# Roughness in Hypervector Spaces

Authors: Nabilah Abughazalah; Naveed Yaqoob; Muhammad Usman
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This paper examines rough sets in hypervector spaces and provides a few examples and results in this regard. We also investigate the congruence relations-based unification of rough set theory in hypervector spaces. We introduce the concepts of lower and upper approximations in hypervector spaces.


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## Body

## 1. Introduction

Giuseppe Peano [1], an Italian mathematician, was the first to define vector space as an abstract algebraic structure in 1888. However, the theory did not emerge until 1920. This idea gained attraction in the 1930s and was applied to a wide range of mathematical and scientific disciplines. On a fundamental level, the vector space hypothesis is a unifying and summarizing theory, as it increased interest in science while also driving new disclosures. The most basic example of vector space in the plane is ℝ2. More studies extended this to Euclidean space ℝn. This area has now developed, and it establishes various regions, such as discrete variable math, coordination space, and functional spaces.Marty [2] proposed the definition of hypergroups and discussed some of its features in 1934, which is the birth year of hyperstructures. Marty, Krasner, Kuntzmann, Croisot, Dresher, Ore, Eaton, Pall, Campaigne, Griffith, PrenoPitz, Utsumi, Dietzman, Vikrov, and Zappa studied the subject as a general hypothesis and its applications in various areas of mathematics in the 1940s, including geometry, group theory, ring theory, and field theory. In the fields of geometry, graphs, hypergraphs, lattices, fuzzy sets, rough sets, automata, cryptography, artificial intelligence, and many others, this theory has a wide variety of applications, see [3, 4]. In 1982, Pawlak introduced the rough set theory [5]. It is a well-known mathematical method for dealing with imprecise, inconsistent, and incomplete data and knowledge based on illogical relationships. The approximation of sets is one of the primary research concerns that rough sets address, while the algorithm of the analysis or reasoning for connected data is another. Financial and business, science, art, establishing automated computational systems, information and decision systems, and data analysis are just few of the uses. Several writers have investigated roughness in various algebraic structures. Roughness in hemirings was introduced by Ali et al. [6]. In the modules [7] and rings [8], Davvaz looked at roughness. Qurashi and Shabir [9, 10] explored rough subsets in quantales and quantale modules. Shabir and others studied roughness in S-acts [11] and in ordered semigroups [11].In 2009, Wu et al. [12, 13] had explored a new idea of roughness in vector spaces by using congruence relations. In 1983, Krasner had demonstrated the notion of the hyperring and hyperfield [14]. Later in 1988, Scafati-Tallini gave an idea of hypervector space [15, 16] but used the classical field to define hypervector space. Ameri and Dehghan [17] introduced some results on dimensions of hypervector spaces. In 2010, Roy and Samanta [18] compiled a brief review of hypervector space; they used the hyperfield to define hypervector space. Taghavi and Hosseinzadeh [19–21] added very useful results to the theory of hypervector spaces. Several writers have investigated fuzziness in hypervector spaces, for instance, Ameri [22], Ameri and Dehghan [17, 23, 24], Dehghan [25], and Roy and Samanta [26]. Muhiuddin and Al-Roqi [27] studied the concept of double-framed soft sets in hypervector spaces. Muhiuddin [28] applied intersectional soft sets theory to generalized hypervector spaces, see also [29, 30].In this paper, we introduced the concept of lower and upper rough subsets in hypervector spaces. In Section2, we added some very basic definition which will be helpful in our further studies. In Section 3, we provided some results related to congruences in hypervector spaces, and in the last section, we applied roughness to hypervector spaces.


## 2. Preliminaries and Basic Definitions

In this section, we added some basic definitions and examples on vector spaces, rough sets, hyperfields, and hypervector spaces.Definition 1 (see [12]).
An equivalence relationσ in a vector space V over a field F is called a congruence relation if ∀s,t,u∈V and ∀a∈F:s,t∈σ⟹s+u,t+u∈σ and a∗s,a∗t∈σ.Example 1.
Consider the vector spaceV=Z4 over the field F=0,1, consider the following relation as
(1)σ=0,0,1,1,2,2,3,3,0,2,2,0,1,3,3,1.Then,σ is a congruence relation on V.Definition 2 (see [2]).
LetM≠∅. Then, ∘:M×M⟶P∗M is called hyperoperation on M, where P∗M=PM−∅ represents the set of all non-empty subsets of M. For any ∅≠P,Q⊆M, we denote
(2)P∘Q=∪p∈P,q∈Qp∘q.Definition 3 (see [2]).
A hypergroupoidM,∘ is known as semihypergroup, if ∀s,t,u∈M, we have s∘t∘u=s∘t∘u.Definition 4 (see [2]).
A semihypergroup is called hypergroup if reproductive axiom holds inM,∀s∈M,M∘s=s∘M=M.Definition 5 (see [31]).
A polygroup is a systemM,∘,e,−1, where e∈P, −1 is a unitary operation on M, ∘:M×M⟶P∗M, and the following axioms hold for all x,y,z∈M:
(i)
x∘y∘z=x∘y∘z(ii)
e∘x=x=x∘e(iii)
x∈y∘z⇒y∈x∘z−1 and z∈y−1∘xExample 2.
LetM=s,t,u and the hyperoperation ∘ defined in Table 1.
Then,M,∘ forms a polygroup.Definition 6 (see [32]).
Letσ be a relation on hypergroup M and ∅≠P,Q⊆M. Then,
(1)
QσP if ∃q∈Q and p∈P such that qσp(2)
Qσ−P if ∀q∈Q∃p∈P and ∀p∈P∃q∈Q such that qσp(3)
Qσ=P if ∀q∈Q and ∀p∈P such that qσpDefinition 7 (see [33]).
Letσ be an equivalence relation on M. Then, σ is called
(1)
left congruence as ifqσp, then s∘qσ−s∘p,∀s∈M(2)
right congruence as ifqσp, then q∘sσ−p∘s,∀s∈M(3)
strongly left congruence as ifqσp, then s∘qσ=s∘p,∀s∈M(4)
strongly right congruence as ifqσp, then q∘sσ=p∘s,∀s∈MDefinition 8 (see [5]).
The lower approximation ofS⊆M, with respect, to σ is the set
(3)apr¯S=s∈M:sσ⊆S,and the upper approximation is the set
(4)apr¯S=s∈M:sσ∩S≠∅.
Then, the setS in M is called a rough set if apr¯S≠apr¯S, otherwise definable.Theorem 9 (see [5]).
LetS,T⊆M and σ be an equivalence relation on M. Then, the following holds:
(1)
apr¯S⊆S⊆apr¯S(2)
apr¯∅=apr¯∅=∅ and apr¯M=apr¯M=M(3)
apr¯S∪T=apr¯S∪apr¯T(4)
IfS⊆T, then apr¯S⊆apr¯T and apr¯S⊆apr¯T(5)
apr¯S∩T⊆apr¯S∩apr¯T(6)
apr¯S∩apr¯T=apr¯S∩T(7)
apr¯S∪apr¯T⊆apr¯S∪T(8)
apr¯−S=−apr¯S(9)
apr¯−S=−apr¯S(10)
apr¯apr¯S=apr¯apr¯S=apr¯S(11)
apr¯apr¯S=apr¯apr¯S=apr¯STable 1
A polygroupM.A Krasner hyperfield is defined as follows.Definition 10 (see [14]).
A nonempty setF under hyperoperation “⊕” and a binary operation “.” is called a hyperfield if the following holds:
(1)
F,⊕ is commutative polygroup(2)
F\0,. is an abelian group(3)
Left an right distributive laws holds with respect to “⊕” and “.” in FExample 3 (see [14]).
LetF=m,n. Then, F,⊕,. is a hyperfield under hyperoperation “⊕” and binary operation “.” defined in the Cayley Tables 2 and 3.Definition 11 (see [18]).
ConsiderV≠∅ with a hyperoperation “⊞” and V,⊞ be a commutative polygroup. Also let F,⊕,. be a Krasner hyperfield, then V is a hypervector space over F, if ∃ a hyperoperation ∗:F×V⟶P∗V, where P∗V=PV\∅ and following holds:
(1)
m∗α⊞β⊆m∗α⊞m∗β,∀α,β∈V and m∈F(2)
m⊕n∗α⊆m∗α⊞n∗α,∀α∈V and m,n∈F(3)
m.n∗α=m∗n∗α,∀α∈V and m,n∈F(4)
α=1F∗α and τ=0F∗α, where 1F,0F∈F and α,τ∈V (τ is zero vector in V).Table 2
Hyperoperation “⊕” defined on F=m,n.Table 3
Binary operation “.” defined on F=m,n.If equality holds in 1,2, thenV is called a good hypervector space.Example 4.
Consider a commutative polygroupV,⊞, where V=α,β,γ,δ and the hyperoperation ⊞ on V is defined in Table 4.Table 4
Hyperoperation “⊞” defined on V=α,β,γ,δ.Also consider the same Krasner hyperfieldF,⊕,., defined in Example 3. Then, V,F,∗ forms a hypervector space under the hyperoperation ∗:F×V⟶P∗V given as in Table 5.Definition 12 (see [18]).
Let∅≠U is a subset of a hypervector space V over hyperfield F. Then, U is called a hypersubspace of V over F if U is itself a hypervector space under the same hyperoperations “⊞” and “∗”. Therefore, a subset U of a hypervector space V is a hypersubspace of V if and only if the following properties hold:
(1)
∀α,β∈U, α⊞β⊆U(2)
U has a zero element(3)
each element of U has an inverse with respect to⊞(4)
∀a∈F and ∀α∈U, a∗α∈UExample 5.
From Example4, consider U=α,β⊆V. Then, U,F,∗ is a hypersubspace of V,F,∗.Table 5
A hyperoperation∗:F×V⟶P∗V.


## 3. Congruence in Hypervector Spaces

Here, we introduce some results on congruences in hypervector spaces.Definition 13.
Letσ be an equivalence relation on a hypervector space V,F,∗. Then, σ is congruence in V,F,∗ if
(1)
∀α,β∈V, if ασβ, then ∀γ∈V, we have α⊞γσ−β(γ and γ⊞ασ−γ⊞β(2)
∀m∈F and ασβ such that m∗ασ−m∗βFrom above definition we may also writeασ for the representation of σ−congruence class containing element α∈V.Definition 14.
For∅≠S,T⊆V, the linear sum of S and T is given as
(5)S⊞T=∪α⊞β:α∈S,β∈T,and product of S with some k∈F(6)k∗S=k∗α:α∈S.Theorem 15.
Letσ be a congruence relation on V. Then, σ−congruence class τσ, where τ is zero vector in V, is hypersubspace of V, and moreover ασ=α⊞τσ.Proof.
Letα,β∈τσ. Then, α,τ∈σ and β,τ∈σ. Since σ is a congruence relation on hypervector space V, so
(7)α⊞β,τ⊞β∈σ−⇒α⊞β,β∈σ−,and also β,τ∈σ− by transitivity of σ,(8)α⊞β,τ∈σ−⇒α⊞β⊆τσ.
Sinceτ is a zero element of V and τ∈τ]σ. So, τσ has a zero element.
Now, letβ∈τ]σ and −1F∈F. Then,
(9)−1F∗β,τ∈σ−⇒−1F∗β,−1F∗τ∈σ−⇒−β,τ∈σ−.
This implies that−β∈τ]σ.
Also forα∈τ]σ and k∈F, we have
(10)k∗α,k∗τ∈σ−⇒k∗α,τ∈σ−⇒k∗α∈τσ.
Hence,τσ is a hypersubspace of V.
Letβ∈α]σ⇒β,α∈σ since σ is a congruence on V. Then, we can write as
(11)−α⊞β,−α⊞α∈σ−⇒−α⊞β,τ∈σ−.
Therefore,−α⊞β⊆τ]σ. This implies that ασ⊆α⊞τ]σ. Now, if β∈α⊞τ]σ, then ∃γ∈τ]σ, such that β∈α⊞γ, since γ∈τ]σ, so γ,τ∈σ and
(12)−α⊞β,τ∈σ−−α⊞β,−α⊞α∈σ−.
Asσ is a congruence relation so β,α∈σ−. This implies that β∈α]σ. Therefore, α⊞τ]σ⊆τ]σ. Hence, ασ=α⊞τ]σ.Theorem 16.
LetU be a hypersubspace of a hypervector space V, such that α⊞−α⊆U,∀α∈V−U and σU=α,β:α,β∈V,α⊞−β⊆U. Then, σU is a congruence on V defined by U and τσU=U.Proof.
SinceU is hypersubspace of V, then τ∈U and τ∈α⊞−α⊆U⇒α,α∈σU. Hence, σU is reflexive also if α,β∈σU, then α⊞−β⊆U. Since U is itself a hypervector space, so β⊞−α⊆U⇒β,α∈σU, so σU is symmetric. Now if α,β∈σU and β,γ∈σU⇒α⊞−β⊆U and β⊞−γ⊆U. Consider
(13)α⊞−γ=α⊞τ⊞−γ⊆α⊞−β⊞β⊞−γ=α⊞−β⊞β⊞−γ⊆U⊞U=U.
Hence,α,γ∈σU. Therefore, σU is transitive. Thus, σU is an equivalence relation on V. Further let α,β∈σU⇒α⊞−β⊆U. Now,
(14)α⊞γ⊞−β⊞γ=α⊞γ⊞−β⊞−γ=α⊞−β⊞γ⊞−γ⊆U⊞U=U.
Similarly, we can show thatγ⊞α⊞−γ⊞β⊆U. Now, let m∈F and α,β∈σU⇒α⊞−β⊆U. As α,β∈V, so it is clear that m∗α and m∗β will be in V. Thus, by definition of σU,(15)m∗α⊞−m∗β⊆U.
Therefore,σU is a congruence on V. Now as if α∈τ]σU⇒α,τ∈σU⇒α⊞−τ⊆U⇒α∈U. If α∈U, then α,τ∈σU⇒α∈τ]σU. Thus, U=τσU.Corollary 17.
LetU be a hypersubspace of a hypervector space V then ασU=α⊞U.Proof.
Since we know thatτσ is a hypersubspace of V, then we can say that U=τσ. Also since σU is congruence on V and U=τσU particularly if we take σ=σU, then, we have U=τσ. So we have one to one correspondence between set of hypersubspaces of V and set of all congruence relations on V. Hence, ασU=α⊞U.Theorem 18.
LetU be a hypersubspace of a hypervector space V. Then, ∀α,β∈V and 0≠k∈F(16)σUα⊞σUβ=σUα⊞β,k∗σUα=σUk∗α.Proof.
(1)
By using Corollary17, we get
(17)
σ
U
α
⊞
σ
U
β
=
α
⊞
U
⊞
β
⊞
U
=
α
⊞
β
⊞
U
⊞
U
=
α
⊞
β
U
=
σ
U
⊞
α
β
.
Hence,σUα⊞σUβ=σUα⊞β.(2)
By using Corollary17, we get(18)k∗σUα=k∗α⊞U=k∗α⊞k∗U⊆k∗α⊞U=σUk∗α.
Also,(19)σUk∗α=k∗α⊞U=k∗α⊞1F∗U=k∗α⊞k.k−1∗U=k∗α⊞k∗k−1∗U⊆k∗α⊞k∗U=k∗α⊞U=k∗σUα.
Hence,k∗σUα=σUk∗α.
Since for any two hypersubspacesU1 and U2 of hypervector space V,U1⊞U2 and U1∩U2 are also a hypersubspaces of V which leads to following result.Theorem 19.
IfU1 and U2 are hypersubspaces of a hypervector space V, then
(20)σU1⊞U2α⊞β=σU1α⊞σU2α,σU1∩U2α=σU1α∩σU2α.Proof.
(1)
By using Corollary17, we get
(21)
σ
U
1
⊞
U
2
α
⊞
β
=
α
⊞
β
⊞
U
1
⊞
U
2
=
α
⊞
U
1
⊞
β
⊞
U
2
=
σ
U
1
α
⊞
σ
U
2
β
.
Therefore,σU1⊞U2α⊞β=σU1α⊞σU2α.(2)
By using Corollary17, we get(22)σU1∩U2α=α⊞U1∩U2=α⊞U1∩α⊞U2=σU1α∩σU2α.
Therefore,σU1∩U2α=σU1α∩σU2α.


## 4. Rough Subsets in Hypervector Spaces

In this section, we study the properties of lower and upper rough subsets in hypervector spaces.Definition 20.
Letσ be a congruence relation on a hypervector space V. Let ∅≠X⊆V, then the sets
(23)apr¯X=α∈V:σα⊆X,apr¯X=α∈V:σα∩X≠∅,are called, respectively, lower and upper approximations of set X with respect to σ.
Ifapr¯X≠apr¯X, then X is called rough set otherwise X is definable.Definition 21.
LetU be a hypersubspace of a hypervector space V. Let ∅≠X⊆V, then the sets
(24)apr¯UX=α∈V:α⊞U⊆X,apr¯UX=α∈V:α⊞U∩X≠∅,are called repectively, lower and upper approximations of set X with respect to hypersubspace U. If apr¯UX≠apr¯UX, then X is a rough set in approximation space V,U, otherwise definable.Example 6.
Consider a commutative polygroupV,⊞, where V=α,β,γ,δ and the hyperoperation (on V is defined in Table 6.Table 6
Hyperoperation “⊞” defined on V=α,β,γ,δ.Also consider the Krasner hyperfieldF,⊕,., where F=l,m,n. The hyperoperation “⊕” and “.” are, respectively, defined in Tables 7 and 8.Table 7
Hyperoperation “⊕” defined on F=l,m,n.Table 8
Binaryoperation “.” defined on F=l,m,n.Then,V,F,∗ forms a hypervector space under the hyperoperation ∗:F×V⟶P∗V defines as follows:
(25)∗x,y=α,ifx=l,y,otherwise.Now, considerU=α,γ. Then,
(26)α⊞U=U,β⊞U=β,δ,γ⊞U=Uandδ⊞U=β,δ.Now, considerS=α,δ, thenapr¯S=∅ and apr¯S=α,β,γ,δ. Since apr¯S≠apr¯S, so S is a rough subset of V.Theorem 22.
LetU be a hypersubspace of a hypervector space V and S is a non-empty subset of V. Then,
(27)apr¯US=S⊞U.Proof.
Forα∈apr¯US, we have α⊞U∩S≠∅. Then, ∃x∈S and γ∈U such that
(28)x∈α⊞γ⇒α∈x⊞−γ⊆S⊞U,thus, α∈S⊞U. Now, let α∈S⊞U. Then, ∃x∈S and γ∈U for which
(29)α∈x⊞γ⇒x∈α⊞−γ⊆αU.
Thus,α⊞U∩S≠∅. This implies that α∈apr¯US. Hence, apr¯US=S⊞U.Theorem 23.
LetV be a hypervector space and U be its hypersubspace. If ∅≠S,T⊆V. Then, the following holds:
(1)
apr¯US⊆S⊆apr¯US;(2)
apr¯US∩T=apr¯US∩apr¯UT;(3)
apr¯US∪T=apr¯US∪apr¯UT;(4)
apr¯US∪apr¯UT⊆apr¯US∪T;(5)
apr¯US∩T⊆apr¯US∩apr¯UT;(6)
IfS⊆T, then apr¯US⊆apr¯UT and apr¯US⊆apr¯UT.Proof.
(1) The proof is obvious.
(2) Letx∈apr¯US∩T⟹x(U⊆S∩T, so x⊞U⊆S and x(U⊆T. Thus, x∈apr¯US and x∈apr¯UT. This implies that x∈apr¯US∩apr¯UT. Now let y∈apr¯US∩apr¯UT. Then, y∈apr¯US and y∈apr¯UT. So, y⊞U⊆S and y⊞U⊆T. This implies that y⊞U⊆S∩T. Thus, y∈apr¯US∩T. Hence, apr¯US∩T=apr¯US∩apr¯UT
(3) Considerx∈apr¯US∪T, so
(30)x∈apr¯US∪T⇔x⊞U∩S∪T≠∅⇔x⊞U∩S∪x⊞U∩T≠∅⇔x⊞U∩S≠∅orx⊞U∩T≠∅gx∈apr¯USorx∈apr¯UT⇔x∈apr¯US∪apr¯UT.
Thus,apr¯US∪T=apr¯US∪apr¯UT. The proofs of (4) and (5) are similar to (1).
(6) IfS⊆T, then we prove that apr¯US⊆apr¯UT. Let α∈apr¯US⇒α⊞U⊆S⊆T, so α⊞U⊆T⇒α∈apr¯UT. Similarly, we can prove apr¯US⊆apr¯UTTheorem 24.
LetU be a hypersubspace of a hypervector space V and S is non-empty subset of V. If U⊆S, then U⊆apr¯US and apr¯US≠∅.Proof.
Letα∈U since U is hypersubspace of V. Then, we have
(31)α⊞U⊆U⊆S⟹U⊆apr¯US.
As clearlyU⊆apr¯US and U≠∅⇒apr¯US≠∅.Theorem 25.
LetU be hypersubspace of a hypervector space V and S,T are non-empty subsets of V then following holds
(1)
apr¯U⊞ST=apr¯US⊞apr¯UT(2)
apr¯US⊞apr¯UT⊆apr¯U⊞ST(3)
apr¯Uk∗S=k∗apr¯US(4)
k∗apr¯US⊆apr¯Uk∗SProof.
(1)
Using Theorem22, we get
(32)
apr
¯
U
S
⊞
T
=
S
⊞
T
U
=
S
⊞
T
U
U
=
S
⊞
U
⊞
T
⊞
U
=
apr
¯
U
S
⊞
apr
¯
U
T
.
Therefore,apr¯US⊞T=apr¯US⊞apr¯UT.(2)
Letx∈apr¯US⊞apr¯UT. Then, ∃β∈apr¯US and γ∈apr¯UT, such that x∈β⊞γ, as since β∈apr¯US and γ∈apr¯UT, so β⊞U⊆S and γ⊞U⊆T. Now consider(33)x⊞U⊆β⊞γU=β⊞γ⊞U⊞U=β⊞U⊞γ⊞U⊆S⊞T.
This implies thatx∈apr¯US⊞T. Hence, apr¯US⊞apr¯UT⊆apr¯US⊞T.(3)
Using Theorem22, we get(34)apr¯Uk∗S=k∗S⊞U=k∗S⊞1F∗U=k∗S⊞k.k−1∗U=k∗S⊞k∗k−1∗U⊆k∗S⊞k∗U=k∗S⊞U=k∗apr¯US.
Also,(35)k∗apr¯US=k∗S⊞U=k∗S⊞k∗U⊆k∗S⊞U=apr¯Uk∗S.
Hence,apr¯Uk∗S=k∗apr¯US.(4)
Letα∈k∗apr¯US, then ∃β∈apr¯US, such that α∈k∗β. Since β∈apr¯US⟹β⊞U⊆S. Now consider(36)α⊞U⊆k∗β⊞U=k∗β⊞1F∗U=k∗β⊞k.k−1∗U=k∗β⊞k∗k−1∗U=k∗β⊞k−1∗U⊆k∗β⊞U⊆k∗S.
This implies thatα∈apr¯Uk∗S. Hence, k∗apr¯US⊆apr¯Uk∗S.Theorem 26.
LetU1 and U2 be hypersubspaces of a hypervector space V and S,T be non-empty subsets of V. Then, the following holds:
(1)
apr¯U1S∩apr¯U2S⊆apr¯U1∩U2S;(2)
apr¯U1∩U2S⊆apr¯U1S∩apr¯U2S;(3)
apr¯U1⊞U2(ST=apr¯U1S⊞apr¯U2T;(4)
apr¯U1S⊞apr¯U2T⊆apr¯U1⊞U2S⊞T;(5)
IfU1⊆U2 then apr¯U1S⊆apr¯U2S and apr¯U2S⊆apr¯U1S.Proof.
(1)
Considerx∈apr¯U1S∩apr¯U2S, we have
(37)
x
∈
apr
¯
U
1
S
∩
apr
¯
U
2
S
⇒
x
∈
apr
¯
U
1
S
and
x
∈
apr
¯
U
2
S
.
⇒
x
⊞
U
1
⊆
S
and
x
⊞
U
2
⊆
S
⇒
x
⊞
U
1
∩
x
⊞
U
2
⊆
S
⇒
x
⊞
U
1
∩
U
2
⊆
S
⇒
x
∈
apr
¯
U
1
∩
U
2
S
.
Therefore,apr¯U1S∩apr¯U2S⊆apr¯U1∩U2S.(2)
Consider(38)apr¯U1∩U2S=S⊞U1∩U2=S⊞U1∩S⊞U2=apr¯U1S∩apr¯U2S.
Thus,apr¯U1∩U2S⊆apr¯U1S∩apr¯U2S.(3)
Consider(39)apr¯U1U2S⊞T=S⊞T⊞U1U2=S⊞U1⊞TU2=apr¯U1S⊞apr¯U2T.
Thus,apr¯U1(U2S⊞T=apr¯U1S⊞apr¯U2T.(4)
Letx∈apr¯U1S⊞apr¯U2T. Then, ∃β∈apr¯U1S and γ∈apr¯U2T, such that x∈β⊞γ. Since β∈apr¯U1S and γ∈apr¯U2T, this implies that β⊞U1⊆S and γ⊞U2⊆T. Consider(40)x⊞U1⊞U2⊆β⊞γ⊞U1⊞U2=β⊞U1⊞γ⊞U2⊆S⊞T.
This implies thatx∈apr¯U1⊞U2S⊞T. Hence, apr¯U1S⊞apr¯U2T⊆apr¯U1⊞U2S⊞T.(5)
LetU1⊆U2. Then, apr¯U1S=S⊞U1⊆S⊞U2=apr¯U2S. Hence, apr¯U1S⊆apr¯U2S. Next, consider α∈apr¯U2S, then α⊞U2⊆S. Now, α⊞U1⊆α⊞U2⊆S. This implies that α∈apr¯U1S. Hence, apr¯U2S⊆apr¯U1S.Theorem 27.
For approximation spaceV,U and ∀∅≠S,T⊆V,
(1)
apr¯Uapr¯US=apr¯US=apr¯Uapr¯US(2)
apr¯Uapr¯US=apr¯US=apr¯Uapr¯US(3)
apr¯US=apr¯UScc(4)
apr¯US=apr¯UScc(5)
apr¯Ux⊞U=apr¯Ux⊞U=x⊞U,∀x∈VProof.
1 to 4 can be proved easily.
(5) Lety∈apr¯Ux⊞U. Then, y(U⊆x⊞U. This implies that y∈x⊞U. Thus, apr¯Ux⊞U⊆x⊞U. Now let y∈x⊞U, we have y⊞U⊆x⊞U. Thus, y∈apr¯Ux⊞U. Therefore, apr¯Ux⊞U=x⊞U. Now, using Theorem 22, we get
(41)apr¯Ux⊞U=x⊞U⊞U=x⊞U⊞U=x⊞U.
Hence,apr¯Ux⊞U=apr¯Ux⊞U=x⊞U.Theorem 28.
IfS and T are definable sets and 0≠k∈F, then k∗S,S∩T, S∪T, S⊞T are definable sets.Proof.
SinceS,T are definable so apr¯US=apr¯US and apr¯UT=apr¯UT. First, we prove that k∗S is definable. By using the conditions 3 and 4 of Theorem 25, we get
(42)apr¯Uk∗S=k∗apr¯US=k∗apr¯US⊆apr¯Uk∗S.
It is obvious thatapr¯Uk∗S⊆apr¯Uk∗S. Thus, apr¯Uk∗S=apr¯Uk∗S. Hence, k∗S is a definable set. Next, we prove that S∩T is definable. By using the conditions 2 and 5 of Theorem 23, we get
(43)apr¯US∩T⊆apr¯US∩apr¯UT=apr¯US∩apr¯UT=apr¯US∩T.
It is obvious thatapr¯US∩T⊆apr¯US∩T. Thus, apr¯US∩T=apr¯US∩T. Hence, S∩T is a definable set. Similarly, by using conditions 3 and 4 of Theorem 23, we can prove that S∪T is a definable set. Lastly, we prove that S⊞T is a definable set. By using the conditions 1 and 2 of Theorem 25, we get
(44)apr¯US⊞T=apr¯US⊞apr¯UT=apr¯US⊞apr¯UT⊆apr¯US⊞T.
It is obvious thatapr¯US⊞T⊆apr¯US⊞T. Thus, apr¯US⊞T=apr¯US⊞T. Hence, ST is a definable set.


## 5. Conclusion

This paper created a bond between rough sets and hypervector space. Based on a congruence, we defined the lower (upper) approximations of a subset of the hypervector space. To begin, we intended the relationship between a congruence relation and hypersubspaces of a hypervector space. Second, some lower and upper approximation characterizations in hypervector spaces are structured. Because of the close relationship between hypervector spaces and Automata (artificial intelligence) and related disciplines, we believe this research will provide a strong tool in approximate reasoning. We believe that the rough hypervector spaces illustrated here will be useful in hyperstructure theory and rough set application.



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*Source: 5451971-2022-03-11.xml*