key: cord-0047114-glhj3uht
authors: Mao, Lingling; Wang, Jingqian; Yu, Peiqiu
title: The Reduct of a Fuzzy [Formula: see text]-Covering
date: 2020-06-10
journal: Rough Sets
DOI: 10.1007/978-3-030-52705-1_14
sha: ea43df0c0dbfb48ebccd500192200f274fc7bb26
doc_id: 47114
cord_uid: glhj3uht

This paper points some mistakes of three algorithms of updating the reduct in fuzzy [Formula: see text]-covering via matrix approaches while adding and deleting some objects of the universe, and gives corrections of these mistakes. Moreover, we study the reduct of a fuzzy [Formula: see text]-covering while adding and deleting objects further.

Recently, fuzzy covering approximation spaces [1] [2] [3] were generalized to fuzzy βcovering approximation spaces by Ma [4] by replacing 1 with a parameter β, where 1 is a condition in fuzzy covering approximation spaces. Inspired by Ma's work, many researches were done. For example, some fuzzy covering-based rough set models were constructed by Yang and Hu [5] [6] [7] , D'eer et al. [8, 9] studied fuzzy neighborhood operators, and Huang et al. [10] presented a matrix approach for computing the reduct of a fuzzy β-covering.

The research idea of Ref. [10] is very good, but we find that Algorithms 1, 2 and 3 are incorrect after checking the paper carefully. Moreover, the result of a fuzzy βcovering can be studied further while adding and deleting objects. Hence, a further study about Ref. [10] can be done in this paper. Firstly, we explain the mistakes about Algorithms 1, 2 and 3 in Huang et al. (2020) [10] . Then, we give corresponding corrections of them. Finally, we present some new definitions and properties for updating the reduct while adding and deleting objects of a universe. The concepts about a fuzzy β-covering approximation space after adding and deleting objects are presented, respectively. Some new properties about the fuzzy β-covering approximation space and the new fuzzy β-covering approximation space after adding and deleting objects are given.

The rest of this paper is organized as follows. Section 2 reviews some fundamental definitions about fuzzy covering-based rough sets. In Sect. 3, we show some mistakes in [7] . Moreover, we give corresponding corrections of them. In Sect. 4, we present some new definitions and properties for updating the reduct while adding and deleting objects. This paper is concluded and further work is indicated in Sect. 5.

This section recalls some fundamental definitions related to fuzzy covering-based rough sets. Supposing U is a nonempty and finite set called universe.

For any family γ i ∈ [0, 1], i ∈ I, I ⊆ N + (N + is the set of all positive integers), we write ∨ i∈I γ i for the supremum of {γ i : i ∈ I}, and ∧ i∈I γ i for the infimum of {γ i : i ∈ I}. Some basic operations on F (U ) are shown as follows [11] : A, B ∈ F (U ),

Ma [4] presented the notion of fuzzy β-covering approximation space.

.., C m } a fuzzy β-covering of U with C i ∈ F (U ) (i = 1, 2, ..., m). We also call (U, C) a fuzzy β-covering approximation space.

The concept of reducible elements is important for us to deal with some problems in fuzzy covering-based rough sets [5] . Let C be a fuzzy β-covering of U and C ∈ C. If C can be expressed as a union of some elements in C − {C}, then C is called a reducible element in C; otherwise C is called an irreducible element in C.

As shown in [5] , if all reducible elements are deleted from a fuzzy β-covering C, then the remainder is still a fuzzy β-covering and this new fuzzy β-covering does not have any reducible element. We call this new fuzzy β-covering the reduct of the original fuzzy β-covering C. The following definition presents its concept. 

In [10] , we find that Algorithms 1, 2 and 3 have mistakes after checking the paper carefully. Then we give corresponding corrections of the paper in this section. By Algorithm 1 (In [10] ), we know that Γ( C) = ∅ for any fuzzy β-covering, which is incorrect. To explain the incorrect results in Algorithm 1, we show the Algorithm 1 (In [10] ) in Fig. 1 :

In Algorithm 1 (In [10] ), U = {x 1 , x 2 , · · · , x n }. By

Step 2, C(x i ) ← 0. Hence,

Step 1: "i = 1, 2, · · · , m" should be changed as "i = 1, 2, · · · , n".

Step 12, we find Step 11 of Algorithm 1 (In [10] ) is incorrect. By Steps 11 and 12, if C k ⊆ C l then q U kl ← 1. But according to Definition 3 (Definition 5 in [10] ), if C k ⊆ C l and k = l then q U kl ← 1. Hence,

• Step 11: "if C k ⊆ C l then" should be changed as "if C k ⊆ C l and k = l then".

From Steps 11 to 12 of Algorithm 1 (In [10] ), it is to find all C k ∈ C − {C l } which satisfy C k ⊆ C l for any C l ∈ C. From Steps 13 to 16 of Algorithm 1 (In [10] ), if

Hence, Steps 8 and 10 should be swaped places. That is to say, -Step 8: "for k = 1, 2, · · · , m do" should be changed as "for l = 1, 2, · · · , m do". -Step 10: "for l = 1, 2, · · · , m do" should be changed as "for k = 1, 2, · · · , m do".

The result of Algorithm 2 (In [10] ) will be G all the time, which is incorrect. To explain the incorrect results in Algorithm 2, we show the Algorithm 2 (In [10] ):

By Algorithm 2, we find:

Hence, Steps 11 and 13 should be swaped places. That is to say, -Step 11: "for k = 1, 2, · · · , m do" should be changed as "for l = 1, 2, · · · , m do".

-Step 13: "for l = 1, 2, · · · , m do" should be changed as "for k = 1, 2, · · · , m do".

The result of Algorithm 3 (In [10] ) will be G all the time, which is incorrect. To explain the incorrect results in Algorithm 3, we show the Algorithm 3 (In [10] ):

By Algorithm 3, we find:

Hence, Steps 11 and 13 should be swaped places. That is to say, -Step 11: "for k = 1, 2, · · · , m do" should be changed as "for l = 1, 2, · · · , m do".

-Step 13: "for l = 1, 2, · · · , m do" should be changed as "for k = 1, 2, · · · , m do".

This section presents some new properties of reducts in fuzzy β-coverings while adding and deleting some objects, respectively. In this section, t denotes an integer which is more than 1. Firstly, we give some new properties on reducts of fuzzy β-coverings while adding some objects of a universe. The concept of increasing fuzzy β-covering approximation space is presented in the following definition.

The following proposition shows that an increasing fuzzy β-covering approximation space from a fuzzy β-covering approximation space is also a fuzzy β-covering approximation space. 

According to Definition 1, we know C is a fuzzy β-covering of U (0 < β ≤ 0.6). According to Definitions 1 and 4, we know C + is a fuzzy 0.5-covering of U .

We give a relationship about the relation character matrices between a fuzzy βcovering approximation space and it's increasing fuzzy β-covering approximation space in the following proposition. Proposition 2. Let (U, C) and (U + , C + ) be two fuzzy β-covering approximation spaces, where U = {x 1 , x 2 , · · · , x n } and C = {C 1 , C 2 , · · · , C m }. If q U ij = 0, then q U + ij = 0 for any i, j ∈ {1, 2, · · · , m}.

Proof. For any i, j ∈ {1, 2, · · · , m}, we have the following two conditions:

That is to say, q U + ij = 0.

Hence, if q U ij = 0, then q U + ij = 0 for any i, j ∈ {1, 2, · · · , 4}.

We give a relationship about reducible elements between a fuzzy β-covering approximation space and it's increasing fuzzy β-covering approximation space in the following proposition.

Proof. It is immediate by Definition 4 and the concept of reducible element.

The converse of Proposition 3 is not true, i.e., "If C i is a reducible element in C, then C + i is a reducible element in C + for any i ∈ {1, 2, · · · , m}." is not true. Example 1 can explain this. In Example 1, since C 1 = C 2 C 3 , C 1 is a reducible element in C. However, C + 1 is not a reducible element in C + . Based on Proposition 3, we give the following corollary. Corollary 1. Let (U, C) and (U + , C + ) be two fuzzy β-covering approximation spaces, where U = {x 1 , x 2 , · · · , x n } and C = {C 1 , C 2 , · · · , C m }. If C i is a irreducible element in C, then C + i is a irreducible element in C + for any i ∈ {1, 2, · · · , m}.

Example 3. (Continued from Example 1) C 2 , C 3 and C 4 are irreducible elements in C. C + 2 , C + 3 and C + 4 are irreducible elements in C + .

The converse of Corollary 1 is not true, i.e., "If C + i is a irreducible element in C + , then C i is a irreducible element in C for any i ∈ {1, 2, · · · , m}." is not true. Example 1 can explain this. In Example 1, C + 1 is a irreducible element in C + . But C 1 is not a irreducible element in C. Inspired by Corollary 1, we give the following theorem. Proof. By Definition 2, Γ( C) and Γ( C + ) are families of all irreducible elements of C and C + , respectively. Hence, it is immediate by Corollary 1.

Note that |Γ( C)| and |Γ( C + )| denote the cardinality of Γ( C) and Γ( C + ), respectively.

Hence, |Γ( C)| = 3 and |Γ( C + )| = 4. That is to say, |Γ( C)| ≤ |Γ( C + )|.

Then, we give some new properties on reducts of fuzzy β-coverings while deleting some objects of a universe. The concept of declining fuzzy β-covering approximation space is presented in the following definition.

The following proposition shows that a declining fuzzy β-covering approximation space from a fuzzy β-covering approximation space is also a fuzzy β-covering approximation space.

is also a fuzzy β-covering approximation space of U − by Definition 1.

According to Definition 1, we know C is a fuzzy β-covering of U (0 < β ≤ 0.6).

According to Definitions 1 and 5, we know C − is a fuzzy 0.5-covering of U .

We give a relationship about the relation character matrices between a fuzzy βcovering approximation space and it's declining fuzzy β-covering approximation space in the following proposition. Proposition 5. Let (U, C) and (U − , C − ) be two fuzzy β-covering approximation spaces, where U = {x 1 , x 2 , · · · , x n } and C = {C 1 , C 2 , · · · , C m }. If q U − ij = 0, then q U ij = 0 for any i, j ∈ {1, 2, · · · , m}. Proof. For any i, j ∈ {1, 2, · · · , m}, we have the following two conditions:

Hence, there exists k ∈ {1, 2, · · · , n − t} such that C i (x k ) > C j (x k ) according to Definition 5, i.e., there exists k ∈ {1, 2, · · · , n} such that C i (x k ) > C j (x k ). Therefore, C i is not contained in C j . That is to say, q U ij = 0. Example 6. (Continued from Example 5)

Hence, if q U − ij = 0, then q U ij = 0 for any i, j ∈ {1, 2, · · · , 4}. Huang et al. [10] gave a relationship about reducible elements between a fuzzy βcovering approximation space and it's declining fuzzy β-covering approximation space in the following proposition. 

The converse of Lemma 1 is not true, i.e., "If C − i is a reducible element in C, then C i is a reducible element in C for any i ∈ {1, 2, · · · , m}." is not true. Example 5 can explain this. In Example 5, since

is not a reducible element in C. Based on Lemma 1, we give the following corollary.

Corollary 2. Let (U, C) and (U − , C − ) be two fuzzy β-covering approximation spaces, where U = {x 1 , x 2 , · · · , x n } and C = {C 1 , C 2 , · · · , C m }. If C − i is a irreducible element in C − , then C i is a irreducible element in C for any i ∈ {1, 2, · · · , m}.

Proof. By Lemma 1, it is immediate.

Example 7. (Continued from Example 5) C − 2 , C − 3 and C − 4 are irreducible elements in C − . C 2 , C 3 and C 4 are irreducible elements in C.

Based on Corollary 2, we give the following theorem. 

In this paper, we explain the mistakes about Algorithms 1, 2 and 3 in Huang et al.

(2020) [10] . Moreover, we present some new definitions and properties for updating the reduct while adding and deleting objects of a universe. It is helpful for others to investigate the work further. In future, updating the reduct while adding and deleting objects at the same time will be done. Neutrosophic sets and related algebraic structures [12] [13] [14] [15] will be connected with the research content of this paper in further research.

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