China
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, Kuanyun Zhu
• Corresponding author
• School of Information and Mathematics, Yangtze University, Jingzhou, 434023, China
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and Xiaolong Xin

## 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.

## 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  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  investigated the topological characterizations of generalized fuzzy rough sets. Zhu and Hu  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 . 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  and Yamak et al.  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 . In 2014, Xiao et al.  studied rough set model on ideals in lattices. In , 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

 LetLbe a lattice and ∅ ≠ IL. ThenIis called an ideal ofLif it satisfies the following conditions: for anyx, yL,

1. xIandyIimplyxyI;
2. xLandxyimplyxI.

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 .

Definition 2.2

 LetLbe a lattice. A relationRis called an equivalence relation onLif for alla, b, cL,

1. Reflexive: (a, a) ∈ R;
2. Symmetry: (a, b) ∈ Rimplies (b, a) ∈ R;
3. Transitivity: (a, b) ∈ R, (b, c) ∈ Rimplies (a, c) ∈ R.

An equivalence relationRis called a congruence relation onL, if for alla, b, c, dL, (a, b) ∈ R, (c, d) ∈ R, then (ac, bd ) ∈ Rand (ac, bd) ∈ R.

Definition 2.3

 Letμbe a fuzzy set of a latticeL. Thenμis called a fuzzy sublattice ofLifμ(xy) ∧ μ(xy) ≥ μ(x) ∧ μ(y), for allx, yL.

Letμbe a fuzzy sublattice ofL. Thenμis a fuzzy ideal ofL, ifμ(xy) = μ(x) ∧ μ(y) for allx, yL.

Proposition 2.4

 Letμbe a fuzzy sublattice of a latticeL. Thenμis a fuzzy ideal ofLif and only ifxyimplies thatμ(x) ≥ μ(y), for allx, yL.

Proposition 2.5

 Letμbe a fuzzy set of a latticeL. Thenμis a fuzzy ideal ofLif and only if any one of the following sets of conditions is satisfied: for allx, yL,

1. μ(0) = 1 andμ(xy) = μ(x) ∧ μ(y);
2. μ(0) = 1, μ(xy) ≥ μ(x) ∧ μ(y) andμ(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 setμis a fuzzy ideal of a latticeLif and only if every subsetμtis an ideal ofLfor allt ∈ [0, 1].

Definition 2.7

 LetRbe an equivalence relation on the universeUand (U, R) be a Pawlak approximation space. A subsetXUis called definable ifR*X = R*X; otherwise, Xis 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

 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

Letμbe a fuzzy ideal ofL. For eacht ∈ [0, 1], the set

$U(μ,t)=(x,y)∈L×L|⋁{μ(a)|a∨x=a∨y,∃a∈L}≥t$

is called at-level relation ofμ.

Example 3.2

LetL = {0, a, b, c, 1}. We define the binary relationin the following Hasse diagram. It is easy to check thatLis a distributive lattice. Let$μ=10+0.8a+0.6b+0.4c+01.$Then it is clear thatμis a fuzzy ideal ofL. Chooset = 0.9, then we haveU(μ, 0.9) = {(0, 0), (a, a), (b, b), (c, c), (1, 1)}. ThusU(μ, 0.9) is called a 0.9-level relation ofμ.

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

Lemma 3.3

Letμbe a fuzzy ideal ofLandt ∈ [0, 1]. ThenU(μ, t) is a congruence relation onL.

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

$⋁a∨x=a∨yμ(a)≥t,⋁b∨y=b∨zμ(b)≥t,$

and so $(⋁a∨x=a∨yμ(a))∧(⋁b∨y=b∨zμ(b))≥t.$ Since μ is a fuzzy ideal of L, we obtain that

$(⋁a∨x=a∨yμ(a))∧(⋁b∨y=b∨zμ(b))=⋁a∨x=a∨y,b∨y=b∨z(μ(a)∧μ(b))=⋁a∨x=a∨y,b∨y=b∨zμ(a∨b).$

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

$t≤⋁a∨x=a∨y,b∨y=b∨zμ(a∨b)≤⋁c∨x=c∨zμ(c),$

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

$⋁a∨x=a∨yμ(a)≥t,⋁b∨u=b∨vμ(b)≥t,$

and so

$(⋁a∨x=a∨yμ(a))∧(⋁b∨y=b∨zμ(b))≥t.$

Further, we have

$(⋁a∨x=a∨yμ(a))∧(⋁b∨u=b∨vμ(b))=⋁a∨x=a∨y,b∨u=b∨v(μ(a)∧μ(b))=⋁a∨x=a∨y,u∨y=b∨vμ(a∨b).$

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

$t≤⋁a∨x=a∨y,u∨y=b∨vμ(a∨b)≤⋁c∨(x∨u)=c∨(y∨v)μ(c).$

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

$⋁b∨x1=b∨y1μ(b)≥t,⋁c∨x2=c∨y2μ(c)≥t.$

So

$(⋁b∨x1=b∨y1μ(b))∧(⋁c∨x2=c∨y2μ(c))≥t.$

For bx1 = by1 and cx2 = cy2, we have

$(b∨x1)∧(c∨x2)=(b∨y1)∧(c∨y2).$

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

$[(b∧c)∨(x1∧c)∨(x2∧b)]∨(x1∧x2)=[(b∧c)∨(y1∧c)∨(y2∧b)]∨(y1∧y2).$

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

$(b∧c)∨(x1∧c)∨(x2∧b)=(b∧c)∨(y1∧c)∨(y2∧b).$

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

$μ[(b∧c)∨(x1∧c)∨(x2∧b)]=μ(b∧c)∧μ(x1∧c)∧μ(x2∧b).$

It follows from bcb, x1cc, x2bb that

$μ(b∧c)∧μ(x1∧c)∧μ(x2∧b)≥μ(b)∧μ(c).$

Thus

$t≤(⋁b∨x1=b∨y1μ(b))∧(⋁c∨x2=c∨y2μ(c))=⋁b∨x1=b∨y1,c∨x2=c∨y2(μ(b)∧μ(c))≤⋁b∨x1=b∨y1,c∨x2=c∨y2(μ(b∧c)∧μ(x1∧c)∧μ(x2∧b))≤⋁[(b∧c)∨(x1∧c)∨(x2∧b)]∨(x1∧x2)=[(b∧c)∨(y1∧c)∨(y2∧b)]∨(y1∧y2)(μ(b∧c)∧μ(x1∧c)∧μ(x2∧b))≤⋁a∨(x1∧x2)=a∨(y1∧y2)μ(a),$

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 sayxis congruent toymodμ, written xty(modμ) if

$⋁a∨x=a∨yμ(a)≥t.$

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

Letμbe a fuzzy ideal ofLandt ∈ [0, 1]. Then for allx, 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

Letμbe a fuzzy ideal ofL, 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 setXwith respect toμandt, respectively. It is easy to know thatU(μ, t)(X) ⊆ XU(μ, t)(X).

Lemma 3.7

Letμandνbe two fuzzy ideals ofLsuch thatμνandt ∈ [0, 1]. Then [x](μ,t) ⊆ [x](ν,t)for allxL.

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

Letμandνbe two fuzzy ideals ofLsuch thatμν, 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 latticeLin Example 3.2, let$μ=10+0.6a+0.8b+0.4c+01,ν=10+0.8a+0.5b+0.3c+01.$Then it is clear thatμandνare fuzzy ideals ofL. Chooset = 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

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

IfX = {0, c}, then

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

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

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

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

The following definition is from Zadeh’s expansion principle.

Definition 3.10

Letμandνbe two fuzzy sets overL. DefineμνoverLas follows:

$(μ∨ν)(x)=⋁x=a∨b(μ(a)∧ν(b))$

for allxL.

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

Letμandνbe two fuzzy ideals ofL, 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

Letμandνbe two fuzzy ideals ofL, t ∈ [0, 1] and ∅ ⊊ XL. Then

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

Letμandνbe two fuzzy ideals ofL, 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,
$⋁a∨y=a∨xμ(a)≥tand⋁a∨y=a∨xν(a)≥t.$
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.,
$⋁a∨x′=a∨xμ(a)≥tand⋁b∨x′=b∨xν(b)≥t.$
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
$x′∨(a∧b)=x∨(a∧b)andμ(a∧b)≥μ(a),ν(a∧b)≥ν(b),$
i.e.,
$t≤(⋁a∨x′=a∨xμ(a))∧(⋁b∨x′=b∨xν(b))≤⋁x′∨(a∧b)=x∨(a∧b)(μ(a)∧ν(b))≤⋁x′∨(a∧b)=x∨(a∧b)(μ(a∧b)∧ν(a∧b))=⋁x′∨(a∧b)=x∨(a∧b)(μ∩ν)(a∧b).$
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,
$⋁a∨x=a∨x′μ(a)≥tand⋁a∨x=a∨x′ν(a)≥t.$
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

Letμbe a fuzzy ideal ofLandt ∈ [0, 1]. Then

$U(μ,t)_(μt)=μt=U(μ,t)¯(μt).$

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

Letμandνbe two fuzzy ideals ofLandt ∈ [0, 1]. Thenμt = 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

Letμandνbe two fuzzy ideals ofLandt ∈ [0, 1]. ThenνtU(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

Letμbe a fuzzy ideal ofL, t ∈ [0, 1]. Then

1. U(μ, t)(↓ a) = μtfor eachaμt;
2. $⋃a∈μt$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, $⋃a∈μt$U(μ, t)(↓ a) ⊆ μt.□

Theorem 3.18

Letμandνbe two fuzzy ideals ofLandt ∈ [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.,

$⋁b∨a=b∨xμ(b)≥t.$

Since μν, we have

$⋁b∨a=b∨xν(b)≥⋁b∨a=b∨xμ(b)≥t.$

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

Letμ, νandωbe fuzzy ideals ofLsuch thatμωandt ∈ [0, 1]. Then

$U(μ,t)¯(U(ν,t)¯(ωt))=U(ν,t)¯(ωt)=U(ν,t)¯(U(μ,t)¯(ωt)).$

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

Letμ, νandωbe fuzzy ideals ofLsuch thatμωandt ∈ [0, 1]. Then

$U(μ,t)∩U(ν,t)¯(ωt)=U(μ,t)¯(ωt)∩U(ν,t)¯(ωt).$

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

Letμandνbe two fuzzy ideals ofLsuch thatμνandt ∈ [0, 1]. If ∅ ⊊ AL, then

$U(μ,t)¯(νt∩A)=U(μ,t)¯(νt)∩U(μ,t)¯(A).$

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

Letμbe a fuzzy ideal ofLandt ∈ [0, 1]. IfA, Bare ideals ofLandμtAB, 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.,
$⋁y′∨a=y′∨yμ(y′)≥t,⋁z′∨b=z′∨zμ(z′)≥t.$
For y′ ∨ a = y′ ∨ y, z′ ∨ b = z′ ∨ z, we have (y′ ∨ z′) ∨ (ab) = (y′ ∨ z′) ∨ (yz) = (y′ ∨ z′) ∨ x. Thus
$t≤(⋁y′∨a=y′∨yμ(y′))∧(⋁z′∨b=z′∨zμ(z′))≤⋁(y′∨z′)∨(a∨b)=(y′∨z′)∨(y∨z)(μ(y′)∧μ(z′))≤⋁(y′∨z′)∨(a∨b)=(y′∨z′)∨xμ(y′∨z′).$
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:
$U(μ,t)∗U(ν,t)={(x,y)∈L×L|∃z∈Lsuchthat(x,z)∈U(μ,t)and(z,y)∈U(ν,t)}$
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

Letμandνbe two fuzzy ideals ofL, t ∈ [0, 1] andU(μ, t)∗ U(ν, t) = U(ν, t)∗ U(μ, t).

1. IfAis a non-empty subset ofL, thenU(μ, t)∗ U(ν, t)(A) ⊆ U(μ, t)(A) ∩ U(μ, t)(A).
2. IfAis a sublattice ofL, thenU(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.,
$⋁a∨y=a∨xμ(a)≥t,⋁b∨z=b∨xν(a)≥t.$
For ay = ax, bz = bx, we have (zy) ∨ a = (zx) ∨ a, (zx) ∨ b = xb. Hence
$⋁(z∨y)∨a=(z∨x)∨aμ(a)≥⋁a∨y=a∨xμ(a)≥t,$
and
$⋁(z∨x)∨b=x∨bν(a)≥⋁z∨b=x∨bν(b)≥t.$
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$μ=10+0.8a+0.6b+0.4c+01andν=10+0.7a+0.8b+0.3c+01.$Then it is clear thatμandνare fuzzy ideals ofL. Chooset = 0.8, thenμt = {0, a} andνt = {0, b}. Now letA = {a, b}, B = {0, b}. Then we haveμtABandAB = {a, b, c}. Thus

$U(μ,t)_(A)∨U(μ,t)_(B)=∅andU(μ,t)_(A∨B)={b,c}.$

Therefore,

$U(μ,t)_(A)∨U(μ,t)_(B)⊆U(μ,t)_(A∨B).$

LetA = {a, b, c}. ThenU(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).

LetA = {a, c} be a sublattice ofL. ThenU(μ, t)(A) = {0, a, b, c}, U(ν, t)(A) = {a, c}, and

$U(μ,t)¯(A)∩U(μ,t)¯(A)={b,c}andU(μ,t)∗U(ν,t)¯(A)={0,a,b,c}.$

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

Theorem 3.25

Letμandνbe two fuzzy ideals ofL, t ∈ [0, 1], U(μ, t)∗ U(ν, t) = U(ν, t)∗ U(μ, t) andAbe an ideal ofL.

1. IfμtA, thenU(μ, t)∗ U(ν, t)(A) = U(μ, t)(A) ∩ U(ν, t)(A).
2. Ifμt, νtA, thenU(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

Letμbe a fuzzy ideal ofLandt ∈ [0, 1]. Then

1. (μ,t)is an ideal ofL;
2. (μ,t) = μt.

Proof

1. Let x, y ∈ (μ,t). Then xy ∈ (μ,t) ∨ (μ,t) ⊆ [0 ∨ 0](μ,t) = (μ,t). Thus, xy ∈ (μ,t). Now let xL, a ∈ (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 ∈ (μ,t). Therefore, (μ,t) is an ideal of L.
2. We first show that μt ⊆ (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 ∈ (μ,t). Therefore, (μ,t)μt. Now we prove that (μ,t)μt. Let y ∈ (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, (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 $ημt$(x) = {b ∈ [a](μ,t)∣a ∈ η(x)} for all xL. Obviously, $ημt$ is a set-valued mapping from L to 𝒫(J). Further, η(x) ⊆ $ημt$(x) for all xL. Thus, $ημt_$(X) = {xL$ημt$(x) ⊆ X} and $ημt¯(X)={x∈L|ημt(x)∩X≠∅}$ 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

Letη : L → 𝒫(J) be a mapping. Then

1. ηis calleda ∨-homomorphic set-valued mapping ifη(x) ∨ η(y) ⊆ η(xy) for allx, yL.
2. ηis calleda ∧-homomorphic set-valued mapping ifη(x) ∧ η(y) ⊆ η(xy) for allx, 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

Letμandνbe fuzzy ideals ofJ, t ∈ [0, 1] andη: L → 𝒫(J) be a homomorphic set-valued mapping. Then

1. $ημt$is a homomorphic set-valued mapping.
2. $ημt∩ηνt$ is a homomorphic set-valued mapping.

Proof

1. Let x, yL and z$ημt$(x) ∨ $ημt$(y). Then there exist x′ ∈ $ημt$(x) and y′ ∈ $ημt$(y) such that z = x′ ∨ y′. It follows from the definition of $ημt$ that there exist aη(x), bη(y) such that x′ ∈ [a](μ,t) and y′ ∈ [b](μ,t), i.e.,
$⋁x′∨c=a∨cμ(c)≥t,⋁y′∨d=b∨dμ(d)≥t.$
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,
$t≤(⋁y′∨d=b∨dμ(c))∧(⋁y′∨d=b∨dμ(d))≤⋁(x′∨y′)∨(c∨d)=(a∨b)∨(c∨d)(μ(c)∧μ(d))=⋁(x′∨y′)∨(c∨d)=(a∨b)∨(c∨d)μ(c∨d),$
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′ ∈ $ημt$(xy). Therefore, $ημt$(x) ∨ $ημt$(y) ⊆ $ημt$(xy). In a similar way, we have $ημt$(x) ∧ $ημt$(y) ⊆ $ημt$(xy). Hence, $ημt$ is a homomorphic set-valued mapping.
2. Let x, yL and z$(ημt∩ηνt)$(x) ∨ $(ημt∩ηνt)$(y). Then there exist x′ ∈ $(ημt∩ηνt)$(x) and y′ ∈ $(ημt∩ηνt)$(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$(ημt∩ηνt)$(xy), and so $(ημt∩ηνt)$(x) ∨ $(ημt∩ηνt)$(y) ⊆ $(ημt∩ηνt)$(xy). In a similar way, we have $(ημt∩ηνt)$(x) ∧ $(ημt∩ηνt)$(y) ⊆ $(ημt∩ηνt)$(xy). Therefore, $ημt∩ηνt$ is a homomorphic set-valued mapping.□

Theorem 4.3

Letμbe a fuzzy ideal ofJ, t ∈ [0, 1] andη : L → 𝒫(J) be a homomorphic set-valued mapping. If ∅ ⊊ X, YJ, then

1. $ημt$(X) ∨ $ημt$(Y) ⊆ $ημt$(XY);
2. $ημt$(X) ∧ $ημt$(Y) ⊆ $ημt$(XY).

Proof

Let c$ημt$(X) ∨ $ημt$(Y). Then there exist x$ημt$(X) and y$ημt$(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, $ημt$(xy) ∩ (AB) ≠ ∅, i.e., c$ημt$(XY). Therefore, $ημt$(X) ∨ $ημt$(Y) ⊆ $ημt$(XY).

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

Proposition 4.4

Letμandνbe fuzzy ideals ofJ, t ∈ [0, 1] andη: L → 𝒫(J) be a homomorphic set-valued mapping. If ∅ ⊊ XJandμν, then

1. $ηνt_(X)⊆ημt_(X).$
2. $ημt¯(X)⊆ηνt¯(X).$

Proof

It is straightforward.□

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

Corollary 4.5

Letμandνbe fuzzy ideals ofJ, t ∈ [0, 1] andη : L → 𝒫(J) be a homomorphic set-valued mapping. If ∅ ⊊ XJ, then

1. $ημt_(X)∪ηνt_(X)⊆ημ∩νt_(X).$
2. $ημ∩νt¯(X)⊆ημt¯(X)∩ηνt¯(X).$

Lemma 4.6

Letμandνbe fuzzy ideals ofJ, t ∈ [0, 1] andη: L → 𝒫(J) be a homomorphic set-valued mapping. Then

$ημ∩νt(x)⊆ημt(x)∩ηνt(x)$

for allxL.

Proof

Let xL and a$ημ∩νt(x).$ Then there exists bη(x) such that a ∈ [b](μν,t), i.e., $⋁a∨c=b∨c$ (μν)(c) ≥ t. On the other hand,

$t≤⋁a∨c=b∨c(μ∩ν)(c)=⋁a∨c=b∨c(μ(c)∧ν(c))=(⋁a∨c=b∨cμ(c))∧(⋁a∨c=b∨cν(c)),$

that is,

$⋁a∨c=b∨cμ(c)≥tand⋁a∨c=b∨cν(c)≥t,$

which means that a ∈ [b](μ,t) and a ∈ [b](ν,t). And so, $a∈ημt(x)∩ηνt(x).$ Therefore, $ημ∩νt(x)⊆ημt(x)∩ηνt(x).$

From Lemma 4.6, we get the following result.

Theorem 4.7

Letμandνbe fuzzy ideals ofJ, t ∈ [0, 1] andη: L → 𝒫(J) be a homomorphic set-valued mapping. If ∅ ⊊ XJ, then

1. $ημ∩νt_(X)⊇ημt∩ημt_(X).$
2. $ημ∩νt¯(X)⊆ημt∩ημt¯(X).$

Lemma 4.8

Letμbe a fuzzy ideal ofJ, t ∈ [0, 1] andη : L → 𝒫(J) be a homomorphic set-valued mapping. LetxL. Then the following statements are equivalent:

1. η(x) ⊆ μt;
2. $ημt$(x) = μt.

Proof

(1) ⇒ (2) Let a$ημt$(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, $ημt$ (x) ⊆ μt. Next we show that μt$ημt$ (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$ημt$(x), i.e., μt$ημt$(x). Therefore, $ημt$(x) = μt.

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

Theorem 4.9

Letμandνbe fuzzy ideals ofJ, t ∈ [0, 1] andη: L → 𝒫(J) be a homomorphic set-valued mapping. IfμtXJandη(x) ⊆ μtfor allxL, then$ημt_(x)=ημt¯(x)=L.$

Proof

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

Theorem 4.10

Letμandνbe fuzzy ideals ofJ, t ∈ [0, 1], μνandη: L → 𝒫(J) be a homomorphic set-valued mapping. Ifxη(x) for allxL, then the following are equivalent:

1. η(x) ⊆ νtfor allxνt;
2. $ημt_$(νt) = νt.

Proof

(1) ⇒ (2) Let x$ημt_$(νt). Then $ημt$(x) ⊆ νt. Since xη(x) ⊆ $ημt$(x), we have xνt. Now let a′ ∈ νt. Then for any y$ημt$(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, $ημt_$(νt) ⊆ νt. Therefore, $ημt_$(νt) = νt.

(2) ⇒ (1) Let xνt and yη(x). Since η(x) ⊆ $ημt$(x), we have y$ημt$(x). On the other hand, $ημt_$(νt) = νt, we have $ημt$(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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W. Yao, Y.H. She, and L.X. Lu, Metric-based L-fuzzy rough sets: Approximation operators and definable sets, Knowl.-Based Syst. 163 (2019), 91–102, .

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J. Dai, Y. Yan, Z. Li, et al., Dominance-based fuzzy rough set approach for incomplete interval-valued data, J. Intell. Fuzzy Syst., 34 (2018), no. 1, 423–436, .

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J. Zhan and Q. Wang, Certain types of soft coverings based rough sets with applications, Int. J. Mach. Learn. Cyb. 10 (2019), no. 5, 1065–1076, .

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B. Yang, B.Q. Hu, and J. Qiao, Three-way decisions with rough membership functions in covering approximation space, Fund. Inform. 165 (2019), no. 2, 157–191, .

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C.Y. Wang, Topological characterizations of generalized fuzzy rough sets, Fuzzy Sets and Systems 312 (2017), 109–125, .

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K.Y. Zhu and B.Q. Hu, A novel Z-soft rough fuzzy BCI-algebras (ideals) of BCI-algebras, Soft Comput. 22 (2018), no. 22, 3649–3662, .

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S. Shao, X. Zhang, C. Bo, et al., Multi-granulation rough filters and rough fuzzy filters in pseudo-BCI algebras, J. Intell. Fuzzy Syst. 34 (2018), no. 6, 4377–4386, .

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S. Mirvakili, S.M. Anvariyeh, and B. Davvaz, Generalization of Pawlak’s approximations in hypermodules by set-valued homomorphisms, Found. Comput. Math. 42 (2017), no. 1, 59–81, .

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V. Leoreanu-Fotea, The lower and upper approximations in a hypergroups, Inform. Sci. 178 (2008), no. 18, 3605–3615, .

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Ş. Yılmaz and O. Kazancı, Approximations in a hyperlattice by using set-valued homomorphisms, Hacet. J. Math. Stat. 45 (2016), no. 6, 1755–1766, .

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B. Yang and B.Q. Hu, A fuzzy covering-based rough set model and its generalization over fuzzy lattice, Inform. Sci. 367 (2016), 463–486, .

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L. Ma, Two fuzzy covering rough set models and their generalizations over fuzzy lattices, Fuzzy Sets and Systems 294 (2016), 1–17, .

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B. Davvaz, A short note on algebraic T-rough sets, Inform. Sci. 178 (2008), no. 16, 3247–3252, .

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S. Yamak, O. Kazani, and B. Davvaz, Generalized lower and upper approximations in a rings, Inform. Sci. 180 (2010), no. 9, 1759–1768, .

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B. Davvaz, Roughness based on fuzzy ideals, Inform. Sci. 176 (2006), no. 16, 2417–2437, .

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Q.M. Xiao, Q.G. Li, and L.K. Guo, Rough sets induced bu ideals in lattices, Inform. Sci. 271 (2014), 82–92, .

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B.A. Davey and H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge, 2002.

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A.A. Estaji, S. Khodaii, and S. Bahrami, On rough set and fuzzy sublattice, Inform. Sci. 181 (2011), no. 18, 3981–3994, .

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• 

U.M. Swamy and D.V. Raju, Fuzzy ideals and congruences of lattices, Fuzzy Sets and Systems 95 (1998), no. 2, 249–253, .

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• Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

• 

Z. Pawlak, Rough sets, Int. J. Comput. Math. Inf. Sci. 11 (1982), no. 5, 341–356, .

• Crossref
• Export Citation
• 

W. Yao, Y.H. She, and L.X. Lu, Metric-based L-fuzzy rough sets: Approximation operators and definable sets, Knowl.-Based Syst. 163 (2019), 91–102, .

• Crossref
• Export Citation
• 

X. Zhang, D. Miao, C. Liu, and M. Le, Constructive methods of rough approximation operators and multigranuation rough sets, Knowl.-Based Syst. 91 (2016), 114–125, .

• Crossref
• Export Citation
• 

J. Dai, Y. Yan, Z. Li, et al., Dominance-based fuzzy rough set approach for incomplete interval-valued data, J. Intell. Fuzzy Syst., 34 (2018), no. 1, 423–436, .

• Crossref
• Export Citation
• 

J. Zhan and Q. Wang, Certain types of soft coverings based rough sets with applications, Int. J. Mach. Learn. Cyb. 10 (2019), no. 5, 1065–1076, .

• Crossref
• Export Citation
• 

B. Yang, B.Q. Hu, and J. Qiao, Three-way decisions with rough membership functions in covering approximation space, Fund. Inform. 165 (2019), no. 2, 157–191, .

• Crossref
• Export Citation
• 

C.Y. Wang, Topological characterizations of generalized fuzzy rough sets, Fuzzy Sets and Systems 312 (2017), 109–125, .

• Crossref
• Export Citation
• 

K.Y. Zhu and B.Q. Hu, A novel Z-soft rough fuzzy BCI-algebras (ideals) of BCI-algebras, Soft Comput. 22 (2018), no. 22, 3649–3662, .

• Crossref
• Export Citation
• 

S. Shao, X. Zhang, C. Bo, et al., Multi-granulation rough filters and rough fuzzy filters in pseudo-BCI algebras, J. Intell. Fuzzy Syst. 34 (2018), no. 6, 4377–4386, .

• Crossref
• Export Citation
• 

S. Mirvakili, S.M. Anvariyeh, and B. Davvaz, Generalization of Pawlak’s approximations in hypermodules by set-valued homomorphisms, Found. Comput. Math. 42 (2017), no. 1, 59–81, .

• Crossref
• Export Citation
• 

V. Leoreanu-Fotea, The lower and upper approximations in a hypergroups, Inform. Sci. 178 (2008), no. 18, 3605–3615, .

• Crossref
• Export Citation
• 

Ş. Yılmaz and O. Kazancı, Approximations in a hyperlattice by using set-valued homomorphisms, Hacet. J. Math. Stat. 45 (2016), no. 6, 1755–1766, .

• Crossref
• Export Citation
• 

B. Yang and B.Q. Hu, A fuzzy covering-based rough set model and its generalization over fuzzy lattice, Inform. Sci. 367 (2016), 463–486, .

• Crossref
• Export Citation
• 

L. Ma, Two fuzzy covering rough set models and their generalizations over fuzzy lattices, Fuzzy Sets and Systems 294 (2016), 1–17, .

• Crossref
• Export Citation
• 

B. Davvaz, A short note on algebraic T-rough sets, Inform. Sci. 178 (2008), no. 16, 3247–3252, .

• Crossref
• Export Citation
• 

S. Yamak, O. Kazani, and B. Davvaz, Generalized lower and upper approximations in a rings, Inform. Sci. 180 (2010), no. 9, 1759–1768, .

• Crossref
• Export Citation
• 

B. Davvaz, Roughness based on fuzzy ideals, Inform. Sci. 176 (2006), no. 16, 2417–2437, .

• Crossref
• Export Citation
• 

Q.M. Xiao, Q.G. Li, and L.K. Guo, Rough sets induced bu ideals in lattices, Inform. Sci. 271 (2014), 82–92, .

• Crossref
• Export Citation
• 

B.A. Davey and H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge, 2002.

• 

A.A. Estaji, S. Khodaii, and S. Bahrami, On rough set and fuzzy sublattice, Inform. Sci. 181 (2011), no. 18, 3981–3994, .

• Crossref
• Export Citation
• 

U.M. Swamy and D.V. Raju, Fuzzy ideals and congruences of lattices, Fuzzy Sets and Systems 95 (1998), no. 2, 249–253, .

• Crossref
• Export Citation
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