A new study on soft rough fuzzy lattices (ideals,filters) over lattices

Abstract

In this paper, we investigate the relationship among soft sets, rough sets, fuzzy sets and lattices. The notion of soft rough fuzzy lattices (ideals, filters) over lattices is introduced, which is an extended notion of soft rough lattices (ideals, filters) and rough fuzzy lattices (ideals, filters) over lattices. Moreover, we study roughness in lattices with respects to a soft approximation space. Some new soft rough fuzzy operations over lattices are explored. In particular, lower and upper soft rough fuzzy lattices (ideals, filters) over lattices with respect to another fuzzy soft set are investigated.

Keywords

Soft rough set fuzzy sublattice (ideal,filter)soft rough fuzzy lattice (ideal,filter)

1 Introduction

Rough set theory was first introduced by Pawlak [20], a mathematical tool for dealing with uncertainty. It follows from the definition of rough sets that any subset of a universe can be characterized by equivalence relations. As far as known, an equivalence relation on a set partitions the set into disjoint classes and vice versa. We know that a subset can be written as union of these classes, which is called definable, otherwise it is not definable. In this case, it can be approximated by two definable subsets called lower and upper approximations of the set. However, these equivalence relations in Pawlak rough sets are restrictive to some areas of applications. Thus, some more general models have been proposed, such as [27 –29]. Nowadays, rough set theory has been applied to many areas, such as knowledge discovery, machine learning, data analysis, approximate classification, conflict analysis, and so on, see [4 , 24]. On the other hand, some researchers applied this theory to algebraic structures, such as Kuroki [14] proposed the concept of rough ideals in a semigroup, Davvaz [6, 7] applied this theory to rings. Further, Yamak and Kazançi et al. [23] discussed generalized lower and upper approximations in a ring. Moreover, some authors also apply rough set theory to other algebraic structures, see [2 , 17].

It is worth noting that the mathematical modelling and manipulating of various types of uncertainties has become an increasingly important issue in solving complicated problems arisings in a wide range of areas such as economy, engineering, environmental science, medicine and social science. We known that fuzzy set theory [25], rough set theory [20] are all effective tools for dealing with vagueness and uncertainty. However, each of them has certain inherent limitations. Based on this reason, Molodtsov [19] proposed soft set theory, as a new mathematical tool for dealing with uncertainties, which is free from the difficulty affecting the above mentioned methods. Since then there has been a rapid growth of interest in soft sets and their various applications. In 2011, Ali [1] studied another view on reduction of parameters in soft sets. Afterwards, a wide range of applications of soft sets have been studied in many different fields including game theory, probability theory, smoothness of functions, operation researches, Riemann integrations and measurement theory and so on. Recently, there has been a rapid growth of interest in soft set theory and its applications, such as [3, 22]. In particular, Jun and Park [13] applied this theory to algebraic structures.

Recently, Feng and Li et al. [11, 12] provided a framework to combine rough sets with soft sets, which gives rise to some interesting new concepts such as rough soft sets, soft rough sets and soft rough fuzzy sets. In particular, in 2013, Shabir and Ali et al. [21] pointed out that there exist some problems on Feng’s soft rough set as follows: (1) An upper approximation of a non-empty set may be empty. (2) The upper approximation of a subset X may not contain the set X. Based on these reasons, Shabir modified the concept of soft rough set, which is called an MSR-set. The underlying concepts are very close to Pawlak rough sets. Further, Li and Xie [16] investigated the relationship among soft sets, soft rough sets and topologies.

In particular, Feng and Li et al. [11] proposed a novel concept of soft rough fuzzy sets by combining rough sets, soft sets with fuzzy sets, we call it Feng-soft rough fuzzy set. In 2011, Meng and Zhang et al. [18] further discussed the Feng-soft rough fuzzy sets and put forward another kind of soft rough fuzzy sets, we call it Meng-soft rough fuzzy set. However, Feng-soft rough fuzzy sets and Meng-soft rough fuzzy sets are all limited on full soft sets which is a rigorous restrictive condition. Based on the above reason, Zhan and Zhu [26] established a novel concept of soft rough fuzzy sets, called a Z-soft rough fuzzy set. Under this definition, the restrictive condition, full soft set, can be removed. Furthermore, it is worth nothing that Z-soft rough fuzzy sets are more precise than Feng-soft rough fuzzy sets and Meng-soft rough fuzzy sets. This means that in the practical application, more precise information can be obtained.

Based on the above idea, in this paper, we apply this novel soft rough fuzzy set theory to lattices, which propose the concept of soft rough fuzzy lattices (ideals, filters) over lattices. This paper is organized as follows: In Section 2, we recall some concepts and results on lattices, soft sets, fuzzy set and rough sets. In Section 3, some new soft rough fuzzy operations over lattices are explored. Further, lower and upper soft rough fuzzy lattices (ideals, filters) over lattices are investigated in Section 4. In particular, in Section 5, we discuss soft rough fuzzy lattices (ideals, filters) over lattices based on another fuzzy soft set.

2 Preliminaries

In this section, we recall some basic definitions and results about lattices, soft sets, fuzzy sets and rough sets.

A poset (L, ≤) is called a lattice if it satisfies the condition that for any x, y ∈ L both x ∨ y and x ∧ y exist, where x ∨ y= sup{x, y} and x ∧ y= inf {x, y}, respectively. Throughout this paper, L is always a lattice.

Definition 2.1. [5] Let L be a lattice and ∅ ⊊ X ⊆ L. Then X is a sublattice over L if x, y ∈ L, x ∨ y ∈ X and x ∧ y ∈ X.

Definition 2.2. [5] Let ∅ ⊊ I ⊆ L. Then I is called an ideal over L if

a, b ∈ I implies x ∨ y ∈ I,

a ∈ L, b ∈ I and a ≤ b imply a ∈ I.

Definition 2.3 [5] Let ∅ ⊊ F ⊆ L. Then F is called a filter over L if

a, b ∈ F implies x ∧ y ∈ F,

a ∈ L, b ∈ F and a ≥ b imply a ∈ F.

Definition 2.4. [19] A pair $S = (F, A)$ is called a soft set over U, where A ⊆ E and F : A → P (U) is a set-valued mapping.

Definition 2.5. [11] A soft set $S = (F, A)$ over U is called a full soft set if ⋃_a∈AF (a) = U.

Definition 2.6. [9] Let μ be a fuzzy set over L. Then μ is called a fuzzy sublattice over L if μ (x ∧ y) ∧ μ (x ∨ y) ≥ μ (x) ∧ μ (y) for all x, y ∈ L.

Definition 2.7. [9] Let μ be a fuzzy sublattice over L. Then

μ is called a fuzzy ideal over L if μ (x ∨ y) = μ (x) ∧ μ (y) for all x, y ∈ L.

μ is called a fuzzy filter over L if μ (x ∧ y) = μ (x) ∧ μ (y) for all x, y ∈ L.

Definition 2.8. [9] Let μ be a fuzzy sublattice over L. Then

μ is called a fuzzy ideal over L if and only if x ≤ y implies that μ (x) ≥ μ (y) for all x, y ∈ L.

μ is called a fuzzy filter over L if and only if x ≤ y implies that μ (x) ≤ μ (y) for all x, y ∈ L.

Definition 2.9. Let $(\tilde{F}, A)$ be a fuzzy soft set over L. Then $(\tilde{F}, A)$ is called a fuzzy soft lattice (ideal, filter) if $\tilde{F} (x)$ is a fuzzy sublattice (ideal, filter) over L for all x ∈ A.

Remark 2.10. Let μ be a fuzzy set over L. Then μ_t = {x ∈ L : μ (x) ≥ t}, t ∈ [0, 1]. μ is a fuzzy sublattice (ideal, filter) over L if and only if every non-empty set μ_t is a sublattice (ideal, filter) over L for all t ∈ [0, 1].

Definition 2.11. [20] Let R be an equivalence relation on the universe U and (U, R) be a Pawlak approximation space. A subset X ⊆ U is called definable if $\underline{R} (X) = \bar{R} (X)$ ; otherwise, i.e., $\underline{R} (X) - \bar{R} (X) \neq \emptyset$ , X is said to be a rough set, where the two operators are defined as: $\begin{matrix} \underline{R} (X) & = & {x \in U : [x]_{R} \subseteq X}, \\ \bar{R} (X) & = & {x \in U : [x]_{R} \cap X \neq \emptyset} . \end{matrix}$

Definition 2.12. [10] Let $S = (F, A)$ be a soft set over U. Then the pair $P = (U, S)$ is called a soft approximation space. Based on P, we define the following two operators: $\begin{matrix} {\underline{apr}}_{P} (X) & = & {u \in U : \exists a \in A, u \in F (a) \subseteq X}, \\ {\bar{apr}}_{P} (X) & = & {u \in U : \exists a \in A, u \in F (a), F (a) \\ \cap X \neq \emptyset}, \end{matrix}$ assigning to every subset X ⊆ U.

Two sets ${\underline{apr}}_{P} (X)$ and ${\bar{apr}}_{P} (X)$ are called the lower and upper soft rough approximations of X in P, respectively. If ${\underline{apr}}_{P} (X) = {\bar{apr}}_{P} (X)$ , X is said to be soft definable; otherwise, X is called a soft rough set.

Definition 2.13. [10] Let $S = (F, A)$ be a full soft set over U and $S = (U, S)$ be a soft approximation space. For a fuzzy set μ ∈ F (U), the lower and upper soft rough approximations of μ with respect to S are denoted by ${\underline{apr}}_{S} (μ)$ and ${\bar{apr}}_{S} (μ)$ , respectively, which are fuzzy sets in U given by $\begin{matrix} {\underline{apr}}_{S} (μ) (x) & = & ⋀ {μ (y) : \exists a \in A, {x, y} \subseteq F (a)}, \\ {\bar{apr}}_{S} (μ) (x) & = & ⋁ {μ (y) : \exists a \in A, {x, y} \subseteq F (a)} \end{matrix}$ for all x ∈ U .

The operators ${\underline{apr}}_{S}$ and ${\bar{apr}}_{S}$ are called the lower and upper soft rough approximation operators on fuzzy set μ, respectively. If ${\underline{apr}}_{S} (μ) = {\bar{apr}}_{S} (μ)$ , μ is said to be soft definable; otherwise, μ is called a soft rough fuzzy set. In what follows, to facilitate the expression, we call it Feng-soft rough fuzzy set.

In [18], Meng and Zhang et al. further discussed Feng-soft rough fuzzy sets and defined a new kind of lower and upper soft rough approximation operators as follows.

Definition 2.14. [18] Let $S = (F, A)$ be a full soft set over U and $S = (U, S)$ be a soft approximation space. For a fuzzy set μ ∈ F (U), the lower and upper soft rough approximations of μ with respect to S are denoted by ${\underline{apr}}_{S}^{'} (μ)$ and ${\bar{apr}}_{S}^{'} (μ)$ , respectively, which are fuzzy sets in U given by $\begin{matrix} {\underline{apr}}_{S}^{'} (μ) (x) & = & ⋁_{x \in F (a)} ⋀_{y \in F (a)} μ (y), \\ {\bar{apr}}_{S}^{'} (μ) (x) & = & ⋀_{x \in F (a)} ⋁_{y \in F (a)} μ (y) \end{matrix}$ for all x ∈ U .

The operators ${\underline{apr}}_{S}$ and ${\bar{apr}}_{S}^{'}$ are called the lower and upper soft rough approximation operators on fuzzy set μ, respectively. If ${\underline{apr}}_{S}^{'} (μ) = {\bar{apr}}_{S}^{'} (μ)$ , μ is said to be soft definable; otherwise, μ is called a soft rough fuzzy set. In what follows, to facilitate the expression, we call it Meng-soft rough fuzzy set.

In the above definitions, Feng-soft rough fuzzy sets and Meng-soft rough fuzzy sets are all limited on full soft sets which is a rigorous restrictive condition. Based on the above reason, Zhan and Zhu [26] established a novel concept of soft rough fuzzy sets. Under this definitions, the restrictive condition, full soft set, can be removed.

Definition 2.15. [26] Let (F, A) be a soft set over U and δ : U → P (A) be a mapping defined as δ (x) = {a : x ∈ F (a)}. Then the pair (U, δ) is called a soft approximation space. For any fuzzy set μ ∈ F (U), the lower and upper soft rough approximations of μ are denoted by $\underline{μ_{δ}}$ and $\bar{μ_{δ}}$ , respectively, which are fuzzy sets in U given by $\begin{matrix} \underline{μ_{δ}} (x) & = & ⋀ {μ (y) : y \in U, δ (y) = δ (x)}, \\ \bar{μ_{δ}} (x) & = & ⋁ {μ (y) : y \in U, δ (y) = δ (x)} \end{matrix}$ for every x ∈ U.

The operators $\underline{μ_{δ}}$ and $\bar{μ_{δ}}$ are called the lower and upper soft rough approximation operators on a fuzzy set, respectively. In particular, if $\underline{μ_{δ}} = \bar{μ_{δ}}$ , μ is said to be soft definable; otherwise, μ is called a soft rough fuzzy set.

3 Some operations of soft rough fuzzy sets over lattices

In this section, we investigate some operations and some properties of soft rough fuzzy sets over lattices are discussed. In order to illustrate the roughness in lattice L with respect to soft approximation spaces, we first introduce two special kinds of soft sets over L.

Definition 3.1. Let $S = (F, A)$ be a soft set over L and δ : L → P (A) be a mapping defined as δ (x) = {a : x ∈ F (a)}. Then $S$ is called a C-soft set over L if δ (a) = δ (b) and δ (c) = δ (d) imply δ (a ∨ c) = δ (b ∨ d) and δ (a ∧ c) = δ (b ∧ d) for all a, b, c, d ∈ L.

Definition 3.2. Let $S = (F, A)$ be a C-soft set over L and δ : L → P (A) be a mapping defined as δ (x) = {a : x ∈ F (a)}. Then $S$ is called a CC-soft set over L if for all c ∈ L,

δ (c) = δ (x ∨ y) for x, y ∈ L, there exist a, b ∈ L such that δ (x) = δ (a) and δ (y) = δ (b) satisfying c = a ∨ b;

δ (c) = δ (x ∧ y) for x, y ∈ L, there exist a, b ∈ L such that δ (x) = δ (a) and δ (y) = δ (b) satisfying c = a ∧ b.

The following definition is from Zadeh’s expansion principle.

Definition 3.3. Let μ and ν be two fuzzy sets over L. Define μ ∨ ν and μ ∧ ν over L as follows: (μ ∨ ν) (x) = ⋁ _x=a∨b (μ (a) ∧ ν (b)), and (μ ∧ ν) (x) = ⋁ _x=a∧b (μ (a) ∧ ν (b)) for all x ∈ L, respectively.

Proposition 3.4. Let $S = (F, A)$ be a C-soft set over L. If μ and ν are any two fuzzy sets over L, then $\bar{μ_{δ}} \lor \bar{ν_{δ}} \subseteq \bar{(μ \lor ν)_{δ}} .$

Proof. Let x = a ∨ b, a, b ∈ L. Then $\begin{matrix} (\bar{μ_{δ}} \lor \bar{ν_{δ}}) (x) & = & ⋁_{x = a \lor b} (\bar{μ_{δ}} (a) \land \bar{ν_{δ}} (b)) \\ = & ⋁_{x = a \lor b} (⋁ {μ (c) | c \in L, δ (c) = δ (a)}) \\ \land (⋁ {ν (d) | d \in L, δ (d) = δ (b)}) . \end{matrix}$

Since $S = (F, A)$ is a C-soft set over L, we have $\begin{matrix} ⋁_{x = a \lor b} (⋁ {μ (c) | c \in L, δ (c) = δ (a)}) \\ \land (⋁ {ν (d) | d \in L, δ (d) = δ (b)}) \\ = ⋁_{x = a \lor b} ⋁ {μ (c) \land ν (d) | δ (c) = δ (a), δ (d) = δ (b)} \\ \leq ⋁_{x = a \lor b} ⋁ {μ (c) \land ν (d) | δ (c \lor d) \\ = δ (a \lor b) = δ (x)} \\ = ⋁ {μ (c) \land ν (d) | δ (c \lor d) = δ (x)} \\ \leq ⋁ {(μ \lor ν) (y) | δ (y) = δ (x)} \\ = \bar{(μ \lor ν)_{δ}} (x) . \end{matrix}$

It follows that $(\bar{μ_{δ}} \lor \bar{ν_{δ}}) (x) \leq \bar{(μ \lor ν)_{δ}} (x)$ , i.e., $\bar{μ_{δ}} \lor \bar{ν_{δ}} \subseteq \bar{(μ \lor ν)_{δ}} .$ □

Proposition 3.5. Let $S = (F, A)$ be a C-soft set over L. If μ and ν are any two fuzzy sets over L, then $\bar{μ_{δ}} \land \bar{ν_{δ}} \subseteq \bar{(μ \land ν)_{δ}} .$

Proof. Let x = a ∧ b, a, b ∈ L. Then $\begin{matrix} (\bar{μ_{δ}} \land \bar{ν_{δ}}) (x) & = & ⋁_{x = a \land b} (\bar{μ_{δ}} (a) \land \bar{ν_{δ}} (b)) \\ = & ⋁_{x = a \land b} (⋁ {μ (c) | c \in L, δ (c) = δ (a)}) \\ \land (⋁ {ν (d) | d \in L, δ (d) = δ (b)}) . \end{matrix}$

Since $S = (F, A)$ is a C-soft set over L, we have $\begin{matrix} ⋁_{x = a \land b} (⋁ {μ (c) | c \in L, δ (c) = δ (a)}) \\ \land (⋁ {ν (d) | d \in L, δ (d) = δ (b)}) \\ = ⋁_{x = a \land b} ⋁ {μ (c) \land ν (d) | δ (c) = δ (a), δ (d) = δ (b)} \\ \leq ⋁_{x = a \land b} ⋁ {μ (c) \land ν (d) | δ (c \land d) \\ = δ (a \land b) = δ (x)} \\ = ⋁ {μ (c) \land ν (d) | δ (c \land d) = δ (x)} \\ \leq ⋁ {(μ \land ν) (y) | δ (y) = δ (x)} \\ = \bar{(μ \land ν)_{δ}} (x) . \end{matrix}$

It follows that $(\bar{μ_{δ}} \land \bar{ν_{δ}}) (x) \leq \bar{(μ \land ν)_{δ}} (x)$ , i.e., $\bar{μ_{δ}} \land \bar{ν_{δ}} \subseteq \bar{(μ \land ν)_{δ}} .$ □

The following example shows that the containment in Propositions 3.4 and 3.5 is proper.

Example 3.6. Let L = {0, a, b, c, d, 1}. We define the binary relation ≤ in the Fig. 1. $S = (F, A)$ is a soft set over L which is given by Table 1. Then the mapping δ : L → P (A) in soft approximation space (L, δ) is given by δ (0) = δ (a) = {e₁, e₃}, δ (b) = {e₁}, δ (c) = {e₂, e₃}, δ (d) = δ (1) = {e₁, e₂, e₃}. Then we can check that $S$ is a C-soft set over L. If we take $μ = {\frac{0.3}{b}}$ and $ν = {\frac{0.4}{c}}$ , then $\bar{μ_{δ}} = {\frac{0.3}{b}}$ and $\bar{ν_{δ}} = {\frac{0.4}{b}}$ . So $\bar{μ_{δ}} \lor \bar{ν_{δ}} = {\frac{0.4}{d}}$ , $\bar{μ_{δ}} \land \bar{ν_{δ}} = {\frac{0.3}{a}}$ and $\bar{(μ \lor ν)_{δ}} = {\frac{0.3}{d}, \frac{0.3}{1}}$ , $\bar{(μ \land ν)_{δ}} = {\frac{0.3}{0}, \frac{0.3}{a}}$ . Thus, $\bar{μ_{δ}} \lor \bar{ν_{δ}} ⫋ \bar{(μ \lor ν)_{δ}}$ , $\bar{μ_{δ}} \land \bar{ν_{δ}} ⫋ \bar{(μ \land ν)_{δ}}$ .

Table 1
Soft set $S$

0 a b c d 1

e ₁ 1 1 1 0 1 1

e ₂ 0 0 0 1 1 1

e ₃ 1 1 0 1 1 1

	0	a	b	c	d	1
e ₁	1	1	1	0	1	1
e ₂	0	0	0	1	1	1
e ₃	1	1	0	1	1	1

If we strength the condition, we can obtain the following result.

Proposition 3.7. Let $S = (F, A)$ be a CC-soft set over L. If μ and ν are any two fuzzy sets over L, then $\bar{μ_{δ}} \lor \bar{ν_{δ}} = \bar{(μ \lor ν)_{δ}} .$

Proof. It follows from Proposition 3.4 that we only need to show $\bar{(μ \lor ν)_{δ}} \subseteq \bar{μ_{δ}} \lor \bar{ν_{δ}}$ . $\begin{matrix} \bar{(μ \lor ν)_{δ}} (x) & = & ⋁ {(μ \lor ν) (y) | δ (y) = δ (x)} \\ = & ⋁ {⋁_{y = a \lor b} μ (a) \land ν (b) | δ (a \lor b) \\ = & δ (x)} . \end{matrix}$

Since $S = (F, A)$ is a CC-soft set over L, there exist c, d ∈ L such that δ (a) = δ (c) , δ (b) = δ (d) satisfying x = c ∨ d. So we have $\begin{matrix} ⋁ {⋁_{y = a \lor b} μ (a) \land ν (b) | δ (a \lor b) = δ (x)} \\ = ⋁_{y = a \lor b} {⋁ μ (a) \land ν (b) | δ (a) = δ (c), δ (b) = δ (d)} \\ = ⋁_{y = a \lor b} (⋁ {μ (a) | δ (a) = δ (c)}) \land (⋁ {ν (b) | δ (b) \\ = δ (d)}) \\ = (⋁ {μ (a) | δ (a) = δ (c)}) \land (⋁ {ν (b) | δ (b) \\ = δ (d)}) \\ = \bar{μ_{δ}} (c) \land \bar{ν_{δ}} (d) \\ \leq ⋁_{x = a^{'} \lor b^{'}} \bar{μ_{δ}} (a^{'}) \land \bar{ν_{δ}} (b^{'}) \\ = (\bar{μ_{δ}} \lor \bar{ν_{δ}}) (x) . \end{matrix}$

Thus, $\bar{(μ \lor ν)_{δ}} \subseteq \bar{μ_{δ}} \lor \bar{ν_{δ}}$ . □

Proposition 3.8. Let $S = (F, A)$ be a CC-soft set over L. If μ and ν are any two fuzzy sets over L, then $\bar{μ_{δ}} \land \bar{ν_{δ}} = \bar{(μ \land ν)_{δ}} .$

Proof. It follows from Proposition 3.5 that we only need to show $\bar{(μ \land ν)_{δ}} \subseteq \bar{μ_{δ}} \land \bar{ν_{δ}}$ . $\begin{matrix} \bar{(μ \land ν)_{δ}} (x) & = & ⋁ {(μ \land ν) (y) | δ (y) = δ (x)} \\ = & ⋁ {⋁_{y = a \land b} μ (a) \land ν (b) | δ (a \land b) \\ = & δ (x)} . \end{matrix}$

Since $S = (F, A)$ is a CC-soft set over L, there exist c, d ∈ L such that δ (a) = δ (c) , δ (b) = δ (d) satisfying x = c ∧ d. So we have $\begin{matrix} ⋁ {⋁_{y = a * b} μ (a) \land ν (b) | δ (a \land b) = δ (x)} \\ = ⋁_{y = a \land b} {⋁ μ (a) \land ν (b) | δ (a) = δ (c), δ (b) \\ = δ (d)} \\ = (⋁_{y = a \land b} ⋁ {μ (a) | δ (a) = δ (c)}) \land (⋁ {ν (b) | δ (b) \\ = δ (d)}) \\ = (⋁ {μ (a) | δ (a) = δ (c)}) \land (⋁ {ν (b) | δ (b) \\ = δ (d)}) \\ = \bar{μ_{δ}} (c) \land \bar{ν_{δ}} (d) \\ \leq ⋁_{x = a^{'} \land b^{'}} \bar{μ_{δ}} (a^{'}) \land \bar{ν_{δ}} (b^{'}) \\ = (\bar{μ_{δ}} \land \bar{ν_{δ}}) (x) . \end{matrix}$

Thus, $\bar{(μ \land ν)_{δ}} \subseteq \bar{μ_{δ}} \land \bar{ν_{δ}}$ . □

Next, we consider lower soft rough fuzzy approximations over L.

Proposition 3.9. Let $S = (F, A)$ be a CC-soft set over L. If μ and ν are any two fuzzy sets over L, then $\underline{μ_{δ}} \lor \underline{ν_{δ}} \subseteq \underline{(μ \lor ν)_{δ}} .$

Proof. Let x = a ∨ b, a, b ∈ L. Then $\begin{matrix} (\underline{μ_{δ}} \lor \underline{ν_{δ}}) (x) & = & ⋁_{x = a \lor b} (\underline{μ_{δ}} (a)) \land (\underline{ν_{δ}} (b)) \\ = & ⋁_{x = a \lor b} (⋀ {μ (c) | c \in L, δ (c) = δ (a)}) \\ \land (⋀ {ν (d) | d \in L, δ (d) = δ (b)}) . \end{matrix}$

Since $S = (F, A)$ is a CC-soft set over L, we have $\begin{matrix} ⋁_{x = a \lor b} (⋀ {μ (c) | c \in L, δ (c) = δ (a)}) \\ \land (⋀ {ν (d) | d \in L, δ (d) = δ (b)}) \\ = ⋁_{x = a \lor b} ⋀ {μ (c) \land ν (d) | δ (c) = δ (a), δ (d) = δ (b)} \\ \leq ⋁_{x = a \lor b} ⋀ {μ (c) \land ν (d) | δ (c \lor d) = δ (a \lor b) \\ = δ (x)} \\ = ⋀ {(μ \lor ν) (y) | δ (y) = δ (x)} \\ = \underline{(μ \lor ν)_{δ}} (x) . \end{matrix}$

Thus, $(\underline{μ_{δ}} \lor \underline{ν_{δ}}) (x) \leq \underline{(μ \lor ν)_{δ}} (x)$ , i.e., $\underline{μ_{δ}} \lor \underline{ν_{δ}} \subseteq \underline{(μ \lor ν)_{δ}} .$ □

Proposition 3.10. Let $S = (F, A)$ be a CC-soft set over L. If μ and ν are any two fuzzy sets over L, then $\underline{μ_{δ}} \land \underline{ν_{δ}} \subseteq \underline{(μ \land ν)_{δ}} .$

Proof. Let x = a ∧ b, a, b ∈ L. Then $\begin{matrix} (\underline{μ_{δ}} \land \underline{ν_{δ}}) (x) & = & ⋁_{x = a \land b} (\underline{μ_{δ}} (a) \land \underline{ν_{δ}} (b)) \\ = & ⋁_{x = a \land b} (⋀ {μ (c) | c \in L, δ (c) = δ (a)}) \\ \land (⋀ {ν (d) | d \in L, δ (d) = δ (b)}) . \end{matrix}$

Since $S = (F, A)$ is a CC-soft set over L, we have $\begin{matrix} ⋁_{x = a \land b} (⋀ {μ (c) | c \in L, δ (c) = δ (a)}) \\ \land (⋀ {ν (d) | d \in L, δ (d) = δ (b)}) \\ = ⋁_{x = a \land b} ⋀ {μ (c) \land ν (d) | δ (c) = δ (a), δ (d) = δ (b)} \\ \leq ⋁_{x = a \land b} ⋀ {μ (c) \land ν (d) | δ (c \land d) = δ (a \land b) \\ = δ (x)} \\ = ⋀ {(μ \land ν) (y) | δ (y) = δ (x)} \\ = \underline{(μ \land ν)_{δ}} (x) . \end{matrix}$

Thus, $(\underline{μ_{δ}} \land \underline{ν_{δ}}) (x) \leq \underline{(μ \land ν)_{δ}} (x)$ , i.e., $\underline{μ_{δ}} \land \underline{ν_{δ}} \subseteq \underline{(μ \land ν)_{δ}} .$ □

Fig.1

A lattice L.

The following example shows that the containments in Propositions 3.9 and 3.10 are proper.

Example 3.11. Let L = {0, a, b, c, 1}. We define the binary relation ≤ in the Fig. 2. $S = (F, A)$ is a soft set over L which is given by Table 2. Then the mapping δ : L → P (A) in soft approximation space (L, δ) is given by δ (0) = δ (b) = {e₁}, δ (a) = δ (c) = {e₁, e₂}, δ (1) = {e₁, e₂, e₃}. Then we can check that $S$ is a CC-soft set over L. If we take $μ = {\frac{0.4}{b}}$ and $ν = {\frac{0.2}{a}, \frac{0.5}{c}}$ , then $\underline{μ_{δ}} = \emptyset$ , $\underline{ν_{δ}} = {\frac{0.2}{a}, \frac{0.5}{c}}$ and $\underline{μ_{δ}} \lor \underline{ν_{δ}} = \emptyset$ , $\underline{μ_{δ}} \land \underline{ν_{δ}} = \emptyset$ . On the other hand, $\underline{(μ \lor ν)_{δ}} = {\frac{0.2}{a}, \frac{0.3}{c}}$ , $\underline{(μ \land ν)_{δ}} = {\frac{0.2}{0}, \frac{0.2}{b}}$ . This means that $\underline{μ_{δ}} \lor \underline{ν_{δ}} ⫋ \underline{(μ \lor ν)_{δ}}$ and $\underline{μ_{δ}} \land \underline{ν_{δ}} ⫋ \underline{(μ \land ν)_{δ}}$

Table 2

Soft set $S$

	0	a	b	c	1
e ₁	1	1	1	1	1
e ₂	0	1	0	1	1
e ₃	0	0	0	0	1

Fig.2

A lattice L.

4 Characterizations of soft rough fuzzy lattices (ideals, filters) over lattices

In this section, we characterize soft rough fuzzy lattices (ideals, filters) over lattices. First, we give the concept of soft rough fuzzy lattices (ideals, filters) over L.

Definition 4.1. Let (F, A) be a soft set over L and δ : L → P (A) be a mapping defined as δ (x) = {a : x ∈ F (a)}. Then the pair (L, δ) is called a soft approximation space. For any fuzzy set μ ∈ F (L), the lower and upper soft rough approximations of μ are denoted by $\underline{μ_{δ}}$ and $\bar{μ_{δ}}$ , respectively, which are fuzzy sets in L given by $\begin{matrix} \underline{μ_{δ}} (x) & = & ⋀ {μ (y) : y \in L, δ (y) = δ (x)}, \\ \bar{μ_{δ}} (x) & = & ⋁ {μ (y) : y \in L, δ (y) = δ (x)} \end{matrix}$ for every x ∈ L. if ${\underline{μ}}_{δ} \neq {\bar{μ}}_{δ}$ ,

μ is called a lower (upper) soft rough fuzzy lattice (ideal, filter) w.r.t. $S$ over L, if ${\underline{μ}}_{δ}$ ( ${\bar{μ}}_{δ}$ ) is a sublattice (ideal, filter) over L;

μ is called a soft rough fuzzy lattice (ideal, filter) w.r.t. $S$ over L, if ${\underline{μ}}_{δ}$ and ${\bar{μ}}_{δ}$ are fuzzy sublattices (ideals, filters) over L.

Example 4.2. Consider the lattice L and the soft set $S = (F, A)$ in Example 3.11. It follows from Definition 4.1 that for a fuzzy set $μ = {\frac{0.3}{0}, \frac{0.4}{b}, \frac{0.2}{c}},$ $\begin{matrix} \underline{μ_{δ}} & = & {\frac{0.2}{0}, \frac{0.3}{b}} and \\ \bar{μ_{δ}} & = & {\frac{0.4}{0}, \frac{0.2}{a}, \frac{0.4}{b}, \frac{0.2}{c}} . \end{matrix}$

It is easy to check that $\underline{μ_{δ}}$ and $\bar{μ_{δ}}$ are fuzzy sublattices (ideals) over L. In other words, μ is a soft rough fuzzy lattice (ideal) over X.

Proposition 4.3. Let $S = (F, A)$ be a C-soft set over L. If μ is a fuzzy sublattice over L, then μ is an upper soft rough fuzzy lattice over L.

Proof. For any x, y ∈ L, since $S = (F, A)$ is a C-soft set over L and μ is a fuzzy sublattice of X, $\begin{matrix} \bar{μ_{δ}} (x) \land \bar{μ_{δ}} (y) \\ = (⋁ {μ (a) | δ (a) = δ (x)}) \\ \land (⋁ {μ (b) | δ (b) = δ (y)}) \\ \leq ⋁ {μ (a) \land μ (b) | δ (a \lor b) = δ (x \lor y)} \\ = ⋁ {μ (a \lor b) | δ (a \lor b) = δ (x \lor y)} \\ = \bar{μ_{δ}} (x \lor y) . \\ \bar{μ_{δ}} (x) \land \bar{μ_{δ}} (y) \\ = (⋁ {μ (a) | δ (a) = δ (x)}) \\ \land (⋁ {μ (b) | δ (b) = δ (y)}) \\ \leq ⋁ {μ (a) \land μ (b) | δ (a \land b) = δ (x \land y)} \\ = ⋁ {μ (a \land b) | δ (a \land b) = δ (x \land y)} \\ = \bar{μ_{δ}} (x \land y) . \end{matrix}$

This implies that $\bar{μ_{δ}} (x \lor y) \land \bar{μ_{δ}} (x \land y) \geq \bar{μ_{δ}} (x) \land \bar{μ_{δ}} (y)$ . So $\bar{μ_{δ}}$ is a fuzzy sublattice over L. It follows from Definition 4.1 that μ is an upper soft rough fuzzy lattice over L. □

Proposition 4.4. Let $S = (F, A)$ be a CC-soft set over L. If μ is a fuzzy sublattice over L, then μ is a lower soft rough fuzzy lattice over L.

Proof. For any x, y ∈ L, since $S = (F, A)$ is a CC-soft set over L and μ is a fuzzy sublattice of L, $\begin{matrix} \underline{μ_{δ}} (x) \land \underline{μ_{δ}} (y) \\ = (⋀ {μ (a) | δ (a) = δ (x)}) \\ \land (⋀ {μ (b) | δ (b) = δ (y)}) \\ = ⋀ {μ (a) \land μ (b) | δ (a \lor b) = δ (x \lor y)} \\ \leq ⋀ {μ (a \lor b) | δ (a \lor b) = δ (x \lor y)} \\ = \underline{μ_{δ}} (x \lor y) . \\ \underline{μ_{δ}} (x) \land \underline{μ_{δ}} (y) \\ = (⋀ {μ (a) | δ (a) = δ (x)}) \\ \land (⋀ {μ (b) | δ (b) = δ (y)}) \\ = ⋀ {μ (a) \land μ (b) | δ (a \land b) = δ (x \land y)} \\ \leq ⋀ {μ (a \land b) | δ (a \land b) = δ (x \land y)} \\ = \underline{μ_{δ}} (x \land y) . \end{matrix}$

This implies that $\underline{μ_{δ}} (x \lor y) \land \underline{μ_{δ}} (x \land y) \geq \underline{μ_{δ}} (x) \land \bar{μ_{δ}} (y)$ . So $\underline{μ_{δ}}$ is a fuzzy sublattice over L. It follows from Definition 4.1 that μ is a lower soft rough fuzzy lattice over L. □

Definition 4.5. Let $S = (F, A)$ be a C-soft set over L and δ : L → P (A) be a mapping defined as δ (x) = {a : x ∈ F (a)}. Then $S$ is called a BC-soft set over L if I is an ideal over L, ∀a, b ∈ L, δ (a) = δ (b) if and only if there exist i₁, i₂ ∈ I such that a ∨ i₁ = b ∨ i₂.

Theorem 4.6. Let $S = (F, A)$ be a BC-soft set over L. If μ is a fuzzy ideal over L, then μ is an upper soft rough fuzzy ideal over L.

Proof. Let μ be a fuzzy ideal over L. Then μ is a fuzzy sublattice over L. It follows from Proposition 4.3 that $\bar{μ_{δ}}$ is a fuzzy sublattice over L. Now let x ≤ y, x, y ∈ L but $\bar{μ_{δ}} (x) < \bar{μ_{δ}} (y)$ . Then $\bar{μ_{δ}} (x) < \bar{μ_{δ}} (x \lor y) \land \bar{μ_{δ}} (y)$ . Choose t ∈ [0, 1] such that $\bar{μ_{δ}} (x) < t < \bar{μ_{δ}} (x \lor y) \land \bar{μ_{δ}} (y)$ . Since $μ \subseteq \bar{μ_{δ}}$ , we have $μ (x) \leq \bar{μ_{δ}} (x) < t$ . Then $\begin{matrix} \bar{μ_{δ}} (x \lor y) \land \bar{μ_{δ}} (y) \\ = (⋁ {μ (a) | δ (a) = δ (x \lor y)}) \\ \land (⋁ {μ (b) | δ (b) = δ (y)}) \\ = ⋁ {μ (a) \land μ (b) | δ (a) \\ = δ (x \lor y), δ (b) = δ (y)} . \end{matrix}$

Since μ is a fuzzy ideal of L, it follows from Remark 2.10 that μ_t is an ideal over L. Because δ (a) = δ (x ∨ y) , δ (b) = δ (y) and $S = (F, A)$ is a BC-soft set over L, there exist i₁, i₂, i₃, i₄ ∈ μ_t such that a ∨ i₁ = (x ∨ y) ∨ i₂, b ∨ i₃ = y ∨ i₄. Since [(x ∨ y) ∨ i₂] ∨ i₄ = [(x ∨ y) ∨ i₂] ∨ i₄, we have [(x ∨ y) ∨ i₄] ∨ i₂ = [(x ∨ y) ∨ i₂] ∨ i₄, that is $\begin{matrix} [x \lor (y \lor i_{4})] \lor i_{2} & = & [(x \lor y) \lor i_{2}] \lor i_{4}, \\ [x \lor (b \lor i_{3})] \lor i_{2} & = & (a \lor i_{1}) \lor i_{4}, \\ [(x \lor b) \lor i_{3}] \lor i_{2} & = & (a \lor i_{1}) \lor i_{4}, \\ (x \lor b) \lor (i_{3} \lor i_{2}) & = & a \lor (i_{1} \lor i_{4}) . \end{matrix}$

It follows from Definition 4.5 that δ (x ∨ b) = δ (a), where i₃ ∨ i₂ ∈ μ_t, i₁ ∨ i₄ ∈ μ_t. Thus $\begin{matrix} \bar{μ_{δ}} (x \land y) \land \bar{μ_{δ}} (y) \\ = ⋁ {μ (a) \land μ (b) | δ (a) = δ (x \lor y), δ (b) = δ (y)} \\ = ⋁ {μ (a) \land μ (b) | (x \lor b) \lor (i_{3} \lor i_{2}) \\ = a \lor (i_{1} \lor i_{4})} \\ = ⋁ {μ (a) \land μ (b) | δ (x \lor b) = δ (a)} \\ > t . \end{matrix}$

This means that there exist a₀, b₀ ∈ L such that μ (a₀) ∧ μ (b₀) > t satisfying δ (x ∨ b₀) = δ (a₀). So there exist i₅, i₆ ∈ μ_t such that (x ∗ b₀) ∨ i₅ = a₀ ∨ i₆ and μ (a₀) ∧ μ (b₀) > t. That is x ∨ (b₀ ∨ i₅) = a₀ ∨ i₆ and μ (a₀) ∧ μ (b₀) > t. So x ≤ a₀ ∨ i₆. Since μ is a fuzzy ideal over L, μ (x) ≥ μ (a₀ ∨ i₆). Thus $\begin{matrix} μ (x) & \geq & μ (x) \land μ (b_{0} \lor i_{5}) \\ \geq & μ (a_{0} \lor i_{6}) \land μ (b_{0} \lor i_{5}) \\ = & μ (a_{0}) \land μ (i_{6}) \land μ (b_{0}) \land μ (i_{5}) \\ \geq & t . \end{matrix}$

This is a contradicts with μ (x) < t.

Hence $\bar{μ_{δ}} (x) \geq \bar{μ_{δ}} (y)$ for all x ≤ y, x, y ∈ L. This implies that $\bar{μ_{δ}}$ is a fuzzy ideal over L, that is, μ is an upper soft rough fuzzy ideal over L.

Remark 4.7. Let $S = (F, A)$ be a CC-soft set over L and δ : L → P (A) be a mapping defined as δ (x) = {a : x ∈ F (a)}. If for all c ∈ L, then the following hold.

δ (c) = δ (x ∨ y) for x, y ∈ L if and only if for δ (x) = δ (a) and δ (y) = δ (b) we have c = a ∨ b, a, b ∈ L.

δ (c) = δ (x ∧ y) for x, y ∈ L if and only if for δ (x) = δ (a) and δ (y) = δ (b) we have c = a ∧ b, a, b ∈ L.

Proof. It is straightforward. □

Theorem 4.8. Let $S = (F, A)$ be a CC-soft set over L. If μ is a fuzzy ideal over L, then μ is a lower soft rough fuzzy ideal over L.

Proof. Let μ be a fuzzy ideal over L. Then μ is a fuzzy sublattice over L. It follows from Proposition 4.4 that $\underline{μ_{δ}}$ is a fuzzy sublattice over L. Now let x ≤ y, x, y ∈ L. $\begin{matrix} \underline{μ_{δ}} (y) & = & \underline{μ_{δ}} (x \lor y) \land \underline{μ_{δ}} (y) \\ = & (⋀ {μ (a) | δ (a) = δ (x \lor y)}) \\ \land (⋀ {μ (b) | δ (b) = δ (y)}) \\ = & ⋀ {μ (a) \land μ (b) | δ (a) \\ = & δ (x \lor y), δ (b) = δ (y)} . \end{matrix}$

Since $S = (F, A)$ is a CC-soft set over L, it follows from Remark 4.7 that a = c ∨ d for δ (x) = δ (c) , δ (y) = δ (d), where c, d ∈ L. Thus $\begin{matrix} \underline{μ_{δ}} (x \lor y) \land \underline{μ_{δ}} (y) \\ = ⋀ {μ (a) \land μ (b) | δ (x) = δ (c), \\ δ (d) = δ (b), a = c \lor d} \\ = ⋀ {μ (c \lor d) \land μ (b) | δ (x) = δ (c), \\ δ (b) = δ (d), a = c \lor d} \\ = ⋀ {μ (c \lor d) | δ (x) = δ (c)} \\ \land ⋀ {μ (b) | δ (b) = δ (d)} \\ = (⋀ {μ (c \lor d) | δ (x) = δ (c)}) \land \underline{μ_{δ}} (d) . \end{matrix}$

Since $\underline{μ_{δ}} \subseteq μ$ and μ is a fuzzy ideal over L, c ∨ d ≥ c, we have μ (c) ≥ μ (c ∨ d). Thus, $\begin{matrix} (⋀ {μ (c \lor d) | δ (x) = δ (c)}) \land \underline{μ_{δ}} (d) \\ \leq ⋀ {μ (c \lor d) \land μ (d) | δ (x) = δ (c)} \\ \leq ⋀ {μ (c) \land μ (d) | δ (c) = δ (x)} \\ \leq ⋀ {μ (c) | δ (c) = δ (x)} \\ = \underline{μ_{δ}} (x) . \end{matrix}$

Thus, $\underline{μ_{δ}} (x) \geq \underline{μ_{δ}} (y)$ for all x ≤ y, x, y ∈ L. This implies that $\underline{μ_{δ}}$ is a fuzzy ideal over L, that is, μ is a lower soft rough fuzzy ideal over L. □

Definition 4.9. Let $S = (F, A)$ be a C-soft set over L and δ : L → P (A) be a mapping defined as δ (x) = {a : x ∈ F (a)}. Then $S$ is called a B′C-soft set over L if D is a filter over L, ∀a, b ∈ Lδ (a) = δ (b) if and only if there exist i₁, i₂ ∈ D such that a ∧ i₁ = b ∧ i₂.

Theorem 4.10. Let $S = (F, A)$ be a B′C-soft set over L. If μ is a fuzzy filter over L, then μ is an upper soft rough fuzzy filter over L.

Proof. Let μ be a fuzzy ideal over L. Then μ is a fuzzy sublattice over L. It follows from Proposition 4.3 that $\bar{μ_{δ}}$ is a fuzzy sublattice over L. Let x ≤ y, x, y ∈ L but $\bar{μ_{δ}} (y) < \bar{μ_{δ}} (x)$ . Then $\bar{μ_{δ}} (y) < \bar{μ_{δ}} (x \land y) \land \bar{μ_{δ}} (x)$ . Choose t ∈ [0, 1] such that $\bar{μ_{δ}} (y) < t < \bar{μ_{δ}} (x \land y) \land \bar{μ_{δ}} (x)$ . Since $μ \subseteq \bar{μ_{δ}}$ , we have $μ (y) \leq \bar{μ_{δ}} (y) < t$ . Then $\begin{matrix} \bar{μ_{δ}} (x \land y) \land \bar{μ_{δ}} (x) \\ = (⋁ {μ (a) | δ (a) = δ (x \land y)}) \\ \land (⋁ {μ (b) | δ (b) = δ (x)}) \\ = ⋁ {μ (a) \land μ (b) | δ (a) \\ = δ (x \land y), δ (b) = δ (x)} . \end{matrix}$

Since μ is a fuzzy filter over L, it follows from Remark 2.10 that μ_t is a filter over L. Because δ (a) = δ (x ∧ y) , δ (b) = δ (y) and $S = (F, A)$ is a B′C-soft set over L, there exist i₁, i₂, i₃, i₄ ∈ μ_t such that a ∧ i₁ = (x ∨ y) ∧ i₂, b ∧ i₃ = y ∧ i₄. Since [(x ∧ y) ∧ i₂] ∨ i₄ = [(x ∧ y) ∧ i₂] ∨ i₄, we have [(x ∧ y) ∧ i₄] ∧ i₂ = [(x ∧ y) ∧ i₂] ∧ i₄, that is $\begin{matrix} [y \lor (x \land i_{4})] \land i_{2} & = & [(x \land y) \land i_{2}] \land i_{4}, \\ [y \land (b \lor i_{3})] \land i_{2} & = & (a \land i_{1}) \land i_{4}, \\ [(y \lor b) \land i_{3}] \land i_{2} & = & (a \land i_{1}) \land i_{4}, \\ (y \land b) \land (i_{3} \land i_{2}) & = & a \land (i_{1} \land i_{4}) . \end{matrix}$

It follows from Definition 4.9 that δ (y ∧ b) = δ (a), where i₃ ∧ i₂ ∈ μ_t, i₁ ∧ i₄ ∈ μ_t. Thus $\begin{matrix} \bar{μ_{δ}} (x \land y) \land \bar{μ_{δ}} (x) \\ = ⋁ {μ (a) \land μ (b) | δ (a) \\ = δ (x \land y), δ (b) = δ (x)} \\ = ⋁ {μ (a) \land μ (b) | (x \land b) \lor (i_{3} \land i_{2}) \\ = a \lor (i_{1} \land i_{4})} \\ = ⋁ {μ (a) \land μ (b) | δ (x \land b) = δ (a)} \\ > t . \end{matrix}$

This means that there exist a₀, b₀ ∈ L such that μ (a₀) ∧ μ (b₀) > t satisfying δ (x ∨ b₀) = δ (a₀). So there exist i₅, i₆ ∈ μ_t such that (x ∧ b₀) ∧ i₅ = a₀ ∨ i₆ and μ (a₀) ∧ μ (b₀) > t. That is y ∧ (b₀ ∨ i₅) = a₀ ∧ i₆ and μ (a₀) ∧ μ (b₀) > t. So y ≥ a₀ ∨ i₆. Since μ is a fuzzy filter over L, μ (y) ≥ μ (a₀ ∧ i₆). Thus $\begin{matrix} μ (y) & \geq & μ (y) \land μ (b_{0} \land i_{5}) \\ \geq & μ (a_{0} \land i_{6}) \land μ (b_{0} \land i_{5}) \\ = & μ (a_{0}) \land μ (i_{6}) \land μ (b_{0}) \land μ (i_{5}) \\ \geq & t . \end{matrix}$

This is a contradicts with μ (y) < t. Hence $\bar{μ_{δ}} (x) \leq \bar{μ_{δ}} (y)$ for all x ≤ y, x, y ∈ L. This implies that $\bar{μ_{δ}}$ is a fuzzy filter over L, that is, μ is an upper soft rough fuzzy filter over L. □

Theorem 4.11. Let $S = (F, A)$ be a CC-soft set over L. If μ is a fuzzy filter over L, then μ is a lower soft rough fuzzy filter over L.

Proof. Let μ be a fuzzy ideal over L. Then μ is a fuzzy sublattice over L. It follows from Proposition 4.4 that $\underline{μ_{δ}}$ is a fuzzy sublattice over L. Let x ≤ y, x, y ∈ L. $\begin{matrix} \underline{μ_{δ}} (x) \\ = \underline{μ_{δ}} (x \land y) \land \underline{μ_{δ}} (x) \\ = (⋀ {μ (a) | δ (a) = δ (x \land y)}) \land (⋀ {μ (b) | δ (b) \\ = δ (x)}) \\ = ⋀ {μ (a) \land μ (b) | δ (a) = δ (x \land y), δ (b) = δ (x)} . \end{matrix}$

Since $S = (F, A)$ is a CC-soft set over L, it follows from Remark 4.7 that a = c ∧ d for δ (x) = δ (d) , δ (y) = δ (c), where c, d ∈ L. Thus $\begin{matrix} \underline{μ_{δ}} (x \land y) \land \underline{μ_{δ}} (y) \\ = ⋀ {μ (a) \land μ (b) | δ (y) = δ (c), \\ δ (d) = δ (b), a = c \land d} \\ = ⋀ {μ (c \land d) \land μ (b) | δ (y) = δ (c), \\ δ (b) = δ (d), a = c \land d} \\ = (⋀ {μ (c \land d) | δ (y) = δ (c)}) \\ \land (⋀ {μ (b) | δ (b) = δ (d)}) \\ = (⋀ {μ (c \land d) | δ (y) = δ (c)}) \land \underline{μ_{δ}} (d) . \end{matrix}$

Since $\underline{μ_{δ}} \subseteq μ$ and μ is a fuzzy filterl over L, c ∧ d ≤ c, we have μ (c) ≥ μ (c ∧ d). Thus, $\begin{matrix} ⋀ {μ (c \land d) | δ (x) \\ = δ (c)} \land \underline{μ_{δ}} (d) \\ \leq ⋀ {μ (c \land d) \land μ (d) | δ (y) = δ (c)} \\ \leq ⋀ {μ (c) \land μ (d) | δ (c) = δ (y)} \\ \leq ⋀ {μ (c) | δ (c) = δ (y)} \\ = \underline{μ_{δ}} (y) . \end{matrix}$

Thus, $\underline{μ_{δ}} (x) \leq \underline{μ_{δ}} (y)$ for all x ≤ y, x, y ∈ L. This implies that $\underline{μ_{δ}}$ is a fuzzy filter over L, that is, μ is a lower soft rough fuzzy filter over L. □

Remark 4.12. The above theorems show that any soft rough fuzzy lattice (ideal, filter) is a generalization of a fuzzy lattice (ideal, filter) over lattices.

5 Soft rough fuzzy lattices (ideals, filters) over lattices with respect to another fuzzy soft set

In this section, we introduce the concept of approximations on an information system with respect to another information system, and lower and upper soft rough fuzzy lattices (ideals, filters) with respect to another fuzzy soft set.

Definition 5.1. Let $S = (F, A)$ be a soft set over L and δ : L → P (A) be a mapping defined as δ (x) = {a : x ∈ F (a)}. Let $T = (G, B)$ be a fuzzy soft set defined over L. The lower and upper soft rough approximations of $T$ with respect to $S$ are denoted by $\underline{(G, B)_{δ}} = (\underline{G_{δ}}, B)$ and $\bar{(G, B)_{δ}} = (\bar{G_{δ}}, B)$ , respectively, which are two operators defined as $\begin{matrix} \underline{G (e)_{δ}} (x) & = & ⋀ {G (e) (y) | y \in L, δ (y) = δ (x)}, \\ \bar{G (e)_{δ}} (x) & = & ⋁ {G (e) (y) | y \in L, δ (y) = δ (x)} \end{matrix}$ for all e ∈ B, x ∈ L.

If $\underline{(G, B)_{δ}} = \bar{(G, B)_{δ}}$ , then $T$ is called soft definable.

If $\underline{(G, B)_{δ}} \neq \bar{(G, B)_{δ}}$ and $\underline{G (e)_{δ}}$ $(\bar{G (e)_{δ}})$ is a fuzzy sublattice (ideal, filter) over L for all e ∈ B, then $T$ is called a lower (upper) soft rough fuzzy sublattice (ideal, filter) with respect to $S$ over L. Moreover, $T$ is called a lower (upper) soft rough fuzzy lattice (ideal, filter) with respect to $S$ over L if $\bar{G (e)_{δ}}$ and $\underline{G (e)_{δ}}$ are fuzzy sublattices (ideals, filter) with respect to $S$ over L for all e ∈ B.

Example 5.2. We consider the lattice L and soft set $S = (F, A)$ in Example 3.11. Define a soft set $T = (G, B)$ as the following Table 3. By calculating, $\underline{G (e_{1})_{δ}} = {\frac{0.2}{1}}$ , $\bar{G (e_{1})_{δ}} = {\frac{0.8}{0}, \frac{0.8}{b}, \frac{0.8}{1}}$ , $\underline{G (e_{2})_{δ}} = {\frac{0.1}{1}}$ , $\bar{G (e_{1})_{δ}} = {\frac{0.3}{0}, \frac{0.3}{b}, \frac{0.1}{1}}$ , $\underline{G (e_{3})_{δ}} = {\frac{0.3}{0}, \frac{0.3}{b}}$ , $\bar{G (e_{3})_{δ}} = {\frac{0.7}{0}, \frac{0.5}{a}, \frac{0.7}{b}, \frac{0.5}{0}}$ . It is easy to check that ${\underline{(G, B)}}_{δ}$ and ${\bar{(G, B)}}_{δ}$ are sublattices (ideas) over L for all e ∈ B. In other words, $T$ is a soft rough fuzzy lattice (ideal) with respect to $S$ over L.

Table 3
Soft set $S$

0 a b c 1

e ₁ 0.8 0 0 0 0.2

e ₂ 0 0 0 0.3 0.1

e ₃ 0.7 0.5 0.3 0 0

	0	a	b	c	1
e ₁	0.8	0	0	0	0.2
e ₂	0	0	0	0.3	0.1
e ₃	0.7	0.5	0.3	0	0

Definition 5.3. Let $T = (G, B)$ and $I = (H, C)$ be two fuzzy soft sets over L with D = B∩ C ≠ ∅. The ∨-operation and ∧-operation of $T \lor I$ and $T \land I$ are defined as $T \lor I = (G, B) \lor (H, C) = (K, D)$ and $T \land I = (G, B) \lor (H, C) = (L, D)$ , respectively, where K (a) = G (a) ∨ H (a) and L (a) = G (a) ∧ H (a) for all a ∈ D.

Proposition 5.4. Let $S = (F, A)$ be a C-soft set over L. Let $T_{1} = (G_{1}, B)$ and $T_{2} = (G_{2}, C)$ be two fuzzy soft sets over L with D = B∩ C ≠ ∅. Then

$\bar{(G_{1}, B)_{δ}} \lor \bar{(G_{2}, C)_{δ}} \subseteq \bar{(G_{1} \lor G_{2}, D)_{δ}} .$

$\bar{(G_{1}, B)_{δ}} \land \bar{(G_{2}, C)_{δ}} \subseteq \bar{(G_{1} \land G_{2}, D)_{δ}} .$

Proof. The proof is similar to that of Propositions 3.4 and 3.5. □

Proposition 5.5. Let $S = (F, A)$ be a CC-soft set over L. Let $T_{1} = (G_{1}, B)$ and $T_{2} = (G_{2}, C)$ be two fuzzy soft sets over L with D = B∩ C ≠ ∅. Then

$\bar{(G_{1}, B)_{δ}} \lor \bar{(G_{2}, C)_{δ}} = \bar{(G_{1} \lor G_{2}, D)_{δ}} .$

$\bar{(G_{1}, B)_{δ}} \land \bar{(G_{2}, C)_{δ}} = \bar{(G_{1} \land G_{2}, D)_{δ}} .$

Proof. The proof is similar to that of Propositions 3.7 and 3.8. □

Proposition 5.6. Let $S = (F, A)$ be a CC-soft set over L. Let $T_{1} = (G_{1}, B)$ and $T_{2} = (G_{2}, C)$ be two fuzzy soft sets over X with D = B∩ C ≠ ∅. Then

$\underline{(G_{1}, B)_{δ}} \lor \underline{(G_{2}, C)_{δ}} \subseteq \underline{(G_{1} \lor G_{2}, D)_{δ}} .$

$\underline{(G_{1}, B)_{δ}} \land \underline{(G_{2}, C)_{δ}} \subseteq \underline{(G_{1} \land G_{2}, D)_{δ}} .$

Proof. The proof is similar to that of Propositions 3.9 and 3.10. □

Finally, we investigate the lower and upper soft rough fuzzy sublattice (ideal, filter) with respect to a fuzzy soft set.

Proposition 5.7. Let $S = (F, A)$ be a C-soft set over L. If $T = (G, B)$ is a fuzzy soft sublattice over L, then $T = (G, B)$ is a upper soft rough fuzzy lattice with respect to $S$ over L.

Proof. The proof is similar to that of Proposition 4.3. □

Proposition 5.8. Let $S = (F, A)$ be a CC-soft set over L. If $T = (G, B)$ is a fuzzy soft sublattice over L, then $T = (G, B)$ is a lower soft rough fuzzy lattice with respect to $S$ over L.

Proof. The proof is similar to that of Proposition 4.4. □

Definition 5.9. Let $S = (F, A)$ be a C-soft set over L and δ : L → P (A) be a mapping defined as δ (x) = {a : x ∈ F (a)}. Then $S$ is called a BC-soft set over L if (G, B) is a fuzzy soft ideal over L and ∀a, b ∈ L, δ (a) = δ (b) if and only if there exist i₁, i₂ ∈ G (e) _t such that a ∨ i₁ = b ∨ i₂, where G (e) _t = {x ∈ L : G (e) (x) ≥ t}, for all e ∈ B, t ∈ [0, 1].

Theorem 5.10. Let $S = (F, A)$ be a BC-soft set over L. If $T = (G, B)$ is a fuzzy ideal over L, then $T$ is a upper soft rough fuzzy ideal over L.

Proof. The proof is similar to that of Proposition 4.6. □

Theorem 5.11. Let $S = (F, A)$ be a CC-soft set over L. If $T = (G, B)$ is a fuzzy soft ideal over L, then $T$ is a lower soft rough fuzzy ideal over L.

Proof. The proof is similar to that of Proposition 4.8. □

Definition 5.12. Let $S = (F, A)$ be a C-soft set over L and δ : L → P (A) be a mapping defined as δ (x) = {a : x ∈ F (a)}. Then $S$ is called a B′C-soft set over L if (G, B) is a fuzzy soft filter over L and ∀a, b ∈ L, δ (a) = δ (b) if and only if there exist i₁, i₂ ∈ G (e) _t such that a ∧ i₁ = b ∧ i₂, where G (e) _t = {x ∈ L : G (e) (x) ≥ t}, for all e ∈ B, t ∈ [0, 1].

Theorem 5.13. Let $S = (F, A)$ be a B′C-soft set over L. If $T = (G, B)$ is a fuzzy soft filter over L, then $T$ is a upper soft rough fuzzy filter over L.

Proof. The proof is similar to that of Proposition 4.10. □

Theorem 5.14. Let $S = (F, A)$ be a CC-soft set over L. If $T = (G, B)$ is a fuzzy soft filter over L, then $T$ is a lower soft rough fuzzy filter over L.

Proof. The proof is similar to that of Proposition 4.11. □

6 Conclusion

This paper aims at providing a framework to combine soft sets, rough sets, fuzzy sets with lattices all together, which propose the concept of soft rough fuzzy lattices (ideals, filters) over lattices. The main conclusions in this paper and the further work to do are listed as follows.

Some operations and fundamental properties of soft rough fuzzy sets over lattices are investigated, which can provide a theoretical research method for other algebraic structures.

By characterizations of lattices, we studied roughness in lattices with respect to soft approximation spaces. In particular, lower and upper soft rough fuzzy lattices (ideals, filters) over lattices have been investigated.

We introduce the concept of approximations on an information system with respect to another information system, lower and upper soft rough fuzzy lattices (ideals, filters) with respect to another fuzzy soft set are investigated.

As an extension of this work, the following topics maybe considered:

Constructing soft rough fuzzy sets to other algebras, such as hyperrings, BL-algebras and so on.

Studying soft fuzzy rough lattices.

Investigating decision making methods based on soft rough fuzzy lattices.

Footnotes

Acknowledgments

This research is partially supported by a grant of National Natural Science Foundation of China (Grant nos. 11571010 and 61179038).

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A new study on soft rough fuzzy lattices (ideals,filters) over lattices

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Some operations of soft rough fuzzy sets over lattices

Table 1 Soft set S 0 a b c d 1 e 1 1 1 1 0 1 1 e 2 0 0 0 1 1 1 e 3 1 1 0 1 1 1

5 Soft rough fuzzy lattices (ideals, filters) over lattices with respect to another fuzzy soft set

Table 3 Soft set S 0 a b c 1 e 1 0.8 0 0 0 0.2 e 2 0 0 0 0.3 0.1 e 3 0.7 0.5 0.3 0 0

Footnotes

Acknowledgments

References

Table 1
Soft set $S$

0 a b c d 1

e ₁ 1 1 1 0 1 1

e ₂ 0 0 0 1 1 1

e ₃ 1 1 0 1 1 1

Table 3
Soft set $S$

0 a b c 1

e ₁ 0.8 0 0 0 0.2

e ₂ 0 0 0 0.3 0.1

e ₃ 0.7 0.5 0.3 0 0