Fuzzy rough sets with a fuzzy ideal

In this paper, we defined the fuzzy upper, fuzzy lower, and fuzzy boundary sets of a rough fuzzy set λ in a fuzzy approximation space (X , R). Based on λ and R, we introduced the fuzzy ideal approximation interior operator intlambdaR and the fuzzy ideal approximation closure operator clλ. We joined the fuzzy ideal notion with the fuzzy approximation spaces, and then introduced the fuzzy ideal approximation closure and interior operators associated to a rough fuzzy set λ. Fuzzy ideal approximation connectedness and the fuzzy ideal approximation continuity between fuzzy ideal approximation spaces are introduced.


Introduction
The notion of rough sets was given by Pawlak [1] referring to uncertainty of intelligent systems characterized by insufficient and incomplete information. Basically, rough sets are defined depending on some equivalence relation R on a universal finite set X. The pair (X, R) was called an approximation space based on an equivalence relation imposed on X.
In any approximation space, the notions of lower approximation, upper approximation, and boundary region operators of some subset could be induced. Many types of generalizations of Pawlak's rough set has been obtained by replacing equivalence relation with an arbitrary binary relation. On the other hand, the relationships between rough sets and topological spaces were studied in [2]. A lot of fuzzy generalizations of rough approximation have been proposed in the literature [3][4][5][6][7]. Irfan in [8] studied the connections between fuzzy set, rough set, and soft set ( [9]) notions. Many papers studied the relationship between fuzzy rough set notion and fuzzy topologies [10,11]. Recently, many researchers have used topological approaches in the study of rough sets and its applications. In [12], it was used the notion of ideal in soft rough ordinary topological space, and in [13], the authors introduced fuzzy soft connectedness in sense of Chang [14].
The motivation of this paper is to introduce a new improved fuzzy lower and fuzzy upper sets by which we get a more reliable fuzzy boundary region set of a fuzzy set λ. From these fuzzy lower and fuzzy upper sets, we can define new fuzzy interior and fuzzy closure operators associated with a specific fuzzy set λ ∈ I X in sense of Chang [14] and that fuzzy Ibedou and  relation R on X. In the fuzzy approximation space (X, R), based on these fuzzy interior and fuzzy closure operators, we defined fuzzy approximation connectedness. Defining a fuzzy ideal on X generates a fuzzy ideal approximation space in which a fuzzy local function was defined and many results are proved. Connectedness in fuzzy ideal approximation spaces are defined and compared with connectedness in fuzzy approximation spaces. Also, fuzzy ideal approximation continuity between two fuzzy ideal approximation spaces were discussed. The author in [12] defined ordinary lower and upper sets but regardless to any relation on X. Liu in [6] introduced the fuzzy lower and fuzzy upper sets and his computations were different from our results. Through the paper, let X be a finite set of objects and I the closed unit interval [ 0, 1]. I X denotes all the fuzzy subsets of X, and λ c (x) = 1 − λ(x) ∀x ∈ X, ∀λ ∈ I X . A constant fuzzy set t for all t ∈ I is defined by t(x) = t ∀x ∈ X. Infimum and supremum of a fuzzy set λ ∈ I X are given as: inf λ = x∈X λ(x) and sup λ = x∈X λ(x).

Assume a fuzzy relation
That is, R is a fuzzy equivalence relation on X. (X, R) is called a fuzzy approximation space based on the fuzzy equivalence relation R on X.

Definition 1
For each x ∈ X, define a fuzzy coset [x] : X → I by: All elements y ∈ X with fuzzy relation value R(x, y) > 0 are elements having a membership value in the fuzzy coset [x], and any element y ∈ X with R(x, y) = 0 is not included in the fuzzy coset [x]. Any fuzzy coset [x] surely include the element x ∈ X, and consequently Note that [x] = 0 ∀x ∈ X since there is at least x ∈ X itself such that [x] (x) = 1, while may be all elements z ∈ X are given such that [x] (z) > 0 ∀z ∈ X. The fuzzy cosets could be such that [x] (x) = 1 and [x] (z) = 0 ∀z = x, which means (X, R) is fuzzy partitioned into completely disjoint fuzzy cosets. Putting I = {0, 1} as a crisp case, we get exactly the usual meaning of partitioning of a set X based on an ordinary equivalence relation R on X.
Recall that the fuzzy difference between two fuzzy sets was defined ( [15]) as: (2)

Fuzzy lower, fuzzy upper, and fuzzy boundary region sets
Definition 2 Let λ ∈ I X and R a fuzzy equivalence relation on X and the fuzzy cosets are defined as in (1). Then, the fuzzy lower set λ R , the fuzzy upper set λ R and the fuzzy boundary region set λ B are defined as follows: λ R , λ R and λ B are then called fuzzy lower, fuzzy upper, and fuzzy boundary region sets associated with the fuzzy set λ in I X and based on the fuzzy equivalence relation R in a fuzzy approximation space (X, R).
From (3) and (4), we get that λ R ≤ λ ≤ λ R ∀λ ∈ I X . Whenever λ R be so that λ R ≤ λ R , we get that λ = λ R = λ R , and then from (5), we have The fuzzy accuracy α R (λ) of approximation of the fuzzy set λ could be characterized numerically by , then λ is crisp with respect to R (λ R = λ R and λ is precise with respect to R), and otherwise, if α R (λ) < 1, λ is rough with respect to R (λ is vague with respect to R).

Lemma 1
For any fuzzy set λ ∈ I X we get that: [6] has defined the fuzzy lower and the fuzzy upper sets Rλ, Rλ : X → I of a fuzzy set λ ∈ I X as follows:

Remark 1 Liu in
In the following example, we will see that the fuzzy lower and the fuzzy upper sets of some fuzzy set will be different if we used the above notations and our definitions. Assume that λ = {0.3, 0.4, 1, 0.2}. Then, the fuzzy cosets are as follows: Then, the fuzzy cosets are as follows: Here, λ R = λ R = λ = 0.5 and then λ B = 0. Associated with a fuzzy set λ in a fuzzy approximation space (X, R), we can define a fuzzy interior operator int λ R : I X → I X as follows:

Lemma 2
The following conditions are satisfied.
Thus, this is called a fuzzy interior associated with λ in the fuzzy approximation space (X, R) generating a fuzzy topology defined by: Also, we can define a fuzzy closure operator cl λ R : I X → I X as follows: Note that:

Lemma 3
The fuzzy closure operator satisfy the following conditions: Proof similar to Lemma 2.
Hence, from cl λ R (ν c ) = (int λ R (ν)) c , it is a fuzzy closure operator generating the same fuzzy topology given above (from (8) in Lemma 1) as follows:
(2020) 28:36 Page 6 of 13 If 1 and 2 are fuzzy ideals on X, we have 1 is finer than 2 ( 2 is coarser than 1 ) if 1 ⊇ 2 . The triple (X, R, ) is called a fuzzy ideal approximation space. Denote the trivial fuzzy ideal • as a fuzzy ideal including only 0. (X, R, ) be a fuzzy ideal approximation space and λ ∈ I X . Then, the fuzzy local set μ * λ (R, ) of a set μ ∈ I X is defined by:
Corollary 1 Let (X, R, • ) be a fuzzy ideal approximation space, λ ∈ I X where • is the trivial fuzzy ideal on X. Then, for each μ ∈ I X , we have μ * λ = cl λ R (μ).

Definition 4
Let (X, R, ) be a fuzzy ideal approximation space and λ ∈ I X . Then, (cl λ R ) * λ and (int λ R ) * λ are fuzzy operators from I X into I X based on a specific fuzzy set λ and a fuzzy ideal in the fuzzy approximation space (X, R). Now, if = • , then from Equation 2.7, Corollary 1, Lemma 2 and Lemma 3, Proposition 2 Let (X, R, ) be a fuzzy ideal approximation space with λ ∈ I X fixed. Then, for any μ, ν ∈ I X , we have: Proof (1): From Equations 3.2, 3.3, we get the proof.

Ibedou and Abbas Journal of the Egyptian Mathematical Society
(2020) 28:36 Page 8 of 13 (2): From (8) in Lemma 1, we get that By the same way, you can prove that (int λ

Connectedness in fuzzy ideal approximation spaces
Definition 5 Let (X, R) be a fuzzy approximation space and λ ∈ I X . Then, (1) The fuzzy sets μ, ν ∈ I X are called fuzzy approximation separated if (2) A fuzzy set η ∈ I X is called fuzzy approximation disconnected set if there exist fuzzy approximation separated sets μ, ν ∈ I X , such that μ ∨ ν = η. A fuzzy set η is called fuzzy approximation connected (FA -connected) if it is not fuzzy approximation disconnected. (3) (X, R) is called fuzzy approximation disconnected space if there exist fuzzy approximation separated sets μ, ν ∈ I X , such that μ ∨ ν = 1. A fuzzy approximation space(X, R) is called fuzzy approximation connected (FA-connected) if it is not fuzzy approximation disconnected.

Definition 6
Let (X, R, ) be a fuzzy ideal approximation space and λ ∈ I X . Then, (1) The fuzzy sets μ, ν ∈ I X are called fuzzy ideal approximation separated if (2) A fuzzy set η ∈ I X is called fuzzy ideal approximation disconnected set if there exist fuzzy ideal approximation separated sets μ, ν ∈ I X , such that μ ∨ ν = η. A fuzzy set η is called fuzzy ideal approximation connected (FIA -connected) if it is not fuzzy ideal approximation disconnected. (3) (X, R, ) is called fuzzy ideal approximation disconnected space if there exist fuzzy ideal approximation separated sets μ, ν ∈ I X , such that μ ∨ ν = 1. A fuzzy ideal approximation space(X, R, ) is called fuzzy ideal approximation connected (FIA -connected) if it is not fuzzy ideal approximation disconnected.

Now, consider the fuzzy ideal is defined by
which means that (cl λ R ) * λ (μ) ∧ ν = {0, 0, 0, 0, 0.3} = 0. Hence, not every fuzzy approximation separated sets are fuzzy ideal approximation separated sets, and moreover, the fuzzy set (μ ∨ ν) will be fuzzy ideal approximation connected set whenever = I X and = {0}, that is, whenever is a proper fuzzy ideal on X.
Proposition 3 Let (X, R, ) be a fuzzy ideal approximation space and λ ∈ I X . Then, the following are equivalent.

Definition 7
Let (X, R), (Y , R * ) be two fuzzy approximation spaces and λ ∈ I X , μ ∈ I Y are fuzzy sets. Then, the mapping f : (X, R) → (Y , R * ) is called fuzzy approximation continuous (FA -cont.) if Equivalently, f is called fuzzy approximation continuous (FA -cont.) if Definition 8 A mapping f : (X, R, ) → (Y , R * ) is called fuzzy ideal approximation continuous (FIA -cont.) if Equivalently, f is called fuzzy ideal approximation continuous (FIA -cont.) if Clearly, every fuzzy ideal approximation continuous mapping will be fuzzy approximation continuous as well (from (1) in Proposition 2) but not converse. In the following, we give an example.