# On Some Types of Multigranulation Covering Based on Binary Relations

Authors: Ashraf Nawar; E. A. Elsakhawy
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Recently, the notions of right and left covering rough sets were constructed by right and left neighborhoods to propose four types of multigranulation covering rough set (MGCRS) models. These models were constructed using the granulations as equivalence relations. In this paper, we introduce four types of multigranulation covering rough set models under arbitrary relations using theq-minimal and q-maximal descriptors of objects in a given universe. We also study the properties of these new models. Thus, we explore the relationships between these models. Then, we put forward an algorithm to illustrate the method of reduction based on the presented model. Finally, we give an illustrative example to show its efficiency and importance.


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## Body

## 1. Introduction

The notion of rough set theory originated by Pawlak in 1982 [1, 2] to deal with uncertain information and knowledge. It is a tool concerned with the approximation of sets described by a single binary relation. In the view of granular computing suggested by Zadeh [3], a general concept described by a set is characterized via the upper and lower approximations under a single granulation (always equivalence relation) on the universe. This tool has been widely used in many subjects including machine learning, data mining, decision support, and analysis. In the past 20 years, many authors have proposed several extensions of the rough set model [4–19]. In some cases, it is important to use multiequivalence relations on the universe to describe precisely a target concept. Recently, more attention is given to multigranulation rough set (MGRS) models and, also, to multigranulation covering rough set (MGCRS) models in which a target concept is approximated by employing the maximal or minimal descriptors of objects in the given universe. In [20, 21], Qian et al. developed a multigranulation rough set (MGRS) model by using equivalence relations. Several scholars worked on MGRS such as the MGRS model through multiple tolerance relations in incomplete information systems, MGRS via the fuzzy approximation space, the hierarchical structures of MGRS, the topological and lattice-theoretic properties of MGRS, and the efficient rough feature selection algorithm with MGRS [22–30]. Moreover, Liu and Miao and Liu and Wang [31, 32] introduced the multigranulation covering via rough set (MGCRS) and fuzzy rough set (MGCFRS). Lin et al. studied two types of the neighborhood via MGRS [33] and three new types of MGCRS [34]. Also, three types of MGRS via the tolerance, ordered, and generalized relations are investigated and developed the multigranulation decision-theoretic rough set [35–38]. In addition, Liu et al. [39] proposed four new types of MGCRS using the minimal and maximal descriptions and discussed relevant characteristics. For more details about MGRS, see, for instance, [40–44].The notions of left and right covering rough sets proposed by Abd El-Monsef et al. [45] are important tool to make an extension of Liu et al. [39]. The objective of this paper is to develop new models of MGCRS using the notions of left and right covering using the concepts of q-minimal and q-maximal descriptions. Also, we discuss the properties of these models. The relationships between these models are studied. Then, we present the reduction method over our proposed work and establish a numerical example to show its performance. The paper consists of six sections and is organized as follows: Section 1 deals with a brief history to the subject. Section 2 includes the preliminary concepts. Section 3 is the main core of the paper and consists of the new models. In Section 4, the properties and differences between the proposed models are introduced. Section 5 explores new criteria to make a reduction with a test example. We end up with conclusion in the last section.


## 2. Basic Terminologies and Results

This section provides a short survey of some notions used throughout the article.Definition 1 [26].
Letℚ be an universal set and ∅≠E=E~1,E~2,⋯,E~m⊆ℚ. We call E as a covering of ℚ, if ⋃i=1mE~iw=ℚ for anyw∈ℚ. Also, ℚ,E is called a covering approximation space (briefly, CAS).Definition 2 [46].
LetR be a binary relation on an universe set ℚ, and for every w∈ℚ, we have the following two classes. Define the after and fore sets as follows:(1)wR=v∈ℚ:wRv,Rw=v∈ℚ:vRw.Definition 3 [45].
LetR be a binary relation on an universe set ℚ. For each w∈ℚ, define the right covering Cr (resp., the left covering Cl) as follows:(2)Cr=wR:ℚ=⋃w∈ℚwR,Cl=Rw:ℚ=⋃w∈ℚRw.Definition 4 [45].
LetR be a binary relation on an universe set ℚ and Eq be a q-cover of ℚ, where q∈r,l. Then, ℚ,R,Eq is said to be Eq covering approximation space (briefly, Eq-CAS).Definition 5 [45].
Letℚ,R,Eq be Eq-CAS. For every w∈ℚ, define the right neighborhood ℕr, the left neighborhood ℕl, the intersection neighborhood ℕi, and the union neighborhood ℕu, respectively, as follows:(3)ℕrw=C∈Cr:w∈C,ℕlw=C∈Cl:w∈C,ℕiw=ℕrw∩ℕlw,ℕuw=ℕrw∪ℕlw.Definition 6 [45].
Letℚ,R,Eq be Eq-CAS and ∀p∈r,l,i,u and Z⊆ℚ. Define the p-lower approximation, p-upper approximation, p-boundary, p-positive, p-negative, and p-accuracy of Z, respectively, as follows:(4)LpZ=w∈Z:ℕpw⊆Z,UpZ=w∈Z:ℕpw∩Z≠∅,BpZ=UpZ−LpZ,⊕pZ=LpZ,⊖pZ=ℚ−UpZApZ=LpZUpZ,whereUpZ≠0.
Pawlak’s [1, 2] rough set properties are given as follows:
(L1)LZ⊆Z,H1Z⊆UZ.
(L2) Lℚ=ℚ,H2U∅=∅.
(L3) L∅=∅,H3Uℚ=ℚ.
(L4) IfZ1⊆Z2,thenLZ1⊆LZ2,H4UZ1⊆UZ2.
(L5) LZ1∩Z2=LZ1∩LZ2.H5UZ1∪Z2=UZ1∪UZ2.
(L6) LZ1∪Z2⊇LZ1∪LZ2.H6UZ1∩Z2⊆UZ1∩UZ2.
(L7) LZc=UZc,H7UZc=LZc.
(L8) LLZ=LZ,H8UUZ=UZ.
(L9) LLZc=LZc,L9UUZc=UZc.Definition 7 [47].
Letℚ,E be a CAS and Z⊆ℚ. For any w∈ℚ, define the minimal and maximal descriptions of w, respectively, as follows:(5)HE=C∈E:w∈C∧∀S∈E∧w∈S∧S⊆C⇒S=C,DE=C∈E:w∈C∧∀S∈E∧w∈S∧S⊇C⇒S=C.Definition 8 [39].
Letℚ,E be MGCAS and Z⊆ℚ. For any w∈ℚ, define four types of the lower and upper approximations, respectively, as follows: (6)L∑d=1nEd1Z=w∈ℚ:∩HE1w⊆Zor∩HE2w⊆Zor⋯or∩HEnw⊆Z,U∑d=1nEd1Z=w∈ℚ:∩HE1w∩Z≠∅and∩HE2w∩Z≠∅and⋯and∩HEnw∩Z≠∅,LEd2Z=w∈ℚ:∪HE1w⊆Zor∪HE2w⊆Zor⋯or∪HEnw⊆Z,UEd2Z=w∈ℚ:∪HE1w∩Z≠∅and∪HE2w∩Z≠∅and⋯and∪HEnw∩Z≠∅,LEd3Z=w∈ℚ:∩DE1w⊆Zor∩DE2w⊆Zor⋯or∩DEnw⊆Z,UEd3Z=w∈ℚ:∩DE1w∩Z≠∅and∩DE2w∩Z≠∅and⋯and∩DEnw∩Z≠∅,LEd4Z=w∈ℚ:∪DE1w⊆Zor∪DE2w⊆Zor⋯or∪DEnw⊆Z,UEd4Z=w∈ℚ:∪DE1w∩Z≠∅and∪DE2w∩Z≠∅and⋯and∪DEnw∩Z≠∅,
IfLEd1Z (resp., LEd2Z, LEd3Z, and LEd4Z) ≠UEd1Z (resp., UEd2Z, UEd3Z, and UEd4Z), then Z is called the first kind of a multigranulation covering rough set (briefly, type 1-MGCRS) (resp., type 2-MGCRS, type 3-MGCRS, and type 4-MGCRS), else it is definable.Definition 9 [48].
Letℚ,E be a covering information system. For any Z⊆ℚ and w∈ℚ, define the first type of optimistic multigranulation covering lower approximation (briefly, 1-OMGCLA) 1L∑d=1ndEOZ and the first type of optimistic multigranulation covering upper approximation (briefly, 1-OMGCUA) 1U∑d=1ndEOZ as follows:(7)1L∑d=1ndEOZ=w∈ℚ:wE1⊆Z∨wE2⊆Z∨⋯∨wEn⊆Z,1L∑d=1ndEOZ=w∈ℚ:wE1∩Z≠∅∧wE2∩Z≠∅∧⋯∧wEn∩Z≠∅.Definition 10 [48].
Letℚ,E be a covering information system. For any Z⊆ℚ and w∈ℚ, define the first type of pessimistic multigranulation covering lower approximation (briefly, 1-PMGCLA) 1L∑d=1ndEqPZ and the first type of pessimistic multigranulation covering upper approximation (briefly, 1-PMGCUA) 1`U∑d=1ndEqPZ as follows:(8)1L∑d=1nEPZ=w∈ℚ:wE1⊆Z∧wE2⊆Z∧⋯∧wEn⊆Z,1U∑d=1ndEPZ=w∈ℚ:wE1∩Z≠∅∨wE2∩Z≠∅∨⋯∨wEn∩Z≠∅.Next, we have the following definitions using the notion ofEq-CAS.Definition 11.
Letℚ,R,Eq be Eq-CAS and Z⊆ℚ. For any w∈ℚ, define the q-minimal and q-maximal descriptions of w, respectively, as follows: (9)HEq=C∈Eq:w∈C∧∀S∈Eq∧w∈S∧S⊆C⇒S=C,DEq=C∈Eq:w∈C∧∀S∈Eq∧w∈S∧S⊇C⇒S=C.We give the following example to illustrate the above definition.Example 1.
Letℚ,R,Eq be Eq-CAS, ℚ=k1,k2,k3,k4 and R=k1,k4,k2,k2,k2,k3,k3,k2,k4,k1,k4,k3. Then, we have the following results:(10)HErk1=k1,k3,HErk2=k2,HErk3=k1,k3,k2,k3,HErk4=k4,HElk1=k1,HElk2=k2,k3,k2,k4,HElk3=k2,k3,HElk4=k4,DErk1=k1,k3,DErk2=k2,k3,DErk3=k1,k3k2,k3,DErk4=k4,DElk1=k1,DElk2=k2,k3,k2,k4,DElk3=k2,k3,DElk4=k2,k4.Definition 12.
Letℚ,R,Eq be Eq-CAS and Z⊆ℚ. For any w∈ℚ, define the lower and upper approximations, respectively, as follows: (11)LEqZ=w∈Z:∩HEqw⊆Z,UEqZ=w∈ℚ:∩DEqw∩Z≠∅.To explain the above definition, we give the following example.Example 2.
Consider Example1, if Z=k1,k2,k4, then we have the following results.(12)LErZ=k2,k4,UErZ=k1,k2,k4,LElZ=k1,k2,k4,UElZ=ℚ.


## 3. Multi-Eq-Covering Approximation Space

Presume thatℚ is an universal set, R is a family of binary relations on ℚ, and Eq is q-cover of ℚ depending on R, where q∈l,r. Thus, ℚ,R,Eq is called a multi-Eq-covering approximation space (briefly, MEqCAS).Definition 13.
Assume thatℚ,R,Eq is a MEqCAS and R=R1,R2,⋯,RS,∀S∈I, for any Z⊆ℚ and w∈ℚ. Then, we have four novel kinds of lower and upper approximations written as follows:
Style 1
The 1-MCLA1L∑d=1ndEqZ and the 1-MCUA 1L∑d=1ndEqZ are shown as follows:(13)1L∑d=1ndEqZ=w∈ℚ:∩HEqR1w⊆Zor∩HEqR2w⊆Zor⋯or∩HEqRnw⊆Z,1U∑d=1ndEqZ=w∈ℚ:∩HEqR1w∩Z≠∅and∩HEqR2w∩Z≠∅and⋯and∩HEqRnw∩Z≠∅.
If1L∑d=1ndEqZ≠1U∑d=1ndEqZ, then Z is said to be the first kind of q-covering multigranulation rough set (briefly, 1-qMGCRS), else it is definable.
Style 2
The 2-MCLA2L∑d=1ndEqZ and the 2-MCUA 2L∑d=1ndEqZ are seen as follows:(14)2L∑d=1ndEqZ=w∈ℚ:∪HEqR1w⊆Zor∪HEqR2w⊆Zor⋯or∪HEqRnw⊆Z,2U∑d=1ndEqZ=w∈ℚ:∪HEqR1w∩Z≠∅and∪HEqR2w∩Z≠∅and⋯and∪HEqRnw∩Z≠∅.
If2L∑d=1ndEqZ≠2U∑d=1ndEqZ, then Z is said to be the second kind of q-covering multigranulation rough set (briefly, 2-qMGCRS), else it is definable.
Style 3
The 3-MCLA3L∑d=1ndEqZ and the 3-MCUA 3U∑d=1ndEqZ are seen as follows:(15)3L∑d=1ndEqZ=w∈ℚ:∩DEqR1w⊆Zor∩DEqR2w⊆Zor⋯or∩DEqRnw⊆Z,3U∑d=1ndEqZ=w∈ℚ:∩DEqR1w∩Z≠∅and∩DEqR2w∩Z≠∅and⋯and∩DEqRnw∩Z≠∅.
If3L∑d=1ndEqZ≠3U∑d=1ndEqZ, then Z is said to be the third kind of q-covering multigranulation rough set (briefly, 3-qMGCRS), else it is definable.
Style 4
The 4-MCLA4L∑d=1ndEqZ and the 4-MCUA 4U∑d=1ndEqZ are seen as follows:(16)4L∑d=1ndEqZ=w∈ℚ:∪DEqR1w⊆Zor∪DEqR2w⊆Zor⋯or∪DEqRnw⊆Z,4U∑d=1ndEqZ=w∈ℚ:∪DEqR1w∩Z≠∅and∪DEqR2w∩Z≠∅and⋯and∪DEqRnw∩Z≠∅.
If4L∑d=1ndEqZ≠4U∑d=1ndEqZ, then Z is said to be the fourth kind of q-covering multigranulation rough set (briefly, 4-qMGCRS), else it is definable.Example 3.
Considerℚ,R,Eq is a MEqCAS, ℚ=k1,k2,k3,k4 and R=R1,R2, where R1=k1,k4,k2,k2,k2,k3,k3,k2,k4,k1,k4,k3 and R2=k1,k1,k1,k2,k2,k3,k2,k4,k3,k1,k4,k1. Take Z=k1,k3; then, we have the presented outcomes:
(1r)1L∑d=12dErZ=k1,k3,1U∑d=12dErZ=k1,k31l1L∑d=12dElZ=k1,1U∑d=12dElZ=k1,k3.
(2r)2L∑d=12dErZ=k1,2U∑d=12dErZ=k1,k32l2L∑d=12dElZ=k1,2U∑d=12dElZ=k1,k3.
(3r)3L∑d=12dErZ=k1,k3,3U∑d=12dErZ=k1,k2,k33l3L∑d=12dElZ=k1,3U∑d=12dElZ=k1,k3.
(4r)4L∑d=12dErZ=k1,4U∑d=12dErZ=k1,k2,k34l4L∑d=12dElZ=k1,4U∑d=12dElZ=k1,k3.Theorem 14.
Suppose thatℚ,R,Eq is a MEqCAS. For any Z⊆ℚ, we get the following properties:(1)
1L∑d=1ndEqZc=1U∑d=1ndEqZc,1U∑d=1ndEqZc=1L∑d=1ndEqZc(2)
2L∑d=1ndEqZc=2U∑d=1ndEqZc,2U∑d=1ndEqZc=2L∑d=1ndEqZc(3)
3L∑d=1ndEqZc=3U∑d=1ndEqZc,3U∑d=1ndEqZc=3L∑d=1ndEqZc(4)
4L∑d=1ndEqZc=4U∑d=1ndEqZc,4U∑d=1ndEqZc=4L∑d=1ndEqZcProof.
Here, we want to set (1) only.(1)
(17)1L∑d=1ndEqZc=w∈ℚ:∩HEqR1w⊆Zcor∩HEqR2w⊆Zcor⋯or∩HEqRnw⊆Zc=w∈ℚ:∩HEqR1w∩Z=∅or∩HEqR2w∩Z=∅or⋯or∩HEqRnw∩Z=∅=w∈ℚ:∩HEqR1w∩Z≠∅and∩HEqR2w∩Z≠∅and⋯and∩HEqRnw∩∅c=1U∑d=1ndEqZc.
Also, it is easy to see1U∑d=1ndEqZc=1L∑d=1ndEqZc.Proposition 15.
Suppose thatℚ,R,Eq is a MEqCAS. For any Z⊆ℚ, we get the following properties:(1)
1L∑d=1ndEq1L∑d=1ndEqZ=1L∑d=1ndEqZ,1U∑d=1ndEq1U∑d=1ndEqZ=1U∑d=1ndEqZ(2)
2L∑d=1ndEq2L∑d=1ndEqZ=2L∑d=1ndEqZ,2U∑d=1ndEq2U∑d=1ndEqZ=2U∑d=1ndEqZ(3)
3L∑d=1ndEq3L∑d=1ndEqZ=3L∑d=1ndEqZ,3U∑d=1ndEq3U∑d=1ndEqZ=3U∑d=1ndEqZProof.
Here, we want to set (1) only.(1)
It is obvious that1L∑d=1ndEq1L∑d=1ndEqZ⊆1L∑d=1ndEqZ. On the other hand, we have 1L∑d=1ndEqZ=1L1EqZ∪1L2EqZ∪⋯∪1LnEqZ. Thus, we get that(18)1L∑d=1ndEq1L∑d=1ndEqZ=1L1Eq1L∑d=1ndEqZ∪1L2Eq1L∑d=1ndEqZ∪⋯∪1LnEq1L∑d=1ndEqZ=1L1Eq1L1EqZ∪1L2EqZ∪⋯∪1LnEqZZ∪1L2Eq1L1EqZ∪1L2EqZ∪⋯∪1LnEqZZ∪⋯∪1LnEq1L1EqZ∪1L2EqZ∪⋯∪1LnEqZZ⊇1L1Eq1L1EqZ∪1L2Eq1L2EqZ∪11LnEq1LnEqZ=1L1EqZ∪1L2EqZ∪⋯∪1LnEqZ=1L∑d=1ndEqZ
Also, it is clear that1U∑d=1ndEq1U∑d=1ndEqZ⊆1U∑d=1ndEqZ. Consequently, we have 1U∑d=1ndEqZ=1U1EqZ∩1U2EqZ∩⋯∩1UnEqZ. So, we have(19)1U∑d=1ndEq1U∑d=1ndEqZ=1U1Eq1U∑d=1ndEqZ∩1U2Eq1U∑d=1ndEqZ∩⋯∩1UnEq1U∑d=1ndEqZ=1U1Eq1U1EqZ∩1U2EqZ∩⋯∩1UnEqZ∩1U2Eq1U1EqZ∩1U2EqZ∩⋯∩1UnEqZ∩⋯∩1UnEq1U1EqZ∩1U2EqZ∩⋯∩1UnEqZ⊇1U1Eq1U1EqZ∩1U2Eq1U2EqZ∩⋯∩1UnEq1UnEqZ=1U1EqZ∩1U2EqZ∩⋯∩1UnEqZ=1U∑d=1ndEqZ.
Hence,1L∑d=1ndEq1L∑d=1ndEqZ=1L∑d=1ndEqZ and 1U∑d=1ndEq1U∑d=1ndEqZ=1U∑d=1ndEqZ.The above Proposition15 is not true for 4-qMGCRS as in the following example.Example 4.
Consider thatℚ,R,Eq is a MEqCAS, ℚ=k1,k2,k3,k4,k5 and R=R1,R2, where R1=k1,k1,k1,k3,k1,k5,k2,k2,k3,k3,k4,k2,k4,k4,k4,k5,k5,k5 and R2=k1,k1,k1,k3,k2,k2,k2,k5,k3,k3,k4,k3,k4,k4,k5,k2,k5,k5.
TakeZ1=k1,k3 and Z2=k4,k5; then, we have the presented outcomes.
(1r) 4L∑d=12dErZ1=k1and4L∑d=12dEr4U∑d=12dErZ1=∅. Then, 4L∑d=12dEr4L∑d=12dErZ1≠4L∑d=12dErZ1.
(2r) 4U∑d=12dErZ2=k2,k3,k4,k5 and 4U∑d=12dEr4U∑d=12dErZ2=ℚ. Then, 4U∑d=12dEr4U∑d=12dErZ2≠4U∑d=12dErZ2.
(1l) 4L∑d=12dElZ1=k3 and 4L∑d=12dEl4L∑d=12dElZ1=∅. Then, 4L∑d=12dEl4L∑d=12dE1Z1≠4L∑d=12dElZ1.
(2l) 4U∑d=12dElZ2=k1,k2,k4,k5 and 4U∑d=12dEl4U∑d=12dElZ2=ℚ. Then, 4U∑d=12dEl4U∑d=12dElZ2≠4U∑d=12dElZ2.Next, we will establish new properties in Proposition16. These characteristics are done for 1-qMGCRS, 2-qMGCRS, 3-qMGCRS, and 4-qMGCRS, though we demonstrate it in the case of 1-qMGCRS and others are similar.Proposition 16.
Suppose thatℚ,R,Eq is a MEqCAS. For any Z1,Z2⊆ℚ, we get the following properties:(1)
IfZ1⊆Z2, then 1L∑d=1ndEqZ1⊆1L∑d=1ndEqZ2(2)
IfZ1⊆Z2, then 1U∑d=1ndEqZ1⊆1U∑d=1ndEqZ2(3)
1L∑d=1ndEqZ1∩Z2⊆1L∑d=1ndEqZ1∩1L∑d=1ndEqZ2(4)
1L∑d=1ndEqZ1∪Z2⊇1L∑d=1ndEqZ1∪1L∑d=1ndEqZ2(5)
1U∑d=1ndEqZ1∪Z2⊇1U∑d=1ndEqZ1∪1U∑d=1ndEqZ2(6)
1U∑d=1ndEqZ1∩Z2⊆1U∑d=1ndEqZ1∩1U∑d=1ndEqZ2Proof.
Now, we just need to show (1) and (2).(1)
From Definition13 and since Z1⊆Z2, then, we obtain the following:(20)1L∑d=1ndEqZ1=w∈ℚ:∩HEqR1w⊆Z1or∩HEqR2w⊆Z1or⋯or∩HEqRnw⊆Z1⊆w∈ℚ:∩HEqR1w⊆Z2or∩HEqR2w⊆Z2or⋯or∩HEqRnw⊆Z2=1L∑d=1ndEqZ2(2)
From Definition13 and since Z1⊆Z2, then, we have the following:(21)1U∑d=1ndEqZ1=w∈ℚ:∩HEqR1w∩Z1≠∅and∩HEqR2w∩Z1≠∅and⋯and∩HEqRnw∩Z1≠∅⊆w∈ℚ:∩HEqR1w∩Z2≠∅and∩HEqR2w∩Z2≠∅and⋯and∩HEqRnw∩Z2≠∅=1U∑d=1ndEqZ2Example 5.
Consider Example4. Then, we have the following:
(1r) Take Z1=k2,k3,k4 and Z2=k2,k4,k5, then we have 1L∑d=12dErZ1=k2,k3,k4,1L∑d=12dErZ2=k2,k4,k5 and 1L∑d=12dErZ1∩Z2=k2. Thus, 1L∑d=12dErZ1∩Z2≠1L∑d=12dErZ1∩1L∑d=12dErZ2
(1l) Take Z1=k2,k4 and Z2=k2,k5, then we have 1L∑d=12dElZ1=k2,k4,1L∑d=12dElZ2=k2,k5 and 1L∑d=12dElZ1∩Z2=∅. Thus, 1L∑d=12dElZ1∩Z2≠1L∑d=12dElZ1∩1L∑d=12dElZ2
(2r) Take Z1=k1 and Z2=k3, then we have 1L∑d=12dErZ1=∅,1L∑d=12dErZ2=k3 and 1L∑d=12dErZ1∪Z2=k1,k3. Thus, 1L∑d=12dErZ1∪Z2≠1L∑d=12dErZ1∪1L∑d=12dErZ2
(2l) Take Z1=k2 and Z2=k4, then we have 1L∑d=12dElZ1=∅,1L∑d=12dElZ2=k4 and 1L∑d=12dElZ1∪Z2=k2,k4. Thus, 1L∑d=12dElZ1∪Z2≠1L∑d=12dElZ1∪1L∑d=12dElZ2
3r Take Z1=k2 and Z2=k3, then we have 1U∑d=12dErZ1=k2,1U∑d=12dErZ2=k1,k3 and 1U∑d=12dErZ1∪Z2=k1,k2,k3,k4. Thus, 1U∑d=12dErZ1∪Z2≠1U∑d=12dErZ1∪1U∑d=12dErZ2
3l Take Z1=k1 and Z2=k2, then we have 1U∑d=12dElZ1=k1,k3,1U∑d=12dElZ2=k2 and 1U∑d=12dElZ1∪Z2=k1,k2,k3,k5. Thus, 1U∑d=12dElZ1∪Z2≠1U∑d=12dElZ1∪1U∑d=12dElZ2
4r Take Z1=k2,k3 and Z2=k2,k4, then we have 1U∑d=12dErZ1=k1,k2,k3,k4,1U∑d=12dErZ2=k2,k4 and 1U∑d=12dErZ1∩Z2=k2. Thus, 1U∑d=12dErZ1∩Z2≠1U∑d=12dErZ1∩1U∑d=12dErZ2
4l Take Z1=k2,k4 and Z2=k2,k5, then we have 1U∑d=12dElZ1=k2,k4,k5,1U∑d=12dElZ2=k2,k5 and 1U∑d=12dElZ1∩Z2=k2. Thus, 1U∑d=12dElZ1∩Z2≠1U∑d=12dElZ1∩1U∑d=12dEl


## 4. Relationships among Different Proposed Models

Next, we present the relationships between the proposed MEqCAS models.By using Definition13, we obtain the following properties.Proposition 17.
Letℚ,R,Eq be a MEqCAS and Z⊆ℚ. Then, we have the following results: (1)
4L∑d=1ndEqZ⊆2L∑d=1ndEqZ⊆1L∑d=1ndEqZ⊆Z(2)
4L∑d=1ndEqZ⊆3U∑d=1ndEqZ⊆1L∑d=1ndEqZ⊆Z(3)
Z⊆1U∑d=1ndEqZ⊆2U∑d=1ndEqZ⊆4U∑d=1ndEqZ(4)
Z⊆1U∑d=1ndEqZ⊆3U∑d=1ndEqZ⊆4U∑d=1ndEqZRemark 18.
Letℚ,Re,Eq be a MEqCAS and Z⊆ℚ. Then, we have the following results: (1)
2L∑d=1ndEqZ⊈3L∑d=1ndEqZand3L∑d=1ndEqZ⊈2L∑d=1ndEqZ(2)
2U∑d=1ndEqZ⊈3U∑d=1ndEqZand3U∑d=1ndEqZ⊈2U∑d=1ndEqZ
This means that 2-qMGCRS and 3-qMGCRS are independent.Proposition 19.
Letℚ,Re,Eq be a MEqCAS and Z⊆ℚ. Then, we have the following results: (1)
1L∑d=1ndEqZ=2L∑d=1ndEqZ∪3L∑d=1ndEqZ(2)
1U∑d=1ndEqZ=2U∑d=1ndEqZ∩3U∑d=1ndEqZTo illustrate the above characteristic, we give the following example.Example 6.
Consider Example4 and let Z=k1,k2. Then, we have the following outcomes:(1)
Forq=r, we have(22)1L∑d=12dEqZ=2L∑d=12dEqZ=k2,3L∑d=12dEqZ=4L∑d=12dEqZ=∅,1U∑d=12dEqZ=2U∑d=12dEqZ=3U∑d=12dEqZ=k1,k2,4U∑d=12dEqZ=k1,k2,k3,k5(2)
Forq=l, we have(23)1L∑d=12=dEqZ=2L∑d=12dEqZ=3L∑d=12dEqZ=k1,4L∑d=12dEqZ=∅,1U∑d=12dEqZ=2U∑d=12dEqZ=3U∑d=12dEqZ=k1,k2,k3,k5,4U∑d=12dEqZ=ℚ
So, you can find the following:(1)
4L∑d=1ndErZ⊆2L∑d=1ndErZ⊆1L∑d=1ndErZ⊆Z(2)
4L∑d=1ndErZ⊆3L∑d=1ndErZ⊆1L∑d=1ndErZ⊆Z(3)
Z⊆1U∑d=1ndErZ⊆2U∑d=1ndErZ⊆4U∑d=1ndErZ(4)
Z⊆1U∑d=1ndErZ⊆3U∑d=1ndErZ⊆4U∑d=1ndErZ(5)
4L∑d=1ndElZ⊆2L∑d=1ndElZ⊆1L∑d=1ndElZ⊆Z(6)
4L∑d=1ndElZ⊆3L∑d=1ndElZ⊆1L∑d=1ndElZ⊆Z(7)
Z⊆1U∑d=1ndElZ⊆2U∑d=1ndElZ⊆4U∑d=1ndElZZ(8)
Z⊆1U∑d=1ndElZ⊆3U∑d=1ndElZ⊆4U∑d=1ndElZTables1 and 2 show the Pawlak characteristics for the lower and upper approximations which are given in Definition 13.Table 1
Table for the lower approximations.
1L∑d=1ndErZ2L∑d=1ndErZ3L∑d=1ndErZ4L∑d=1ndErZ1L∑d=1ndElZ2L∑d=1ndElZ3L∑d=1ndElZ4L∑d=1ndElZL1√√√√√√√√L2√√√√√√√√L3√√√√√√√√L4××××××××L5√√√√√√√√L6√√√√√√√√L7√√√√√√√√L8√√√×√√√×L9××××××××Table 2
Table for the upper approximations.
1U∑d=1ndErZ2U∑d=1ndErZ3U∑d=1ndErZ4U∑d=1ndErZ1U∑d=1ndElZ2U∑d=1ndElZ3U∑d=1ndElZ4U∑d=1ndElZH1√√√√√√√√H2√√√√√√√√H3√√√√√√√√H4××××××××H5√√√√√√√√H6√√√√√√√√H7√√√√√√√√H8√√√×√√√×H9××××××××


## 5. Relative Reduction of a MEqCAS

This section is aimed at discussing a relative reduction of a pessimistic multigranulationq-covering rough sets (briefly, PMEqCRS). First, we give the following couple of definitions.Definition 20.
Letℚ,R,Eq be a MEqCAS and R=R1,R2,⋯,RS,∀S∈I. For any Z⊆ℚ and w∈ℚ, define the pessimistic multigranulation q-covering lower approximation (briefly, PMGEqCLA) L∑d=1ndEqPZ and pessimistic multigranulation q-covering lower approximation (briefly, PMGEqCLA) U∑d=1ndEqPZ as follows:(24)L∑d=1ndEqPZ=w∈ℚ:wEqR1⊆ZandwEqR2⊆Zand⋯andwEqRn⊆Z,U∑d=1ndEqPZ=w∈ℚ:wEqR1∩Z≠∅orwEqR2∩Z≠∅or⋯orwEqRn∩Z≠∅.Definition 21.
Letℚ,R,Eq be a MEqCAS and R=R1,R2,⋯,RS,∀S∈I. Suppose that D=D1,D2,⋯,Dt is a decision partition of ℚ. Then,(25)PLEqRkD=L∑d=1ndEqPD1,L∑d=1ndEqPD2,⋯,L∑d=1ndEqPDt,PUEqRkD=U∑d=1ndEqPD1,U∑d=1ndEqPD2,⋯,U∑d=1ndEqPDt.(i)
BqRk⊆EqRk and PLBqRkD=PLEqRkD, but PLB∧qRkD≠PLEqRkD, for B∧qRk⊆BqRk; then, BqRk is a D reduction of PMEqCLA(ii)
BqRk⊆EqRk and PUBqRkD=PUEqRkD, but PUB∧qRkD≠PUEqRkD, for B∧qRk⊆BqRk, then, BqRk is a D reduction of PMEqCUAWe can illustrate the method of reduction as the following Algorithm1.Algorithm 1: Algorithm for reduction of PMEqCLA.
Input: ℚ,R,Eq with information system.Output: Reduction of PMEqCLA.1: CalculatePLEqRkD.2: RemoveEqRk, BqRk=EqRi−EqRk and PLBqRi−kD=PLBqRiD.3: Remove a covering inBqRk again and get B∧qRi−k. If PLB∧qRi−kD≠PLBqRiD, return BqRk; else, go to Step 2.4: : Repeat the Steps 2 and 3 for each covering inEqRi to get all the relative reduce of the covering family.Example 7.
Presume thatℚ=k1,k2,⋯,k6 is a set of six houses, Z=equallysharedarea,color,price,surroundings is a set of attributes, and D=purchaseopinions is a set of decisions. The values of equally shared area could be {large, ordinary, small}. The values of color could be {excellent, good, bad}. The values of price could be {high, middle, low}. The values of surroundings could be {quiet, noisy, very noisy}. The decision values of purchase opinions could be {support, oppose}, which is randomly chosen from experts. The evaluation results are shown in Table 3.Table 3
Table for house assessment problem.
Equally shared areaColorPriceSurroundingPurchase optionsk1{Large}{Good}{High}{Very noisy}Opposek2{Small, large}{Excellent}{Middle, low}{Quiet, noisy}Supportk3{Small, large}{Excellent, good}{Middle, low}{Noisy}Supportk4{Small, ordinary}{Bad}{High, middle}{Noisy, very noisy}Opposek5{Small, ordinary}{Bad}{High, middle}{Very noisy}Opposek6{Ordinary, large}{Excellent, good}{High, low}{Quiet, noisy}SupportAs for the attribute setZ, the binary relation is obtained as follows ∀k∈Z:(26)Rk=v,w:Fkv⊆Fkw.It is easy to see that theRk is reflexive and transitive but not symmetric.IfD is the decision set, then the nonequivalence relation is defined as follows:(27)RD=v,w:FDv⊆FDw.Then, we can construct the following two covers:(i)
Right covering (r-cover for short)(28)Cr=wRK:∀K∈Z,D,w∈ℚ,ℚ=⋃w∈ℚwRK(ii)
Left covering (l-cover for short)(29)Cl=RKw:∀K∈Z,D,w∈ℚ,ℚ=⋃w∈ℚRKwSo, we have the following results:(30)CrR1=k1,k2,k3,k2,k3,k4,k5,k6,CrR2=k1,k3,k6,k2,k3,k6,k3,k6,k4,k5,CrR3=k1,k4,k5,k6,k2,k3,k4,k5,k6,CrR4=k1,k4,k5,k2,k6,k2,k3,k4,k6,k4,CrRD=k1,k4,k5,k2,k3,k6,ClR1=k1,k1,k2,k3,k4,k5,k6,ClR2=k1,k2,k1,k2,k3,k6,k4,k5,ClR3=k1,k2,k3,k1,k4,k5,k1,k6,ClR4=k1,k5,k2,k3,k6,k3,k1,k3,k4,k5,ClRD=k1,k4,k5,k2,k3,k6.Thus, we can establish Tables4 and 5 for the neighborhood of k as follows.Table 4
Table for house assessment problem.
k1k2k3k4k5k6kCrR1k1,k2,k3k2,k3k2,k3k4,k5k4,k5k6kCrR2k1,k3,k6k2,k3,k6k3,k6k4,k5k4,k5k3,k6kCrR3k1,k4,k5,k6k2,k3k2,k3k4,k5k4,k5k6kCrR4k1,k4,k5k2,k6k2,k3,k6k4k1,k4,k5k2,k6Table 5
Table for house assessment problem.
k1k2k3k4k5k6kClR1k1k1,k2,k3k1,k2,k3k4,k5k4,k5k6kClR2k1k2k1,k2,k3,k6k4,k5k4,k5k1,k2,k3,k6kClR3k1k2,k3k2,k3k1,k4,k5k1,k4,k5k1,k6kClR4k1,k5k2,k3,k6k3k1,k3,k4,k5k1,k5k2,k3,k6Now, we can apply Algorithm1 as follows.Step 1.
PLErRkD=L∑d=14dErPD1,L∑d=14dErPD2=k4,k5,k2,k3,k6.Step 2.
PLBrR1D=k4,k5,k2,k3,k6, PLBrR2D=k4,k5,k2,k3,k6. Therefore, BrRk=CrR3,CrR4 is a reduction of the PMErCRS.Also, we can get the following outcomes of the left covering:(31)PLElRkD=L∑d=14dElPD1,L∑d=14dElPD2=k1,k5,∅,PLBlR1D=k1,k5,k2,PLBlR2D=k1,k5,ϕ,PLBlR3D=k1,k5,k6,PLBlR4D=k1,k4,k5,∅.Therefore,BlRk=ClR1,ClR3,ClR4 is a reduction of the PMElCRS.


## 6. Conclusion

In this article, we present a notion called multi-Eq-covering approximation space (MEqCAS) by using the concept of q-minimal and q-maximal descriptions. Based on these notions, we establish four new types of multigranulation covering rough sets, denoted MEqCAS. We also study the properties of these new models. Further, we put forward a new methodology to make a reduction by the presented work. Then, we demonstrate the reduction method with the help of an illustrative example which shows its effectiveness and reliability. The main differences between our proposed work and the previous one in [39] are that the authors in [39] introduced four types of MGCRSs using the minimal and maximal description based on equivalence relations and here we used the notions of right (resp., left) covering rough sets to investigate four kinds of multigranulation right (resp., left) covering rough sets using the right (resp., left) minimal and right (resp., left) maximal description induced by binary relations. In further research, we hope to use this approach in fuzzy rough covering-based fuzzy neighborhoods [49], fuzzy soft covering-based rough sets [50], and soft fuzzy covering-based rough sets [51].



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