New generalized fuzzy soft rough approximations applied to fuzzy topological spaces

Abstract

In this paper, we introduce a generalized fuzzy soft rough model. A pair of new fuzzy soft rough approximations namely, fuzzy soft $I ℙ$ -lower approximation and fuzzy soft $I ℙ$ -upper approximation is proposed and their related properties are presented. Furthermore, we have succeeded to reduce the fuzzy soft boundary region comparing with already known in the literature. Finally, we obtain the structure of fuzzy soft ideal rough topologies induced by fuzzy soft sets and fuzzy ideals.

Keywords

Fuzzy soft sets Soft rough sets Fuzzy soft rough sets Generalized fuzzy soft rough approximations Fuzzy soft ideal rough topologies

1 Introduction

While probability theory, fuzzy set theory [33], rough set theory [21, 22] and other mathematical tools are well known and often useful approaches to describing uncertainty, each of these theories has its inherent difficulties as pointed out in [19, 20]. Rough set theory is a major mathematical method developed by Pawlak in [21, 22].

Rough set theory has attracted worldwide attention of many researchers and practitioners, who have contributed essentially to its development and applications. Rough set theory overlaps with many other theories. Despite this, rough set theory may be considered as an independent discipline in its own right. The rough set approach seems to be of fundamental importance in artificial intelligence and cognitivesciences, especially in research areas such as machine learning, intelligent systems, inductive reasoning, pattern recognition, image processing, signal analysis, knowledge discovery, decision analysis, and expert systems. Different kinds of generalizations of Pawlak’s rough set model were obtained in [26, 30 –32]. Rough set theory is expressible in terms of S-approximation spaces [9, 28], which is generalized in [27].

Fuzzy ideals in fuzzy topological spaces have been considered by Sarkar [6]. In 1999 Molodtsov [20], introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainties. In [16], Maji et al. introduced the notion of reduct-soft set and described the application of soft set theory to a decision-making problem using rough sets. Chen et al. [4], presented a new definition of soft set parameterization reduction, and compared this definition to the related concept of attributes reduction in rough set theory. Kong et al. [10], introduced the notion of normal parameter reduction of soft sets and constructed a reduction algorithm based on the importance degree of parameters. In [36], Zhan and Davvaz introduced a foundation for providing a rough soft tool in considering many problems that contain uncertainties. Also, they gave an approach to decision making problem based on soft fuzzy rough set model by analyzing the limitations and advantages in the existing literatures. Danjuma et al. [18], introduced a review on different parameter reduction and decision making techniques for soft set and hybrid soft sets under unpleasant set of hypothesis environment as well as performance analysis of the their derived algorithms.

Soft sets combined with fuzzy sets and rough sets were introduced in [7] by F. Feng in 2010. The notions of soft fuzzy rough sets and soft rough fuzzy sets were introduced in [17] in 2011. Sun et al. [29], proposed a new concept of soft fuzzy rough set by combining the fuzzy soft set with the traditional fuzzy rough set in 2014. Li et al. [12] in 2015, have investigated roughness of fuzzy soft sets. They introduced a pair of fuzzy soft rough approximations and studied some of their properties. Based on fuzzy soft rough approximations, the concept of fuzzy soft rough sets is introduced. New types of fuzzy soft sets such as full, intersection complete and union complete fuzzy soft sets are defined. In 2016 Beg et al. [2], introduced a modified soft fuzzy rough set model. It is shown that these new models of approximations are finer than those in [12, 17]. Zhan and Alcantud [34], introduced some different algorithms of parameter reduction based on some types of (fuzzy) soft sets. Ma et al. [13, 14], introduced some decision making methods based on (fuzzy) soft sets, rough soft sets and soft rough sets. Based on the idea in [7], Zhan et al. [37] applied rough soft sets to hemirings, and described some characterizations of rough soft hemirings, which is extended in [35, 38, 39]. Peng and Liu [23], constructed a new axiomatic definition of single-valued neutrosophic similarity measure and give a similarity formula. They introduced neutrosophic soft decision making methods based on EDAS, new similarity measure and level soft set, which is extended in [24]. Abd El-latif [1], generalized the soft rough set theory by using the ideal notion. Moreover, he presented some properties of the soft ideal rough approximation operators and introduced a new soft rough set model, which is an improvement of Feng’s model. He introduced the deviations of some properties of Pawlak’s approximation space and soft (ideal) rough approximation space. He succeeded to reduce the soft boundary region by increasing the lower approximation and decreasing the upperapproximation.

This paper is devoted to generalize the notion of soft fuzzy rough sets [2, 12, 17, 29], by using the fuzzy ideals. So, the fuzzy soft ideal rough approximation operators will be introduced and their basic properties will be given. The deviation between our model and previous models will be shown and clarified by examples and counterexamples. Furthermore, a new generalized fuzzy soft ideal rough approximation will be given. Based on this new approximation, we will generate a structure of fuzzy topologies.

2 Preliminaries

In this section, we will recall some notions and properties of fuzzy sets, rough sets, fuzzy soft sets and fuzzy topological spaces.

Definition 2.1. [33] A fuzzy set A of a non-empty set X is characterized by a membership function μ_A : X ⟶ [0, 1] = I whose value μ_A (x) represents the "degree of membership" of x in A for x ∈ X. We denote family of all fuzzy sets by I^X, the fuzzy set which satisfies $\bar{1} (x) = 1$ for each x ∈ X by $\bar{1}$ and the fuzzy set which satisfies $\bar{0} (x) = 0$ for each x ∈ X by $\bar{0}$ . If A, B ∈ I^X, then some basic set operations for fuzzy sets are given by Zadeh [33], as follows:

A ⊆ B ⇔ μ_A (x) ≤ μ_B (x) ∀ x ∈ X.

A = B ⇔ μ_A (x) = μ_B (x) ∀ x ∈ X.

C = A ∪ B ⇔ μ_C (x) = μ_A (x) ∨ μ_B (x) ∀ x ∈ X.

D = A ∩ B ⇔ μ_D (x) = μ_A (x) ∧ μ_B (x) ∀ x ∈ X.

$M = A^{c} \Leftrightarrow μ_{M} (x) = \bar{1} - μ_{A} (x) \forall x \in X$ .

N = A - B ⇔ μ_N (x) = μ_A (x) ∧ (1 - μ_B (x)) ∀ x ∈ X.

Definition 2.2. [25] A fuzzy set is called a fuzzy point in X, if it takes the value 0 for each x ∈ X except one, say, x ∈ X. If its value at x is λ (λ ∈ (0, 1]), we denote this fuzzy point by x_λ, where the point x is called its support and λ is called its height. For a fuzzy point x_λ and A ∈ I^U, we define x_λ ∈ A if and only if λ ≤ A (x).

Definition 2.3. [3] Let τ ⊆ I^X. Then, τ is called a fuzzy topology on X if

$\bar{1}, \bar{0} \in τ$ ,

the union of any members of τ belongs to τ,

the intersection of any two members of τ belongs to τ.

The pair (X, τ) is called a fuzzy topological space over X. Also, each member A of τ is called fuzzy open in X and its related complement A^c is called a fuzzy closed set in X.

Definition 2.4. [6] A non-empty collection $I$ of fuzzy subsets of a set X is called a fuzzy ideal on X, if it satisfies the following conditions:

$A \in I$ and $B \in I \Rightarrow A \cup B \in I$ ,

$A \in I$ and $B \subseteq A \Rightarrow B \in I$ .

Definition 2.5. [5] Let τ ⊆ I^X. Then, τ is called a generalized fuzzy topology on X if

$\bar{0} \in τ$ ,

the union of any members of τ belongs to τ.

The pair (X, τ) is called a generalized fuzzy topological space over X. Also, each member A of τ is called a generalized fuzzy open in X and its related complement A^c is called a generalized fuzzy closed set in X.

Definition 2.6. [20] Let X be an initial universe and E be a set of parameters. Let $P (X)$ denote the power set of X and A be a non-empty subset of E. A pair (F, A) is called a soft set over X, where F is a mapping given by $F : A \to P (X)$ .

Definition 2.7. [15] Let A ⊆ E. A pair (f, A), denoted by f_A, is called a fuzzy soft set over X, where f is a mapping given by f : A → I^X defined by $f_{A} (e) = μ_{f_{A}}^{e}$ , where $μ_{f_{A}}^{e} = \bar{0}$ if e ∉ A and $μ_{f_{A}}^{e} \neq \bar{0}$ if e ∈ A. We denote family of all fuzzy soft sets byFSS (X) _E.

Definition 2.8. [15] The complement of a fuzzy soft set (f, A), denoted by (f, A) ^c, is defined by (f, A) ^c = (f^c, A), $f_{A}^{c} : E \to I^{X}$ is a mapping given by $μ_{f_{A}^{c}}^{e} = \bar{1} - μ_{f_{A}}^{e} \forall e \in A$ .

Definition 2.9. [15] A fuzzy soft set f_A over X is called a null fuzzy soft set, denoted by ${\tilde{0}}_{A}$ , if for each $e \in A, f_{A} (e) = \bar{0}$ . Also, f_A is called absolute fuzzy soft set, denoted by ${\tilde{1}}_{A}$ , if for each $e \in A, f_{A} (e) = \bar{1}$ .

Definition 2.10. [15] Let f_A, g_B ∈ FSS (X) _E. Then, f_A is fuzzy soft subset of g_B, denoted by f_A ⊆ g_B, if A ⊆ B and $μ_{f_{A}}^{e} (x) \leq μ_{g_{B}}^{e} (x) for each x \in X and e \in A$ .

Definition 2.11. [15]. The union of two fuzzy soft sets f_A and g_B over the common universe X is also a fuzzy soft set h_C, where C = A ∪ B and for all e ∈ C, $h_{C} (e) = μ_{h_{c}}^{e} = μ_{f_{A}}^{e} \lor μ_{g_{B}}^{e} \forall e \in C$ . Here, we write h_C = f_A ∪ g_B.

Definition 2.12. [15]. The intersection of two fuzzy soft sets f_A and g_B over the common universe X is also a fuzzy soft set h_C, where C = A ∩ B and for all e ∈ C, $h_{C} (e) = μ_{h_{c}}^{e} = μ_{f_{A}}^{e} \land μ_{g_{B}}^{e} \forall e \in C$ . Here, we write h_C = f_A ∩ g_B.

Definition 2.13. [8, 12] Let f_E be a fuzzy soft set over X. Then, f_E is called:

Full fuzzy soft set, if $⋃_{e \in E} f (e) = \bar{1}$ ,

Intersection complete fuzzy soft set, if for each e₁, e₂ ∈ E, there exists e₃ ∈ E such that f (e₃) = f (e₁) ∩ f (e₂).

3 Fuzzy soft ideal rough approximation operators

3.1 Some properties of generalized fuzzy soft rough model

In this section, we will generalize the fuzzy soft rough model by using the fuzzy ideal notion. Also, we will present some properties of fuzzy soft ideal rough approximation operators, and introduce a new fuzzy soft rough set model, which is an improvement of previous models [12, 17, 29].

Definition 3.1. [12] Let f_E be a fuzzy soft set over U, u ∈ U, e ∈ E and A ∈ I^U. Denote

${\underline{A}}_{u} = {λ \in I : \exists e \in E such that u_{λ} \in f (e) \subseteq A}$ (1)

$\begin{matrix} {\bar{A}}_{u} & = & {λ \in I : \exists e \in E such that u_{λ} \in f (e) \\ and f (e) \cap A \neq 0} \end{matrix}$ (2)

${\underline{A}}_{u}^{e} = {λ \in I : u_{λ} \in f (e) \subseteq A}$ (3)

${\bar{A}}_{u}^{e} = {λ \in I : u_{λ} \in f (e) and f (e) \cap A \neq 0}$ (4)

${\underline{A}}^{e} (u) = \sup {\underline{A}}_{u}^{e}, {\bar{A}}^{e} (u) = \sup {\bar{A}}_{u}^{e}$ (5)

Definition 3.2. [12] Let f_E be a fuzzy soft set over U. Then, the pair P = (U, f_E) is called a fuzzy soft approximation space. Based on the fuzzy soft approximation space P, we define the following two operations: ${\underline{apr}}_{P} (A) (u) = \sup {\underline{A}}_{u}, u \in U$ , ${\bar{apr}}_{P} (A) (u) = \sup {\bar{A}}_{u}, u \in U$ , assigning to every A ∈ I^U two sets ${\underline{apr}}_{P} (A)$ and ${\bar{apr}}_{P} (A)$ , which are called the fuzzy soft P-lower approximation and the fuzzy soft P-upper approximation of U, respectively. In general, we refer to ${\underline{apr}}_{P} (A)$ and ${\bar{apr}}_{P} (A)$ as fuzzy soft rough approximations of U with respect to P. Moreover, the sets $P_{osP} (A) = {\underline{apr}}_{P} (A)$ , $N_{egP} (A) = \bar{1} - {\bar{apr}}_{P} (A)$ , $B_{ndP} (A) = {\bar{apr}}_{P} (A) - {\underline{apr}}_{P} (A),$ are called the soft P-positive region, the soft P-negative region and the soft P-boundary region of A, respectively. If ${\bar{apr}}_{P} (A) = {\underline{apr}}_{P} (A)$ , then A is said to be a fuzzy soft P-definable. Otherwise, A is called a fuzzy soft P-rough set. Consider the collection $τ_{P} = {A \in I^{U} : {\underline{apr}}_{P} (A) = A}$ . Then, τ_P is a generalized fuzzy topology on U.

Definition 3.3. Let f_E be a fuzzy soft set over U, $I$ be a fuzzy ideal on U and P = (U, f_E) be a fuzzy soft approximation space. Based on the fuzzy soft approximation space P and the fuzzy ideal $I$ , we define the following: $\begin{matrix} {\underline{A}}_{I u} & = & {λ \in I : \exists e \in E such that u_{λ} \in f (e) \\ and f (e) \cap A^{c} \in I} \end{matrix}$ (6) $\begin{matrix} {\bar{A}}_{I u} & = & {λ \in I : \exists e \in E such that u_{λ} \in f (e) \\ and f (e) \cap A \notin I} \end{matrix}$ (7) ${\underline{A}}_{I u}^{e} = {λ \in I : u_{λ} \in f (e) and f (e) \cap A^{c} \in I}$ (8) ${\bar{A}}_{I u}^{e} = {λ \in I : u_{λ} \in f (e) and f (e) \cap A \notin I}$ (9) ${\underline{A}}_{I}^{e} (u) = \sup {\underline{A}}_{I u}^{e}, {\bar{A}}_{I}^{e} (u) = \sup {\bar{A}}_{I u}^{e}$ (10)

Proposition 3.4. Let f_E be a fuzzy soft set over U, $I$ be a fuzzy ideal on U and P = (U, f_E) be a fuzzy soft approximation space. Then, for each A, B ∈ I^U, we have

${\underline{A}}_{I u} = ⋃_{e \in E} {\underline{A}}_{I u}^{e}$ .

${\bar{A}}_{I u} = ⋃_{e \in E} {\bar{A}}_{I u}^{e}$ .

If A ⊆ B, then ${\underline{A}}_{I u} \subseteq {\underline{B}}_{I u}$ .

If A ⊆ B, then ${\bar{A}}_{I u} \subseteq {\bar{B}}_{I u}$ .

$(\bar{A \cup B})_{I u} = {\bar{A}}_{I u} \cup {\bar{B}}_{I u}$ .

$(\underline{A \cap B})_{I u} \subseteq {\underline{A}}_{I u} \cap {\underline{B}}_{I u}$ , the equality holds if f_E is an intersection complete fuzzy soft set.

${\underline{A}}_{I u} ⊈ {\bar{A}}_{I u}$ and ${\bar{A}}_{I u} ⊈ {\underline{A}}_{I u}$ , in general.

Proof. (1)-(2) Obvious from Definition 3.3. (3) If ${\underline{A}}_{I u} = \bar{0}$ , then we get the proof. If ${\underline{A}}_{I u} \neq \bar{0}$ . Let $λ \in {\underline{A}}_{I u}$ . Then, u_λ ∈ I^U, u_λ ∈ f (e) and $f (e) \cap A^{c} \in I$ for some e ∈ E. Since A ⊆ B, B^c ⊆ A^c. It follows, B^c ∩ f (e) ⊆ A^c ∩ f (e). Since $f (e) \cap A^{c} \in I$ , $f (e) \cap B^{c} \in I$ from Definition 2.4. Therefore, $λ \in {\underline{B}}_{I u}$ and hence ${\underline{A}}_{I u} \subseteq {\underline{B}}_{I u}$ . (4) If ${\bar{A}}_{I u} = \bar{0}$ , then we get the proof. If ${\bar{A}}_{I u} \neq \bar{0}$ . Let $λ \in {\bar{A}}_{I u}$ . Then, u_λ ∈ I^U, u_λ ∈ f (e) and $f (e) \cap A \notin I$ for some e ∈ E. Since A ⊆ B, A ∩ f (e) ⊆ B ∩ f (e). Now, $f (e) \cap A \notin I$ and A ∩ f (e) ⊆ B ∩ f (e) implies $f (e) \cap B \notin I$ from Definition 2.4. Therefore, $λ \in {\bar{B}}_{I u}$ . Hence, ${\bar{A}}_{I u} \subseteq {\bar{B}}_{I u}$ . (5) Let $λ \in (\bar{A \cup B})_{I u}$ . Then, u_λ ∈ I^U and there exists e ∈ E such that u_λ ∈ f (e) and $f (e) \cap (A \cup B) \notin I$ . It follows, either $f (e) \cap A \notin I$ or $f (e) \cap B \notin I$ . Hence, $λ \in {\bar{A}}_{I u}$ or $λ \in {\bar{B}}_{I u}$ implies $λ \in {\bar{A}}_{I u} \cup {\bar{B}}_{I u}$ . The other inclusion direct from (4). (6) $(\underline{A \cap B})_{I u} \subseteq {\underline{A}}_{I u} \cap {\underline{A}}_{I u}$ clear from (3). Now, let f_E be an intersection complete fuzzy soft set and ${\underline{A}}_{I u} \cap {\underline{B}}_{I u} \neq \bar{0}$ . Let $λ \in {\underline{A}}_{I u} \cap {\underline{B}}_{I u}$ . Then, $λ \in {\underline{A}}_{I u}$ and $λ \in {\underline{B}}_{I u}$ . So, there exist e₁, e₂ ∈ E such that u_λ ∈ I^U, u_λ ∈ f (e₁), $f (e_{1}) \cap A^{c} \in I$ and u_λ ∈ f (e₂), $f (e_{2}) \cap B^{c} \in I$ implies λ ≤ (f (e₁) ∧ f (e₂)) (u) = (f (e₁) ∩ f (e₂)) (u). Since f_E is an intersection complete soft set, there exists e ∈ E such that $f (e) = f (e_{1}) \cap f (e_{2})$ (11) Also, from Definition 2.4, $[[f (e_{1}) \cap A^{c}] \cup [f (e_{2}) \cap B^{c}]] \in I$ implies $[[f (e_{1}) \cup [f (e_{2}) \cap B^{c}]] \cap [A^{c} \cup [f (e_{2}) \cap B^{c}]]] \in I$ . So, $f (e_{1}) \cap [[f (e_{2}) \cup A^{c}] \cap [A^{c} \cup B^{c}]] \in I$ and then $f (e_{1}) \cap [f (e_{2}) \cap [A^{c} \cup B^{c}]] \in I$ . Thus, $[f (e_{1}) \cap f (e_{2})] \cap [f (e_{1}) \cap [A^{c} \cup B^{c}]] \in I$ . It follows, $[[f (e_{1}) \cap f (e_{2})] \cap [A^{c} \cup B^{c}]] \in I$ and hence $[[f (e_{1}) \cap f (e_{2})] \cap [A \cap B]^{c}] \in I$ . From (11), $[f (e)] \cap [A \cap B]^{c}] \in I$ . Hence, $λ \in {\underline{(A \cap B)}}_{I u}$ . (7) Example 3.6 shall clear our claim.

Remark 3.5. In the above proposition, the equality in part (6) and the inclusion in part (7) don’t hold in general, as shall be shown in the following examples.

Examples 3.6. (1) Let U = {a, b, c, d}, E = {e₁, e₂} and f_E be a fuzzy soft set over U given as a table [Table 1] in the following form.

Table 1

Tabular representation of the fuzzy soft set f_E

E/U	a	b	c	d
e ₁	0.2	1	0.1	0
e ₂	0	1	0.2	1

Let A, B ∈ I^U be two fuzzy sets over U defined as follows: A = {a₀, b₁, c₀, d₁} and B = {a_0.6, b_0.9, c₁, d₀}. Consider the fuzzy ideal

I

over U defined as follows:

I = {\bar{0}, {a_{0.2}, b_{0.1}, c_{0}, d_{0}}, {a_{0.2}, b_{0}, c_{0}, d_{0}}, {a_{0.1}, b_{0.1}, c_{0}, d_{0}}, {a_{0.1}, b_{0}, c_{0}, d_{0}}, {a_{0}, b_{0.1}, c_{0}, d_{0}}, {a_{0.2}, b_{0}, c_{0.1}, d_{0}}, {a_{0.1}, b_{0}, c_{0.1}, d_{0}}, {a_{0}, b_{0}, c_{0.1}, d_{0}}, {a_{0.2}, b_{0.1}, c_{0.1}, d_{0}}, {a_{0}, b_{0.1}, c_{0.1}, d_{0}}, {a_{0.1}, b_{0.1}, c_{0.1}, d_{0}}}

. Thus, we have

f (e_{1}) \cap A^{c} = {a_{0.2}, b_{0}, c_{0.1}, d_{0}} \in I

and

f (e_{2}) \cap A^{c} = {a_{0}, b_{0}, c_{0.2}, d_{0}} \notin I

f (e_{1}) \cap B^{c} = {a_{0.2}, b_{0.1}, c_{0}, d_{0}} \in I

and

f (e_{2}) \cap B^{c} = {a_{0}, b_{0.1}, c_{0}, d_{1}} \notin I

f (e_{1}) \cap (A \cap B)^{c} = {a_{0.2}, b_{0.1}, c_{0.1}, d_{0}} \notin I

and

f (e_{2}) \cap (A \cap B)^{c} = {a_{0}, b_{0.1}, c_{0.2}, d_{1}} \notin I

. It follows,

{\underline{A}}_{I a} = {\underline{A}}_{I a}^{e_{1}} = {λ \in (0, 1] : λ \leq f (e_{1}) (a)} = (0, 0.2]

{\underline{B}}_{I a} = {\underline{B}}_{I a}^{e_{1}} = {λ \in (0, 1] : λ \leq f (e_{1}) (a)} = (0, 0.2]

{\underline{(A \cap B)}}_{I a} = \bar{0}

. Hence,

(\underline{A \cap B})_{I a} = \bar{0} ⊉ {\underline{B}}_{I a} \cap {\underline{A}}_{I a} = (0, 0.2]

. (2) Let U = {a, b, c, d}, E = {e₁, e₂} and f_E be a fuzzy soft set over U given as a table [Table 2] in the following form.

Table 2

Tabular representation of the fuzzy soft set f_E

E/U	a	b	c	d
e ₁	0.2	0	0.1	0
e ₂	0.1	0	0.1	0

Let A ∈ I^U be a fuzzy set over U defined as follows: A = {a_0.5, b_0.6, c_0.4, d_0.7}. Consider the fuzzy ideal

I

over U defined as follows:

I = {\bar{0}, {a_{0.2}, b_{0}, c_{0.1}, d_{0}}, {a_{0.2}, b_{0}, c_{0}, d_{0}}, {a_{0.1}, b_{0}, c_{0}, d_{0}}, {a_{0.1}, b_{0}, c_{0.1}, d_{0}}, {a_{0}, b_{0}, c_{0.1}, d_{0}}}

. Thus, we have

f (e_{1}) \cap A^{c} = f (e_{1}) = {a_{0.2}, b_{0}, c_{0.1}, d_{0}} \in I

and

f (e_{2}) \cap A^{c} = f (e_{2}) = {a_{0.1}, b_{0}, c_{0.1}, d_{0}} \in I

f (e_{1}) \cap A = f (e_{1}) = {a_{0.2}, b_{0}, c_{0.1}, d_{0}} \in I

and

f (e_{2}) \cap A = f (e_{2}) = {a_{0.1}, b_{0}, c_{0.1}, d_{0}} \in I

. It follows,

{\underline{A}}_{I u} = {(a, (0, 0.2])

(b, \bar{0})

,(c, (0, 0.1]),

(d, \bar{0})}

and

{\bar{A}}_{I u} = \bar{0}

. Hence,

{\underline{A}}_{I u} ⊈ {\bar{A}}_{I u}

. Also, if we consider the fuzzy ideal

J

over U as follows:

J = {\bar{0}, {a_{0.1}

,b₀,c₀, d₀}, {a_0.2,b₀, c₀,d₀}}. So, we can get

{\bar{A}}_{J u} = {(a

,(0, 0.2]),

(b, \bar{0})

,(c, (0, 0.1]),

(d, \bar{0})}

and

{\underline{A}}_{J u} = \bar{0}

. Thus

{\bar{A}}_{J u} ⊈ {\underline{A}}_{J u}

Theorem 3.7. Let f_E be a fuzzy soft set over U, $I, J$ be two fuzzy ideals on U and P = (U, f_E) be the corresponding fuzzy soft approximation space. If $I \subseteq J$ , then for each A ∈ I^U, we have:

${\bar{A}}_{J u} \subseteq {\bar{A}}_{I u}$ .

${\underline{A}}_{I u} \subseteq {\underline{A}}_{J u}$ .

Proof. (1) Let ${\bar{A}}_{J u} \neq \bar{0}$ and $λ \in {\bar{A}}_{J u}$ . Then, u_λ ∈ I^U, u_λ ∈ f (e) and $f (e) \cap A \notin J$ for some e ∈ E. Since $I \subseteq J$ , $f (e) \cap A \notin I$ . Hence, $λ \in {\bar{A}}_{I u}$ . Therefore, ${\bar{A}}_{J u} \subseteq {\bar{A}}_{I u}$ . (2) Let ${\underline{A}}_{I u} \neq \bar{0}$ and $λ \in {\underline{A}}_{I u}$ . Then, u_λ ∈ I^U, u_λ ∈ f (e) and $f (e) \cap A^{c} \in I$ for some e ∈ E. Since $I \subseteq J$ , $f (e) \cap A^{c} \in I$ . Therefore, $λ \in {\bar{A}}_{J u}$ . Hence, ${\underline{A}}_{I u} \subseteq {\underline{A}}_{J u}$ .

Definition 3.8. P = (U, f_E) be a fuzzy soft approximation space, we define the following two operations: ${\underline{apr}}_{I ℙ} (A) (u) = \sup {\underline{A}}_{I u}, u \in U$ , ${\bar{apr}}_{I ℙ} (A) (u) = \sup {\bar{A}}_{I u}, u \in U$ , assigning to every A ∈ I^U two sets ${\underline{apr}}_{I ℙ} (A)$ and ${\bar{apr}}_{I ℙ} (A)$ , which are called the fuzzy soft $I ℙ$ -lower approximation and the fuzzy soft $I ℙ$ -upper approximation of U, respectively. If ${\bar{apr}}_{I ℙ} (A) = {\underline{apr}}_{I ℙ} (A)$ , then A is said to be a fuzzy soft $I ℙ$ -definable. Otherwise, A is called a fuzzy soft $I ℙ$ -rough set. In general, we refer to ${\underline{apr}}_{I ℙ} (A)$ and ${\bar{apr}}_{I ℙ} (A)$ as a generalized fuzzy soft rough approximations of U with respect to P and $I$ .

Proposition 3.9. Let f_E be a fuzzy soft set over U, $I$ be a fuzzy ideal on U and P = (U, f_E) be a fuzzy soft approximation space. Then, for each A ∈ I^U, we have (1) If $I = {\bar{0}}$ , then ${\bar{apr}}_{I ℙ} (A) = {\bar{apr}}_{P} (A)$ and ${\underline{apr}}_{I ℙ} (A) = {\underline{apr}}_{P} (A)$ . (2) If $I = I^{U}$ , then ${\bar{apr}}_{I ℙ} (A) = \bar{0}$ and ${\underline{apr}}_{I ℙ} (A) = ⋃_{e \in E} f (e)$ .

Proof. Straightforward.

Proposition 3.10. Let f_E be a fuzzy soft set over U, $I$ be a fuzzy ideal on U and P = (U, f_E) be a fuzzy soft approximation space. Then, for each A ∈ I^U, we have (1) ${\underline{apr}}_{I ℙ} (A) = ⋃_{e \in E} {\underline{A}}_{I u}^{e} = ⋃ {f (e) : f (e) \cap A^{c} \in I}$ . (2) ${\bar{apr}}_{I ℙ} (A) = ⋃_{e \in E} {\bar{A}}_{I u}^{e} = ⋃ {f (e) : f (e) \cap A \notin I}$ .

Proof. Obvious from Proposition 3.4.

Theorem 3.11. Let f_E be a fuzzy soft set over U, $I$ be a fuzzy ideal on U and P = (U, f_E) be the corresponding fuzzy soft approximation space. Then, for each A ∈ I^U, we have: (1) ${\bar{apr}}_{I ℙ} (\bar{1}) \subseteq {\underline{apr}}_{I ℙ} (\bar{1}) = ⋃_{e \in E} f (e) \subseteq \bar{1}$ . (2) ${\bar{apr}}_{I ℙ} (\bar{0}) = \bar{0} \neq {\underline{apr}}_{I ℙ} (\bar{0})$ . (3) If A ⊆ B, then ${\underline{apr}}_{I ℙ} (A) \subseteq {\underline{apr}}_{I ℙ} (B)$ . (4) If A ⊆ B, then ${\bar{apr}}_{I ℙ} (A) \subseteq {\bar{apr}}_{I ℙ} (B)$ . (5) ${\underline{apr}}_{I ℙ} (A \cap B) \subseteq {\underline{apr}}_{I ℙ} (A)$ and ${\underline{apr}}_{I ℙ} (A \cap B) \subseteq {\underline{apr}}_{I ℙ} (B)$ . (6) ${\bar{apr}}_{I ℙ} (A \cup B) \supseteq {\bar{apr}}_{I ℙ} (A)$ and ${\bar{apr}}_{I ℙ} (A \cup B) \supseteq {\bar{apr}}_{I ℙ} (B)$ . (7) If $I \in I$ , then ${\bar{apr}}_{I ℙ} (I) = \bar{0}$ .

Proof. (1) Follows from Proposition 3.10. (2) Immediate. (3) Let u ∈ U and assume that ${\underline{A}}_{I u} \neq \bar{0}$ . Then, for all $λ \in {\underline{A}}_{I u}$ , $λ \leq \sup {\underline{B}}_{I u} = {\underline{apr}}_{I ℙ} (B) (u)$ from Proposition 3.4 (3). It follows, ${\underline{apr}}_{I ℙ} (A) (u) = \sup {\underline{A}}_{I u} \leq {\underline{apr}}_{I ℙ} (B) (u)$ . Hence, ${\underline{apr}}_{I ℙ} (A) \subseteq {\underline{apr}}_{I ℙ} (B) .$ (4) It is similar to (3). (5) Follows from (3). (6) Follows from (4). (7) Let $I \in I$ . Then, ${\bar{apr}}_{I ℙ} (I) = ⋃ {f (e) : f (e) \cap I \notin I}$ . Since $I \in I$ , $f (e) \cap I \in I$ from Definition 2.4. Hence, ${\bar{apr}}_{I ℙ} (I) = \bar{0}$ .

Theorem 3.12. Let f_E be a fuzzy soft set over U, $I$ be a fuzzy ideal on U and P = (U, f_E) be the corresponding fuzzy soft approximation space. Then, for each A ∈ I^U, we have: (1) ${\bar{apr}}_{I ℙ} (A \cup B) = {\bar{apr}}_{I ℙ} (A) \cup {\bar{apr}}_{I ℙ} (B)$ . (2) ${\underline{apr}}_{I ℙ} (A \cup B) \supseteq {\underline{apr}}_{I ℙ} (A) \cup {\underline{apr}}_{I ℙ} (B)$ . (3) ${\underline{apr}}_{I ℙ} (A \cap B) \subseteq {\underline{apr}}_{I ℙ} (A) \cap {\underline{apr}}_{I ℙ} (B)$ . (4) ${\bar{apr}}_{I ℙ} (A \cap B) \supseteq {\bar{apr}}_{I ℙ} (A) \cap {\bar{apr}}_{I ℙ} (B)$ . (5) $A ⊈ {\bar{apr}}_{I ℙ} (A)$ , in general.

Proof. (1) Let u ∈ U. If ${\bar{A}}_{I u} \neq \bar{0}$ . Let $λ \in {\bar{A \cup B}}_{I u}$ . Since $(\bar{A \cup B})_{I u} = {\bar{A}}_{I u} \cup {\bar{B}}_{I u}$ from Proposition 3.4 (5), $λ \in {\bar{A}}_{I u} \cup {\bar{A}}_{I u}$ . So, $λ \in {\bar{A}}_{I u}$ or $λ \in {\bar{B}}_{I u}$ . It follows, $λ \leq \sup {\bar{A}}_{I u} = {\bar{apr}}_{I ℙ} (A) (u)$ or $λ \leq \sup {\bar{B}}_{I u} = {\bar{apr}}_{I ℙ} (B) (u)$ implies $λ \leq {\bar{apr}}_{I ℙ} (A) (u) \lor {\bar{apr}}_{I ℙ} (B) (u)$ = $[{\bar{apr}}_{I ℙ} (A) \cup {\bar{apr}}_{I ℙ} (B)] (u)$ . Therefore, ${\bar{apr}}_{I ℙ} (A \cup B) (u) = \sup \bar{(} A \cup B)_{I u} \leq [{\bar{apr}}_{I ℙ} (A) \cup {\bar{apr}}_{I ℙ} (B)] (u)$ . Hence, ${\bar{apr}}_{I ℙ} (A \cup B)$ $\subseteq {\bar{apr}}_{I ℙ} (A)$ $\cup {\bar{apr}}_{I ℙ} (B)$ . The other inclusion direct from Theorem 3.11(6). [(2)-(4)] The proof is similar to (1). (5) Examples 3.14 (3) shall clear our claim.

Remark 3.13. In the above theorem, the equality in parts (2), (4) and the inclusion in part (5) don’t hold in general, as shall be shown in the followingexamples. Examples 3.14. (1) Let U = {a, b, c, d}, E = {e₁, e₂} and f_E be a fuzzy soft set over U given as a table [Table 3] in the following form.

Table 3

Tabular representation of the fuzzy soft set f_E

E/U	a	b	c	d
e ₁	0.3	0	0.5	0
e ₂	0	0.4	0.7	0

Let A, B ∈ I^U be two fuzzy sets over U defined as follows: A = {a₀, b₁, c₀, d₁} and B = {a₁, b₀, c₁, d₀}. Consider the fuzzy ideal

I

over U defined as follows:

I = {\bar{0}, {a_{0}, b_{0.1}, c_{0}, d_{0}}

, {a₀,b_0.2,c₀,d₀}, {a₀,b_0.3,c₀,d₀}}. Thus, we have

f (e_{1}) \cap A^{c} = {a_{0.3}, b_{0}, c_{0.5}, d_{0}} \notin I

and

f (e_{2}) \cap A^{c} = {a_{0}, b_{0.4}, c_{0.7}, d_{0}} \notin I

f (e_{1}) \cap B^{c} = {a_{0.3}, b_{0}, c_{0.5}, d_{0}} \notin I

and

f (e_{2}) \cap B^{c} = {a_{0}, b_{0.4}, c_{0.7}, d_{0}} \notin I

f (e_{1}) \cap (A \cup B)^{c} = \bar{0} \in I

and

f (e_{2}) \cap (A \cup B)^{c} = \bar{0} \in I

. It follows,

{\underline{A}}_{I c} = \bar{0} = {\underline{B}}_{I c}

{\underline{(A \cup B)}}_{I c} = {\underline{(A \cup B)}}_{I c}^{e_{1}} \cup {\underline{(A \cup B)}}_{I c}^{e_{2}} = {λ \in (0, 1] : λ \leq f (e_{1}) (c)} \cup {λ \in (0, 1] : λ \leq f (e_{2}) (c)} = (0, 0.5] \cup (0, 0.7] = (0, 0.7]

. So,

{\underline{apr}}_{I ℙ} (A) (c) = \sup {\underline{A}}_{I c} = 0 = {\underline{apr}}_{I ℙ} (B) (c)

and so

[{\underline{apr}}_{I ℙ} (A) \cup {\underline{apr}}_{I ℙ} (B)] (c) = 0

. Also

{\underline{apr}}_{I ℙ} (A \cup B) (c) = \sup {\underline{(A \cup B)}}_{I c} = 0.7

. Hence,

{\underline{apr}}_{I ℙ} (A \cup B) ⊈ s {\underline{apr}}_{I ℙ} (A) \cup {\underline{apr}}_{I ℙ} (B)

. (2) In (1), consider the fuzzy ideal

I

over U defined as follows:

I = {\bar{0}, {a_{0.3}, b_{0}, c_{0}, d_{0}}, {a_{0.1}, b_{0}, c_{0}, d_{0}}, {a_{0.2}, b_{0}, c_{0}, d_{0}}}

. Thus, we have

f (e_{1}) \cap A = {a_{0}, b_{1}, c_{0}, d_{0}} \notin I

and

f (e_{2}) \cap A = {a_{0}, b_{1}, c_{0}, d_{1}} \notin I

f (e_{1}) \cap B = {a_{0.2}, b_{0}, c_{0.1}, d_{0}} \notin I

and

f (e_{2}) \cap B = {a_{0}, b_{1}, c_{0.2}, d_{0}} \notin I

f (e_{1}) \cap (A \cap B) = \bar{0} \in I

and

f (e_{2}) \cap (A \cap B) = \bar{0} \in I

. It follows,

{\bar{A}}_{I b} = (0, 0.4]

{\bar{B}}_{I b} = (0, 0.4]

and

{\bar{(A \cap B)}}_{I b} = \bar{0}

. So,

{\bar{apr}}_{I ℙ} (A) (b) = \sup {\bar{(A)}}_{I b} = 0.4 = {\bar{apr}}_{I ℙ} B (b)

and so

[{\bar{apr}}_{I ℙ} (A) \cap {\bar{apr}}_{I ℙ} (B)] (b) = 0.4

. Also,

{\bar{apr}}_{I ℙ} (A \cap B) (b) = \sup {\bar{(A \cap B)}}_{I b} = 0

. Hence,

{\bar{apr}}_{I ℙ} (A \cap B) ⊉ {\bar{apr}}_{I ℙ} (A) \cap {\bar{apr}}_{I ℙ} (B)

. (3) Let U = {a, b, c, d}, E = {e₁, e₂} and f_E be a fuzzy soft set over U given as a table [Table 4] in the following form.

Table 4

Tabular representation of the fuzzy soft set f_E

E/U	a	b	c	d
e ₁	0	0.3	0	0.5
e ₂	0.3	0.9	0.1	0

Let A ∈ I^U be a fuzzy set over U defined as follows: A = {a_0.6, b₀, c_0.5, d_0.1}. Consider the fuzzy ideal

I

over U defined as follows:

I = {\bar{0}, {a_{0}, b_{0}, c_{0}, d_{0.1}}}

. Thus, we have

f (e_{1}) \cap A = {a_{0}, b_{0}, c_{0}, d_{0.1}} \in I

and

f (e_{2}) \cap A = {a_{0.3}, b_{0}, c_{0.1}, d_{0}} \notin I

, It follows,

{\bar{A}}_{I a} = {\bar{A}}_{I a}^{e_{2}} = {λ \in (0, 1] : λ \leq f (e_{2}) (a)} = (0, 0.3]

and so

{\bar{apr}}_{I ℙ} (A) (a) = \sup {\bar{A}}_{I a} = 0.3 < A (a) = 0.6

. Hence,

A ⊈ {\bar{apr}}_{I ℙ} (A)

Theorem 3.15. Let f_E be a fuzzy soft set over U, $I, J$ be two fuzzy ideals on U and P = (U, f_E) be the corresponding fuzzy soft approximation space. If $I \subseteq J$ , then for each A ∈ I^U, we have: (1) ${\bar{apr}}_{J} (A) \subseteq {\bar{apr}}_{I ℙ} (A)$ . (2) ${\underline{apr}}_{I ℙ} (A) \subseteq {\underline{apr}}_{J} (A)$ .

Proof. (1) Let u ∈ U, ${\bar{A}}_{J u} \neq \bar{0}$ and $λ \in {\bar{A}}_{J u}$ . Then, u_λ ∈ I^U, u_λ ∈ f (e) and $f (e) \cap A \notin J$ for some e ∈ E. By Theorem 3.7 (1), $λ \leq \sup {\bar{A}}_{I u} = {\bar{apr}}_{I ℙ} (A) (u)$ . It follows, ${\bar{apr}}_{J ℙ} (A) (u) = \sup {\bar{A}}_{J u} \leq {\underline{apr}}_{I ℙ} (A) (u)$ . Hence, ${\bar{apr}}_{J ℙ} (A) \subseteq {\bar{apr}}_{I ℙ} (A)$ . (2) It is similar to (1).

Theorem 3.16. Let f_E be a fuzzy soft set over U, $I$ be a fuzzy ideal on U and P = (U, f_E) be the corresponding fuzzy soft approximation space. Then, for each A ∈ I^U, we have: (1) ${\bar{apr}}_{I ℙ} (A) \subseteq {\bar{apr}}_{I ℙ} ({\bar{apr}}_{I ℙ} (A))$ . (2) ${\underline{apr}}_{I ℙ} (A) \subseteq {\underline{apr}}_{I ℙ} ({\underline{apr}}_{I ℙ} (A))$ . (3) ${\bar{apr}}_{I ℙ} (A) \subseteq {\underline{apr}}_{I ℙ} ({\bar{apr}}_{I ℙ} (A))$ . (4) ${\underline{apr}}_{I ℙ} (A) \subseteq ({\bar{apr}}_{I ℙ} (A^{c}))^{c}$ . (5) $[{\underline{apr}}_{I ℙ} (A)]^{c} \subseteq {\bar{apr}}_{I ℙ} (A^{c})$ . (6) $[{\bar{apr}}_{I ℙ} (A)]^{c} \subseteq {\underline{apr}}_{I ℙ} (A^{c})$ .

Proof. (1) Let ${\bar{apr}}_{I ℙ} (A) = B$ and u_λ ∈ B. It follows, $\exists e \in E such that u_{λ} \in f (e) and f (e) \cap A \notin I$ . From Proposition 3.10 (2), $B = ⋃ {f (e) : f (e) \cap A \notin I}$ . It follows, $f (e) \cap B = f (e) \notin I$ from Definition 2.4. Therefore, $u_{λ} \in {\bar{apr}}_{I ℙ} (B)$ and hence $B \subseteq {\bar{apr}}_{I ℙ} (B)$ . (2) Let ${\underline{apr}}_{I ℙ} (A) = B$ and u_λ ∈ B. So, $\exists e \in E such that u_{λ} \in f (e) and f (e) \cap A^{c} \in I$ . From Proposition 3.10 (1), $B = ⋃ {f (e) : f (e) \cap A^{c} \in I}$ . It follows, f (e) ⊆ B and so $f (e) \cap B^{c} \in I$ . Hence, $u_{λ} \in {\underline{apr}}_{I ℙ} (B)$ and so $B \subseteq {\underline{apr}}_{I ℙ} (B)$ . (3) Let ${\bar{apr}}_{I ℙ} (A) = B$ and u_λ ∈ B. It follows, $\exists e \in E such that u_{λ} \in f (e) and f (e) \cap A \notin I$ . From Proposition 3.10 (2), $B = ⋃ {f (e) : f (e) \cap A \notin I}$ . It follows, f (e) ⊆ B and so $f (e) \cap B^{c} \in I$ . Hence, $u_{λ} \in {\underline{apr}}_{I ℙ} (B)$ and so $B \subseteq {\underline{apr}}_{I ℙ} (B)$ . (4) $({\bar{apr}}_{I ℙ} (A^{c}))^{c} = [⋃ {u \in U : \exists e \in E such that u \in f (e) and f (e) \cap A^{c} \notin I}]^{c} = ⋂ {u \in U : \exists e \in E such that u \in f (e) and f (e) \cap A^{c} \in I} \subseteq ⋃ {u \in U : \exists e \in E such that u \in f (e) and f (e) \cap A^{c} \in I} = {\underline{apr}}_{I ℙ} (A)$ . [(5)-(6)] It is similar to (4).

Remark 3.17. In the above theorem, the equality in parts (2-3) don’t hold in general, as shall be shown in the following example. Examples 3.18. Let U = {a, b, c, d}, E = {e₁, e₂} and f_E be a fuzzy soft set over U given as a table [Table 5] in the following form.

Table 5

Tabular representation of the fuzzy soft set f_E

E/U	a	b	c	d
e ₁	0.2	0.1	0	0
e ₂	0.1	0.1	0	0

Let A ∈ I^U be a fuzzy set over U defined as follows: A = {a_0.1, b₀, c₀, d₀}. Consider the fuzzy ideal

I

over U defined as follows:

I = {\bar{0}, {a_{0.2}, b_{0.1}, c_{0}, d_{0}}, {a_{0.1}, b_{0.1}, c_{0}, d_{0}}, {a_{0.2}, b_{0}, c_{0}, d_{0}}, {a_{0.1}, b_{0}, c_{0}, d_{0}}, {a_{0}, b_{0.1}, c_{0}, d_{0}}}

. Thus, we have It follows,

{\underline{apr}}_{I ℙ} (A) = {a_{0.2}, b_{0.1}, c_{0}, d_{0}} ⊈ A = {a_{0.1}, b_{0}, c_{0}, d_{0}}

and so

{\underline{apr}}_{I ℙ} (A) ⊉ {\underline{apr}}_{I ℙ} ({\underline{apr}}_{I ℙ} (A))

and

{\bar{apr}}_{I ℙ} (A) ⊉ {\underline{apr}}_{I ℙ} ({\bar{apr}}_{I ℙ} (A))

Proposition 3.19. Let f_E be a fuzzy soft set over U, $I$ be a fuzzy ideal on U and P = (U, f_E) be the corresponding fuzzy soft approximation space. If $I = {\bar{0}}$ , then the following are equivalent: (1) f_E is full fuzzy soft set. (2) For any A ∈ I^U, $A \subseteq {\bar{apr}}_{I ℙ} (A)$ . (3) ${\bar{apr}}_{I ℙ} (\bar{1}) = \bar{1}$ . (4) ${\underline{apr}}_{I ℙ} (\bar{1}) = \bar{1}$ .

Proof. Obvious.

3.2 Deviations of some properties of previous fuzzy soft approximations and the current approximations

In this section, we list the deviations of some properties of fuzzy soft approximations [12, 17, 29] and the current approximations as follows. Let f_E be a fuzzy soft set over U,

I

be a fuzzy ideal on U and P = (U, f_E) be the corresponding fuzzy soft approximation space. Then, for each A, B ∈ I^U. Thus, we notice the following deviations of our approximation comparing with previous approximations. (1)

{\bar{apr}}_{I ℙ} (\bar{1}) ⊉ {\underline{apr}}_{I ℙ} (\bar{1}) \neq \bar{1}

. (2)

{\underline{apr}}_{I ℙ} (\bar{0}) \neq \bar{0}

. (3)

{\underline{apr}}_{I ℙ} (A \cap B) ⊉ {\underline{apr}}_{I ℙ} (A) \cap {\underline{apr}}_{I ℙ} (B)

. (4)

{\underline{apr}}_{I ℙ} (A) ⊈ {\bar{apr}}_{I ℙ} (A)

and

{\bar{apr}}_{I ℙ} (A) ⊈ {\underline{apr}}_{I ℙ} (A)

. (5)

{\underline{apr}}_{I ℙ} (A) ⊈ {\bar{apr}}_{I ℙ} ({\bar{apr}}_{I ℙ} (A))

. (6)

{\underline{apr}}_{I ℙ} (A) ⊈ {\bar{apr}}_{I ℙ} ({\underline{apr}}_{I ℙ} (A))

. The following counterexamples support our claims about the above deviations. Examples 3.20. [(1)-(2)] Let U = {a, b, c, d}, E = {e₁, e₂} and f_E be a fuzzy soft set over U given as a table [Table 6] in the following form.

Table 6
Tabular representation of the fuzzy soft set f_E

E/U	a	b	c	d
e ₁	0	0.1	0.2	0
e ₂	0.1	0.3	0	0.4

Consider the fuzzy ideal

I

over U defined as follows:

I = {\bar{0}, {a_{0}, b_{0.1}, c_{0.2}, d_{0}}, {a_{0}, b_{0.1}, c_{0}, d_{0}}, {a_{0}, b_{0.1}, c_{0.1}, d_{0}}, {a_{0}, b_{0}, c_{0.1}, d_{0}}, {a_{0}, b_{0}, c_{0.2}, d_{0}}}

. Thus, we have

f (e_{1}) \cap {\bar{1}}^{c} \in I

and

f (e_{2}) \cap {\bar{1}}^{c} \in I

f (e_{1}) \cap \bar{1} = f (e_{1}) \in I

and

f (e_{2}) \cap \bar{1} = f (e_{2}) \notin I

. It follows,

{\underline{\bar{1}}}_{I c} = {\underline{\bar{1}}}_{I c}^{e_{1}} \cup {\underline{\bar{1}}}_{I c}^{e_{2}} = {λ \in (0, 1] : λ \leq f (e_{1}) (c)} \cup {λ \in (0, 1] : λ \leq f (e_{2}) (c)} = \bar{0} \cup (0, 0.2] = (0, 0.2]

and so

{\underline{apr}}_{I ℙ} (\bar{1}) (c) = \sup {\underline{\bar{1}}}_{I c} = \sup (0, 0.2] = 0.2

. Also,

{\bar{\bar{1}}}_{I c} = {\bar{\bar{1}}}_{I c}^{e_{2}} = {λ \in (0, 1] : λ \leq f (e_{2}) (c)} = \bar{0}

and so

{\bar{apr}}_{I ℙ} (\bar{1}) (c) = \sup {\underline{\bar{1}}}_{I c} = 0

. Hence,

{\bar{apr}}_{I ℙ} (\bar{1}) ⊉ {\underline{apr}}_{I ℙ} (\bar{1}) ⊉ \bar{1}

. Also, we notice that

{\underline{apr}}_{I ℙ} (\bar{0}) = {a_{0}, b_{0.1}, c_{0.2}, d_{0}} \neq \bar{0}

. (3) In Examples 3.6 (1), we have

{\bar{apr}}_{I ℙ} (A) (a) = \sup {\bar{A}}_{I a} = \sup (0, 0.2] = 0.2 = {\bar{apr}}_{I ℙ} (B) (a)

and so

[{\underline{apr}}_{I ℙ} (A) \cap {\bar{apr}}_{I ℙ} (B)] (a) = 0.2

. Also,

{\underline{apr}}_{I ℙ} (A \cap B) (a) = 0

. Hence,

{\underline{apr}}_{I ℙ} (A \cap B) ⊉ {\underline{apr}}_{I ℙ} (A) \cap {\bar{apr}}_{I ℙ} (B)

. (4) In Examples 3.6 (2),

{\underline{apr}}_{I ℙ} (A) (a) = \sup (0, 0.2] = 0.2

and

{\bar{apr}}_{I ℙ} (A) (a) = 0

. Hence,

{\underline{apr}}_{I ℙ} (A) ⊈ {\bar{apr}}_{I ℙ} (A)

. Also,

{\underline{apr}}_{J} (A) (c) = 0

and

{\bar{apr}}_{J} (A) (c) = 0.1

. Therefore,

{\bar{apr}}_{I ℙ} (A) ⊈ {\underline{apr}}_{I ℙ} (A)

. (5) In Examples 3.6 (2), we have

{\underline{apr}}_{I ℙ} (A) = {a_{0.2}, b_{0}, c_{0.1}, d_{0}}

and

{\bar{apr}}_{I ℙ} ({\bar{apr}}_{I ℙ} (A)) = \bar{0}

. Thus,

{\underline{apr}}_{I ℙ} (A) = {a_{0.2}, b_{0.1}, c_{0}, d_{0}} ⊈ \bar{0} = {\bar{apr}}_{I ℙ} ({\bar{apr}}_{I ℙ} (A))

. (6) Let U = {a, b, c, d}, E = {e₁, e₂} and f_E be a fuzzy soft set over U given as a table [Table 7] in the following form.

Table 7

Tabular representation of the fuzzy soft set f_E

E/U	a	b	c	d
e ₁	0	0.3	0	0.2
e ₂	0.3	0.2	0.1	0

Let A ∈ I^U be a fuzzy set over U defined as follows: A = {a_0.2, b_0.1, c₀, d₀}. Consider the fuzzy ideal

I

over U defined as follows:

I = {\bar{0}, {a_{0}, b_{0.3}, c_{0}, d_{0.2}}, {a_{0}, b_{0.3}, c_{0}, d_{0.1}}, {a_{0}, b_{0.3}, c_{0}, d_{0}}, {a_{0}, b_{0.2}, c_{0}, d_{0.2}}, {a_{0}, b_{0.2}, c_{0}, d_{0.1}}, {a_{0}, b_{0.2}, c_{0}, d_{0}}, {a_{0}, b_{0.1}, c_{0}, d_{0}}, {a_{0}, b_{0.3}, c_{0}, d_{0.2}}, {a_{0}, b_{0}, c_{0}, d_{0.2}}, {a_{0}, b_{0}, c_{0}, d_{0.1}}, {a_{0}, b_{0.1}, c_{0}, d_{0.1}}, {a_{0}, b_{0.1}, c_{0}, d_{0.2}}}

. Thus, we have

f (e_{1}) \cap A^{c} = {a_{0}, b_{0.3}, c_{0}, d_{0.2}} \in I

and

f (e_{2}) \cap A^{c} = {a_{0.3}, b_{0.2}, c_{0.1}, d_{0}} \notin I

. Thus, we have

{\underline{apr}}_{I ℙ} (A) = {a_{0}, b_{0.3}, c_{0}, d_{0.2}}

and

{\bar{apr}}_{I ℙ} ({\underline{apr}}_{I ℙ} (A)) = \bar{0}

. Hence,

{\underline{apr}}_{I ℙ} (A) ⊈ {\bar{apr}}_{I ℙ} ({\underline{apr}}_{I ℙ} (A))

4 New approaches of generalized fuzzy soft rough approximations

4.1 Fuzzy soft rough topological spaces

In this section, we introduce and study the concepts of fuzzy soft rough topology and some of its properties based on the notions of the fuzzy soft P-lower approximation and the fuzzy soft P-upper approximation

Proposition 4.1. Let f_E be a fuzzy soft set over U, A ∈ I^U and P = (U, f_E) be the corresponding fuzzy soft approximation space. Then, the collection $τ_{SR} (A) = {\bar{1}, \bar{0}, {\bar{apr}}_{P} (A), {\underline{apr}}_{P} (A), B_{ndP} (A)}$ (12) satisfies the axioms of Definition 2.3 and hence it forms a fuzzy topology on U called as the fuzzy soft rough topology on U w.r.t A.

Proof. Immediate.

Example 4.2. Let U = {a, b, c, d}, E = {e₁, e₂} and f_E be a fuzzy soft set over U given as a table [Table 8] in the following form.

Table 8

Tabular representation of the fuzzy soft set f_E

E/U	a	b	c	d
e ₁	0	0.3	0	0.5
e ₂	0.3	0.9	0.1	0

Let A ∈ I^U be a fuzzy set over U defined as follows: A = {a_0.6, b₀, c_0.5, d_0.1}. Thus, we have $f (e_{1}) \cap A = {a_{0}, b_{0}, c_{0}, d_{0.1}} \neq \bar{0}$ and $f (e_{2}) \cap A = {a_{0.3}, b_{0}, c_{0.1}, d_{0}} \neq \bar{0}$ , f (e₁) ⊈ A and f (e₂) ⊈ A

It follows, ${\bar{apr}}_{I ℙ} (A) = {a_{0.3}, b_{0.9}, c_{0.1}, d_{0.5}}$ , ${\underline{apr}}_{I ℙ} (A) = \bar{0}$ and B_ndP (A) = {a_0.3, b_0.9, c_0.1, d_0.5}. Hence, $τ_{SR} (A) = {\bar{1}, \bar{0}, {a_{0.3}, b_{0.9}, c_{0.1}, d_{0.5}}}$ is a fuzzy soft rough topology on U w.r.t A.

Proposition 4.3. Let τ_SR (A) be a fuzzy soft rough topology on U w.r.t A. Then, the collection $β_{SR} (A) = {\bar{1}, {\underline{apr}}_{P} (A), B_{ndP} (A)}$ (13) forms a base for τ_SR (A).

Proof. (1) $⋃_{G \in β_{SR}} G = \bar{1}$ . (2) let $\bar{1}, {\underline{apr}}_{P} (A) \in β_{SR}$ . Then, for each $u_{λ} \in \bar{1} \cap {\underline{apr}}_{P} (A)$ , we have $u_{λ} \in {\underline{apr}}_{P} (A) \subseteq \bar{1} \cap {\underline{apr}}_{P} (A)$ . By a similar way if $\bar{1}, B_{ndP} (A) \in β_{SR}$ or ${\underline{apr}}_{P} (A), B_{ndP} (A) \in β_{SR}$ and for each $u_{λ} \in \bar{1} \cap B_{ndP} (A)$ or $u_{λ} \in {\underline{apr}}_{P} (A) \cap B_{ndP} (A)$ , we have, $u_{λ} \in B_{ndP} (A) \subseteq \bar{1} \cap B_{ndP} (A)$ and $u_{λ} \in {\underline{apr}}_{P} (A) \subseteq {\underline{apr}}_{P} (A) \cap B_{ndP} (A)$ . Hence, β_SR (A) is a base for τ_SR (A).

Proposition 4.4. Let (U, τ_SR (A), E) and (V, τ_SR (B), E) be two fuzzy soft rough topologies on U w.r.t A and V w.r.t B, respectively. If β_1SR and β_2SR are the bases for τ_SR (A) and τ_SR (B), respectively such that β_1SR ⊆ β_2SR, then τ_SR (A) ⊆ τ_SR (B).

Proof. Let G ∈ τ_SR (A). Then, G = ⋃ _{H∈β_1SR}H. Since β_1SR ⊆ β_2SR, G = ⋃ _{H∈β_2SR}H. Hence, G ∈ τ_SR (B). Therefore, τ_SR (A) ⊆ τ_SR (B).

Definition 4.5. Let (U, τ_SR (A), E) be a fuzzy soft rough topological space w.r.t. G ∈ I^U. The fuzzy soft rough interior of each G is defined as the union of all fuzzy soft rough open subsets of G and it is denoted by Fsint_R (G). Also, the fuzzy soft rough closure of G is defined as the intersection of all fuzzy soft rough closed subsets containing G and it is denoted by Fscl_R (G). i.e ${Fscl}_{R} (G) = G \cup {\bar{apr}}_{P} (G)$ .

4.2 Fuzzy soft ideal rough topological spaces induced by new generalized fuzzy soft rough approximations

In this section, we introduce new generalized fuzzy soft approximations. Furthermore, we show that the new generalized fuzzy soft approximations are generalizations to those in [12, 17, 29]. Also, we use this new approximation to generate fuzzy topologies named fuzzy soft ideal rough topologies. These fuzzy topologies are finer than fuzzy topologies generated by fuzzy soft approximationsin [12].

Definition 4.6. Let U be the universe, A ∈ I^U and $I$ be a fuzzy ideal on U. The fuzzy soft upper approximation of A is defined by $R_{I}^{*} (A) = A \cup {\bar{apr}}_{I ℙ} (A)$ (14) and the fuzzy soft lower approximation is defined by: $R_{* I} (A) = ⋃_{e \in E} {f (e) : f (e) \cap A^{c} \in I}$ (15)

Moreover, the universe set can be divided into three disjoint regions using the fuzzy soft lower and fuzzy soft upper approximations as follows: $P_{os I ℙ} (A) = R_{* I} (A)$ , $N_{eg I ℙ} (A) = \bar{1} - R_{I}^{*} (A)$ , $B_{nd I ℙ} (A) = R_{I}^{*} (A) - R_{* I} (A)$ .

are called the fuzzy soft $I ℙ$ -positive region, the fuzzy soft $I ℙ$ -negative region and the fuzzy soft $I ℙ$ -boundary region of A, respectively.

Proposition 4.7. Let f_E be a fuzzy soft set over U, $I$ be a fuzzy ideal on U and P = (U, f_E) be a fuzzy soft approximation space. Then, for each A ∈ I^U, we have (1) If $I = {\bar{0}}$ , then $R_{I}^{*} (A) = {\bar{apr}}_{P} (A)$ and $R_{* I} (A) = {\underline{apr}}_{P} (A)$ . (2) If $I = I^{U}$ , then $R_{I}^{*} (A) = A$ .

Proof. Follows from Proposition 3.9.

Corollary 4.8. Let f_E be a fuzzy soft set over U, $I$ be a fuzzy ideal on U and P = (U, f_E) be a fuzzy soft approximation space. If $I = {\bar{0}}$ , then the collection $τ_{P} = {A \in I^{U} : {\underline{apr}}_{I ℙ} (A) = A}$ is a generalized fuzzy topology on U [5]. If f_E is full fuzzy soft set and intersection complete, then τ_P is a fuzzy topology on U.

Proof. Follows from Proposition 4.7 and [Theorem 4.1, [12]].

Theorem 4.9. Let U be the universe, $I$ and $J$ be two fuzzy ideals on U. If $I \subseteq J$ , then $B_{nd J P} (A) \subseteq B_{nd I ℙ} (A)$ for each A ∈ I^U.

Proof. Since $I \subseteq J$ , ${\bar{apr}}_{Jℙ} (A) \subseteq {\bar{apr}}_{I ℙ} (A)$ and ${\underline{apr}}_{I ℙ} (A) \subseteq {\underline{apr}}_{Jℙ} (A)$ from Theorem 3.15. It follows, $R_{J}^{*} (A) \subseteq R_{I}^{*} (A)$ and $[R_{* J} (A)]^{c} \subseteq [R_{* J} (A)]^{c}$ . Thus, $R_{J}^{*} (A) \cap [R_{* J} (A)]^{c} \subseteq R_{I}^{*} (A) \cap [R_{* I} (A)]^{c}$ . Hence, $B_{nd J P} (A) \subseteq B_{nd I ℙ} (A)$ .

Proposition 4.10. Let f_E be a fuzzy soft set over U, $I$ be a fuzzy ideal on U and P = (U, f_E) be the corresponding fuzzy soft approximation space. For each A, B ∈ I^U, the fuzzy soft lower and fuzzy soft upper approximations defined by (10) and (11), respectively satisfy the following properties: (1) $R_{I}^{*} ({\bar{0}}) = {\bar{0}}$ . (2) $A \subseteq R_{I}^{*} (A)$ . (3) If A ⊆ B, then $R_{I}^{*} (A) \subseteq R_{I}^{*} (B)$ . (4) $R_{I}^{*} (A \cup B) = R_{I}^{*} (A) \cup R_{I}^{*} (B)$ . (5) $R_{I}^{*} (R_{I}^{*} (A)) = R_{I}^{*} (A)$ . (6) $R_{I}^{*} (A \cap B) \subseteq R_{I}^{*} (A) \cap R_{I}^{*} (B)$ . (7) If A ⊆ B, then $R_{* I} (A) \subseteq R_{* I} (B)$ . (8) If $I \in I$ , then $R_{I}^{*} (I) = I$ . (9) If $I^{c} \in I$ , then $R_{* I} (I) = ⋃_{e \in E} f (e)$ .

Proof. Obvious from Theorem 3.11 and Theorem 3.12.

Corollary 4.11. On account of Theorem 4.10, the fuzzy soft upper approximation (10) satisfies Kuratowski’s axioms [11] and induces a fuzzy topology on U, denoted by $τ_{R_{I}^{*}}$ , called the fuzzy soft ideal rough topology, given as follows: $τ_{R_{I}^{*}} = {A \in I^{U} : R_{I}^{*} (A^{c}) = A^{c}} .$ (16)

In such case, the fuzzy soft rough closure of A w.r.t. $I$ , denoted by ${Fscl}_{R_{I}^{*}} (A)$ , is identical with $R_{I}^{*} (A)$ (10).

Proposition 4.12. Let f_E be a fuzzy soft set over U, $I$ be a fuzzy ideal on U and P = (U, f_E) be the corresponding fuzzy soft approximation space. If $I = I^{U}$ , then $τ_{R_{I}^{*}}$ is the fuzzy discrete topology on U.

Proof. Follows from Proposition 4.7.

Theorem 4.13. Let $(U, τ_{R_{I}^{*}}, E)$ be a fuzzy soft ideal rough topological space. Then, ${Fscl}_{R_{I}^{*}} (A) \subseteq {Fscl}_{R} (A)$ and $τ_{SR} \subseteq τ_{R_{I}^{*}}$ .

Proof. Let $u_{λ} \in {Fscl}_{R_{I}^{*}} (A)$ . Then, u_λ ∈ A or $u_{λ} \in {\bar{apr}}_{I ℙ} (A)$ from (10). It follows, $λ \in {\bar{A}}_{I u}$ and so $\exists e \in E such that u_{λ} \in f (e) and f (e) \cap A \notin I$ . Thus, $f (e) \cap A \neq \bar{0}$ and so $λ \in {\bar{A}}_{u}$ . Hence, $u_{λ} \in A \cup {\bar{apr}}_{P} (A) = {Fscl}_{R} (A)$ . Therefore, ${Fscl}_{R_{I}^{*}} (A) \subseteq {Fscl}_{R} (A)$ .

On account of Postposition 3.9 Definition 4.6 and Postposition 4.7, we have the following corollary. This corollary shows that, our approximation succeeds to reduce the boundary region comparing with [12, 17, 29], as shall be shown in Example 4.15.

Corollary 4.14. Let f_E be a fuzzy soft set over U, $I$ be a fuzzy ideal on U and P = (U, f_E) be a fuzzy soft approximation space. Then, for each A ∈ I^U, we have (1) ${\bar{apr}}_{I ℙ} (A) \subseteq {\bar{apr}}_{P} (A)$ . (2) ${\underline{apr}}_{P} (A) \subseteq {\underline{apr}}_{I ℙ} (A)$ . (3) $P_{osP} (A) \subseteq P_{os I ℙ} (A)$ . (4) $N_{eg I ℙ} (A) \subseteq N_{egP} (A)$ . (5) $B_{nd I ℙ} (A) \subseteq B_{ndP} (A)$ .

Example 4.15. Let U = {a, b, c, d}, E = {e₁, e₂} and f_E be a fuzzy soft set over U given as a table [Table 9] in the following form.

Table 9

Tabular representation of the fuzzy soft set f_E

E/U	a	b	c	d
e ₁	0.5	0.7	0.1	0.6
e ₂	0.1	0.8	0.7	0.3

Consider the fuzzy ideal

I = I_{W}

over U defined as follows:

I_{W} = {S \in I^{U} : S \subseteq W}

, where W = {a_0.6, b_0.8, c_0.7, d_0.6}. Therefore, the following table [Table 10] shows that our method reduced the fuzzy soft boundary comparing with [12, 17, 29] methods, for arbitrary fuzzy sets A ∈ I^U.

Table 10

Tabular representation of the comparison between [12, 17, 29] methods and the present method

A	[12, 17, 29] models	our model	[12, 17, 29] models	our model	[12, 17, 29] models	our model
	${\underline{a p r}}_{P} (A)$	$R_{* II} (A)$	${\bar{a p r}}_{P} (A)$	$R_{II}^{*} (A)$	B_ndP(A)	$B_{n d II P} (A)$
$\bar{1}$	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.5,b_0.8,c_0.7,d_0.6}	$\bar{1}$	{a_0.5,b_0.2,c_0.3,d_0.4}	{a_0.5,b_0.2,c_0.3,d_0.4}
$\bar{0}$	$\bar{0}$	{a_0.5,b_0.8,c_0.7,d_0.6}	$\bar{0}$	$\bar{0}$	$\bar{0}$	$\bar{0}$
{a_0.2,b_0.3,c_0.4,d_0.5}	$\bar{0}$	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.2,b_0.3,c_0.4,d_0.5}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.2,b_0.3,c_0.4,d_0.5}
{a_0.5,b_0.7,c_0.2,d_0.6}	{a_0.5,b_0.7,c_0.1,d_0.6}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.5,b_0.7,c_0.2,d_0.6}	{a_0.5,b_0.3,c_0.7,d_0.4}	{a_0.5,b_0.2,c_0.2,d_0.4}
{a_0.1,b_0.4,c_0.3,d_0.6}	$\bar{0}$	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.1,b_0.4,c_0.3,d_0.6}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.1,b_0.2,c_0.3,d_0.4}
{a_0.2,b_0.3,c_0.5,d_0.6}	$\bar{0}$	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.2,b_0.3,c_0.5,d_0.6}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.2,b_0.3,c_0.4,d_0.5}
{a_0.2,b_0.8,c_0.7,d_0.4}	{a_0.1,b_0.8,c_0.7,d_0.3}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.2,b_0.8,c_0.7,d_0.4}	{a_0.5,b_0.2,c_0.3,d_0.6}	{a_0.2,b_0.2,c_0.3,d_0.4}
{a_0.3,b_0.2,c_0.4,d_0.1}	$\bar{0}$	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.3,b_0.2,c_0.4,d_0.1}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.3,b_0.2,c_0.3,d_0.1}
{a_0.7,b_0.2,c_0.8,d_0.9}	$\bar{0}$	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.7,b_0.2,c_0.8,d_0.9}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.5,b_0.2,c_0.3,d_0.4}
{a₁,b_0.8,c₁,d_0.7}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a₁,b_0.8,C₁,d_0.7}	{a_0.5,b_0.2,c_0.3,d_0.4}	{a_0.5,b_0.2,c_0.3,d_0.4}
{a_0.8,b_0.3,c_0.7,d_0.5}	$\bar{0}$	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.8,b_0.3,c_0.7,d_0.5}	{a_0.5,b_0.8,c_0.7,d_0.6}	{a_0.5,b_0.2,c_0.3,d_0.4}

5 Conclusion

In this paper, we present a generalized fuzzy soft rough approximations by using fuzzy topological spaces. Meanwhile, the structure of fuzzy soft ideal rough topologies induced by fuzzy soft sets and fuzzy ideals is discussed in detail. The interrelation between our model and previous models [12, 17, 29] are studied and the related results are compared. In future, we will try to apply the presented results in various fields such as covering-based fuzzy soft lower and upper approximation operators and the future research will be undertaken in this direction.

Conflict of interest

The author declares that he has no conflict of interest.

Footnotes

Acknowledgements

The author express his sincere thanks to the anonymous referees, the Editor-in-Chief and managing editors for their valuable suggestions and comments.

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New generalized fuzzy soft rough approximations applied to fuzzy topological spaces

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Fuzzy soft ideal rough approximation operators

3.1 Some properties of generalized fuzzy soft rough model

Table 6 Tabular representation of the fuzzy soft set f E

4.1 Fuzzy soft rough topological spaces

Conflict of interest

Footnotes

Acknowledgements

References

Table 6
Tabular representation of the fuzzy soft set f_E