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International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(2): 125-140

Published online June 25, 2024

https://doi.org/10.5391/IJFIS.2024.24.2.125

© The Korean Institute of Intelligent Systems

Near-Generalized Approximations in Groups Based on a Set-Valued Mapping

Dian Winda Setyawati1, Subiono1, and Bijan Davvaz2

1Department of Mathematics, Institut Teknologi Sepuluh Nopember, Kampus ITS, Sukolilo-Surabaya, Indonesia
2Department of Mathematical Sciences, Yazd University, Yazd, Iran

Correspondence to :
Subiono (subiono2008@matematika.its.ac.id)

Received: June 5, 2023; Revised: August 24, 2023; Accepted: June 5, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

A normal subgroup of a group can partition a group into equivalence classes. Therefore, approximations can be constructed within a group. The near approximations in a group are extensions of the approximations in a group. A set-valued mapping T from group G to the set of all non-empty subsets of group G′ can establish generalized approximations in group G based on the set-valued mapping T. In this study, we introduce the notion of near-generalized approximations in a group based on set-valued mapping, an extension of the concept of generalized approximations in a group based on set-valued mapping and near approximations in a group. We then present some properties of nearby subgroups in a group based on set-valued mapping. Furthermore, we compare these types of near-generalized and generalized approximations in a group based on set-valued mapping.

Keywords: Normal subgroup, Group, Approximations, Near approximations, Generalized approximations, Near-generalized approximations

In the 1980s, Pawlak [1] introduced rough set theory as a mathematical tool for dealing with the problems of vagueness and uncertainty in decision-making. The theory has been used in various fields, including machine learning, data mining, intelligent systems, pattern recognition, decision analysis, inductive reasoning, mereology, etc. [210]. The rough set theory is based on the concept of equivalence relation that is reflexive, symmetric, and transitive. The foundation of this theory is an equivalence class, which is used to construct the lower and upper approximations of a set. The lower approximation of set A is the union of the equivalent classes, subsets of set A, and upper approximation of set A is the union of the equivalent classes that intersect with set A. The concept of lower and upper approximations of a set with respect to the normal subgroups of a group has been extensively studied by many scientists [1114]. In 2004, Davvaz [15] introduced the concept of the lower and upper approximations of a set with respect to the ideals of a ring. Furthermore, in 2006, Davvaz and Mahdavipour [16] introduced the concept of lower and the upper approximations of a set with respect to the submodules of an R-module. Scientists studied the rough set of the ideals of a ring and the submodules of an R-module [1522]. In 2022, Salama et al. [23] constructed the lower and upper approximations of a set using dominance relation that is reflexive, antisymmetric, and transitive.

The algebraic structure of rough sets is generalized to approximation mapping. It is generalized to approximation mappings with respect to an equivalence relation of algebraic structures, such as generalized approximation mapping with respect to the normal subgroups of a group [24], the ideals of a ring [25, 26] and the submodules of an R-module [27]. In 2022, Setyawati and Subiono [24] discussed the relationship between the lower and upper approximations of different sets with different normal subgroups of a group based on set-valued mapping.

The near set was introduced by Peters [28], in which the objects were perceived to be close to each other, with a similarity of description to some degree. The basic idea behind near sets is the description and classification of objects, that is, the members of a universal set based on the perceptual information system. Peters and Wasilewski [29] introduced an approach to the foundations of information science considered in the context of near sets. The algebraic properties of near sets are described in [30]. In 2019, Bağirmaz [31] introduced and derived the properties of near approximations in a group. Furthermore, he compared the near approximations and the approximations in a group. The near approximations in a group are an extension of the rough approximations in a group. Near approximations in a group used two or more normal subgroups of a group, whereas approximations in a group used the normal subgroup of a group. Davvaz et al. [32,33] extended the ideas presented by Bağirmaz [31] by using the ideals of a ring and the submodules of an R-module. In 2023, Setyawati et al. [34] applied the concept of near approximations to the Cayley graphs.

In this study, we introduce the notion of near-generalized approximations in a group based on set-valued mapping (an extension of the concept of generalized approximations in a group based on set-valued mapping [24] and near approximations in a group [31]) and investigate their properties. We then present some properties of nearby subgroups in a group based on set-valued mapping. Furthermore, we compare these types of near-generalized and generalized approximations in a group based on set-valued mapping.

The remainder of this paper is organized as follows. After its introduction, Section 2 reviews fundamental definitions of and near approximations in a group. Section 3 reviews fundamental definitions of approximation in a group based on set-valued mapping. Section 4 introduces the notion of near-generalized approximations in a group based on set-valued mapping and proves their properties. Section 5 presents the properties of lower and upper near subgroups in a group based on set-valued mapping. Section 6 compares the near-generalized and generalized approximations in a group based on a set-valued mapping.

In this section, we define near approximations in a group, which is an extended definition of rough approximations in a group. Let K be a finite set of objects and Q be an equivalence relation on K. A pair (K, Q) is called the approximation space. If S is a normal subgroup of G, then pair (G, S) is an approximation space.

Definition 1 [11]

Let G be a group, S a normal subgroup of G, and A a non-empty subset of G. We define the sets

AprS_(A)={xGxSA},

and

AprS¯(A)={xGxSA},

called lower and upper approximations, respectively. of set A with respect to the normal subgroup S.

Example 1

Let D6 = {ρ0, ρ1, ρ2, ρ3, ρ4, ρ5, μ, ρ1μ, ρ2μ, ρ3μ, ρ4μ, ρ5μ} be a dihedral group with order 12 and S = {ρ0, ρ2, ρ4} be a normal subgroup of D6 where ρ0 is a neutral element of D6. Let A = {ρ1, ρ3, ρ5, μρ2}. Then, we obtain AprS(A) = {ρ1, ρ3,ρ5} and AprS¯(A)={ρ1,ρ3,ρ5,μ,μρ2,μρ4}.

We denote as the set of all normal subgroups in the group G and ℬ as a subset of . A pair (G, ) is called a nearness approximation space. If (G, ) is a nearness approximation space, then

xr=x(SrS),

where ℬr ⊆ ℬ.

Definition 2 [31]

Let (G, ) be a nearness approximation space and A be a non-empty subset of G. We define the sets

Nr()_(A)={xGxrA,for r},

and

Nr()¯(A)={xGxrA,for r},

called near lower and near upper approximations of set A with respect to the normal subgroups of G, respectively. Clearly, Nr()¯(A)=rAprr¯(A) and Nr()¯(A)=rAprr¯(A).

Consider the following example. In this example [31], three different near approximations of the subset A of G defined by families N1(ℬ)(A), N2(ℬ)(A), N3(ℬ)(A).

Example 2

Let ℤ12 be a group of integers modulo 12 and ℬ = {S1, S2, S3} where S1 = {[0]12, [6]12}, S2 = {[0]12, [4]12, [8]12}, S3 = {[0]12, [3]12, [6]12, [9]12} be normal subgroups of ℤ18. Let A = {[0]12, [1]12, [3]12, [4]12, [7]12, [8]12, [10]12}. Then, we get

  • N1()¯(A)=[0]12,[1]12,[4]12,[7]12,[8]12,[10]12,

  • N2()¯(A)=A,

  • N3()¯(A)=A,

  • N1()¯(A)=12,

  • N2()¯(A)={[0]12,[1]12,[2]12,[3]12,[4]12,[6]12,[7]12,[8]12,[9]12,[10]12},

  • N3()¯(A)=A.

From the calculations above, we can conclude that N1()_(A)N2()_(A)=N3()_(A)=N3()¯(A)N2()¯(A)N1()¯(A).

In this study, if N is a non-empty set, then *(N) denotes the set of all non-empty subsets of N. The following definition is a generalization of Definition 1.

Definition 3 [35]

Let M and N be two non-empty sets and T : M*(N) be a set-valued mapping. Let A be a non-empty subset of N. We define the sets

T_(A)={xMT(x)A},

and

T¯(A)={xMT(x)A},

which are called lower and upper approximations of A under T, respectively.

Example 3

Let M = {p, q, r, s, t}, N = {a, b, c, d, e, f, g, h, i, j, k}, and A = {a, b, c, g, h, k}. Let the set-valued mapping T : M*(N) be defined as T(p) = {a, c, h}, T(q) = {b}, T(r) = {e, f, g}, T(s) = {d, i, j}, T(t) = {j, k}. Then, T(A) = {p, q} and (A) = {p, q, r, t}.

In [35], the set-valued homomorphism between the two groups is defined.

Definition 4

Let G and G′ be the two groups. A mapping T : G*(G′) is called a set-valued homomorphism if

  • (1) T(xy) = T(x)T(y),

  • (2) (T(x))−1 = {a−1 | aT(x)} = T (x−1),

for all x, yG.

Example 4 shows that each group homomorphism is a set-valued homomorphism.

Example 4

Let G and G′ be two groups and f : GG′ be a group homomorphism. Subsequently, the set-valued mapping T : G*(G′) defined by T(x) = {f(x)} is a set-valued homomorphism because for every x, yG we have

  • (1) T(xy) = {f(xy)} = {f(x)f(y)} = {f(x)} {f(y)} = T(x)T(y),

  • (2) (T(x))−1 = {(f(x))−1} = {f (x−1)} = T(x−1).

Example 5 [35]

The set ℝ* of non-zero real numbers is a group. Suppose that D={[100a]|a*}. The set-valued mapping T : ℝ**(D) defined by T(a)={[100a],[100-a]} is a set-valued homomorphism.

Furthermore, we introduced the generalized lower and upper approximations of a set with respect to the normal subgroup of a group [24]. This definition is a generalization of Definition 3. The following definition is similar to that in [25].

Definition 5

Let G and G′ be two groups and T : G* (G′) be a set-valued mapping. Let S be a normal subgroup of G′ and A be a non-empty subset of G′. Then,

TS_(A)={xGT(x)SA},

and

TS¯(A)={xG(T(x)S)A},

called the generalized lower and upper approximations of A under T with respect to a normal subgroup S of G′, respectively.

Example 6

Let D6 = {ρ0, ρ1, ρ2, ρ3, ρ4, ρ5, μ, ρ1μ, ρ2μ, ρ3μ, ρ4μ, ρ5μ} be a dihedral group with order 12, S = {ρ0, ρ2, ρ4} be a normal subgroup of D6 where ρ0 is a neutral element of D6. Let T : D6* (D6) be defined as

T(x)={{x}if x{ρ0,ρ1,ρ2,ρ3,ρ4,ρ5},{ρi,μ}if x=ρiμ{μ,ρ1μ,ρ2μ,ρ3μ,ρ4μ,ρ5μ}.

Let A = {ρ1, ρ3, ρ5, μρ2}. Then, we get

  • TS(A) = {ρ1, ρ3, ρ5},

  • TS¯(A)={ρ1,ρ3,ρ5,μ,ρμ,ρ2μ,ρ3μ,ρ4μ,ρ5μ}.

From the above calculations, we conclude that TS_(A)TS¯(A).

Definitions 2 and 5 motivate to form Definition 6. We denote as the set of all normal subgroups in a group G′ and ℬ as a subset of . A pair (G′, ) is defined as nearness approximation space. If (G′, ) is a nearness approximation space, then

T(x)r=T(x)(SrS),

where ℬr ⊆ ℬ.

Definition 6

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued mapping. Let A be a non-empty subset of G′. We define the sets

TNr()_(A)={xGT(x)rA,for r},

and

TNr()¯(A)={xGT(x)rA,for r},

which are called the near-generalized lower and upper approximations of A under T with respect to the normal subgroups ℬ of G′, respectively. Clearly, TNr()¯(A)=rTr¯(A) and TNr()¯(A)=rTr¯(A).

Example 7

Let D4 = {ρ0, ρ1, ρ2, ρ3, μ, ρ1μ, ρ2μ, ρ3μ} be a dihedral group with order 8, HD4 and H = {ρ0, ρ1, ρ2, ρ3} be a group where ρ0 is a neutral element of D4 and H. Let T : ℤ36 × H*(ℤ36 × D4) be defined by

T(([x]36,ρi))={([x]36,ρi),([x+1]36,ρiμ)},

for every ([x]18, ρi) ∈ ℤ36 × H. Let N = {ρ0, ρ2, μ, ρ2μ}, H be normal subgroups of D4 and ℬ = {S1, S2, S3}, where S1 = ⟨[9]36⟩ × H, S2 = ⟨[6]36⟩ × N, S3 = ⟨[4]36⟩ × H be normal subgroups of ℤ36 × D4. Let A ⊂ ℤ36 × D4 where

A={([a]36,b)[a]36Z36-{[1]36,[2]36,[3]36,[4]36,[13]36,[14]36,[15]36,[16]36,[19]36,[20]36,[21]36,[25]36,[26]36,[27]36,[28]36}andb{ρ0,ρ2,ρ3,μ,ρ2μ,ρ3μ}}.

We determine TNr()¯(A) and TNr()¯(A).

  • • The first set of components of A is ℤ36 – {[1]36, [2]36, [3]36, [4]36, [13]36, [14]36, [15]36, [16]36, [19]36, [20]36, [21]36, [25]36, [26]36, [27]36, [28]36}. There are no cosets of ⟨[4]36⟩ contained in the first set of components of A. Cosets of ⟨[6]36⟩ contained in the first set of components of A are [0]18 + ⟨[6]36⟩, [5]18 + ⟨[6]36⟩. There are no cosets of ⟨[9]36⟩ contained in the first set of components of A. Cosets of ⟨[12]36⟩ contained in the first set of components of A are [0]36+⟨[12]36⟩, [5]36+⟨[12]36⟩, [6]36+ ⟨[12]36⟩, [10]36 + ⟨[12]36⟩, [11]36 + ⟨[12]36⟩. Cosets of ⟨[18]36⟩ contained in the first set of components of A are [0]36 + ⟨[18]36⟩, [5]36 + ⟨[18]36⟩, [6]36 + ⟨[18]36⟩, [11]36 + ⟨[18]36⟩, [12]36 + ⟨[18]36⟩, [17]36 + ⟨[18]36⟩.

  • • The second set of components of A is {ρ0, ρ2, ρ3, μ, ρ2μ, ρ3μ}. Cosets of {ρ0, ρ2} contained in the second set of components of A are {ρ0, ρ2}, {μ, ρ2μ}. Cosets of N contained in the second set of components of A are N. There are no cosets of H contained in the second set of components of A.

  • (1) Determine TN1()¯(A).


    For r = 1, the subsets ℬ1 ⊆ ℬ are {S1}, {S2}, {S3}.

    • S1 = ⟨[9]36⟩×H then (36×D4)/S1=36×84×4=18.

      TS1_(A)={([x]36,ρi)36×HT(([x]36,ρi))S1A}={([x]36,ρi)36×H{([x]36,ρi),([x+1]36,ρiμ)}([9]36×H)A}=.

    • S2 = ⟨[6]36⟩×N then (36×D4)/S2=36×86×4=12.

      TS2_(A)={([x]36,ρi)36×HT(([x]36,ρi))S2A}={([x]36,ρi)36×H{([x]36,ρi),([x+1]36,ρiμ)}([6]36×N)A}={([x]36,ρi)36×H[x]36{[5]36,[11]36,[17]36,[23]36,[29]36,[35]36}and ρi{ρ0,ρ2}}.

    • S3 = ⟨[4]36⟩×H then (36×D4)/S3=36×89×4=8.

      TS3_(A)={([x]36,ρi)36×HT(([x]36,ρi))S3A}={([x]36,ρi)36×H{([x]36,ρi),([x+1]36,ρiμ)}([4]36×H)A}=.

    From the above calculations, we obtain

    TN1()_(A)={([x]36,ρi)36×HT(([x]36,ρi))1A,for 1}=TS1_(A)TS2_(A)TS3_(A)={([x]36,ρi)36×H[x]36{[5]36,[11]36,[17]36,[23]36,[29]36,[35]36}and ρi{ρ0,ρ2}}.

  • (2) Determine TN2()¯(A).


    For r = 2, the subsets ℬ2 ⊆ℬ are {S1, S2}, {S1, S3}, {S2, S3}. By the above explanation, we have T(x)ℬ2 = T(x)(∩S∈ℬ2S.)

    • S1S2 = ⟨[18]36⟩×{ρ0, ρ2} then (36×D4)/(S1S2)=36×82×2=72.

      TS1S2_(A)={([x]36,ρi)36×HT(([x]36,ρi))(S1S2)A}={([x]36,ρi)36×H{([x]36,ρi),([x+1]36,ρiμ)}([18]36×{ρ0,ρ2})A}={([x]36,ρi)36×H[x]36{[5]36,[11]36,[17]36,[23]36,[29]36,[35]36}and ρi{ρ0,ρ2}}.

    • S1S3 = {[0]36} × H then (36×D4)/(S1S3)=36×81×4=72.

      TS1S3_(A)={([x]36,ρi)36×HT(([x]36,ρi))(S1S3)A}={([x]36,ρi)36×H{([x]36,ρi),([x+1]36,ρiμ)}({[0]36}×H)A}=.

    • S2S3 = ⟨[12]36⟩×{ρo, ρ2} then (36×D4)/(S2S3)=36×83×2=48.

      TS2S3_(A)={([x]36,ρi)36×HT(([x]36,ρi))(S2S3)A}={([x]36,ρi)36×H{([x]36,ρi),([x+1]36,ρiμ)}([12]36×{ρ0,ρ2})A}={([x]36,ρi)36×H[x]36{[5]36,[10]36,[11]36,[17]36,[22]36,[23]36,[29]36,[34]36,[35]36}and ρi{ρ0,ρ2}}.

    From the above calculations, we obtain

    TN2()_(A)={([x]36,ρi)36×HT(([x]36,ρi))2A,for 2}=TS1S2_(A)TS1S3_(A)TS2S3_(A)={([x]36,ρi)36×H[x]36{[5]36,[10]36,[11]36,[17]36,[22]36,[23]36,[29]36,[34]36,[35]36}and ρi{ρ0,ρ2}}.

  • (3) Determine TN3()¯(A).


    For r = 3, the subsets ℬ3 ⊆ ℬ then {S1, S2, S3}. By the above explanation, we have T(x)ℬ3 = T(x) (∩S∈ℬ3S).

    • S1S2S3 = {[0]36} × {ρo, ρ2} then (36×D4)/(S1S2S3)=36×81×2=144.

      TN3()_(A)=TS1S2S3_(A)={([x]36,ρi)36×HT(([x]36,ρi))(S1S2S3)A}={([x]36,ρi)36×H{([x]36,ρi),([x+1]36,ρiμ)}({[0]36×{ρ0,ρ2})A}={([x]36,ρi)36×H[x]36{[5]36,[6]36,[7]36,[8]36,[9]36,[10]36,[11]36,[17]36,[22]36,[23]36,[29]36,[30]36,[31]36,[32]36,[33]36,[34]36,[35]36}and ρi{ρ0,ρ2}}.

    As a summary, we have TN1()¯(A)TN2()¯(A)TN3()¯(A).

    • • The first set of components of A is ℤ36 – {[1]36, [2]36, [3]36, [4]36, [13]36, [14]36, [15]36, [16]36, [19]36, [20]36, [21]36, [25]36, [26]36, [27]36, [28]36}.


      Every coset of ⟨[4]36⟩ intersects with the first set of components of A.


      Every coset of ⟨[6]36⟩ intersects with the first set of components of A.


      Every coset of ⟨[9]36⟩ intersects with the first set of components of A.


      Every coset of ⟨[12]36⟩ intersects with the first set of components of A except [1]36 +⟨[12]36⟩, [2]36 + ⟨[12]36⟩, [3]36 + ⟨[12]36⟩, [4]36 + ⟨[12]36⟩.


      Every coset of ⟨[18]36⟩ always intersects with the first set of components of A except [1]36 +⟨[18]36⟩, [2]36 + ⟨[18]36⟩, [3]36 + ⟨[18]36⟩.

    • • The second set of components of A is {ρ0, ρ2, ρ3, μ, ρ2μ, ρ3μ}.


      Every coset of {ρ0, ρ2} intersects with the second set of components of A.


      Every coset of N intersects with the second set of components of A.


      Every coset of H intersects with the second set of components of A.

  • (4) Determine TN1()(A)¯.


    For r = 1, the subsets ℬ1 ⊆ ℬ are {S1}, {S2}, {S3}.

    • S1 = ⟨[9]36⟩×H then (36×D4)/S1=36×84×4=18.

      TS1¯(A)={([x]36,ρi)36×HT(([x]36,ρi))S1A}={([x]36,ρi)36×H({([x]36,ρi),([x+1]36,ρiμ)}([9]36×H))A}=36×H.

    From the above calculations, we obtain

    TN1()(A)¯={([x]36,ρi)36×HT(([x]36,ρi)1A,for 1}=TS1(A)¯TS2(A)¯TS3(A)¯=36×H.

  • (5) Determine TN2()(A)¯.


    For r = 2, the subsets ℬ2 ⊆ ℬ are {S1, S2}, {S1, S3}, {S2, S3}. By the above explanation, we have T(x)ℬ2 = T(x) (∩S∈ℬ2S).


    • S1S2 = ⟨[18]36⟩ × {ρo, ρ2} then (36×D4)/(S1S2)=36×82×2=72.

      TS1S2¯(A)={([x]36,ρi)36×HT(([x]36,ρi))(S1S2)A}={([x]36,ρi)36×H({([x]36,ρi),([x+1]36,ρiμ)}([18]36×{ρ0,ρ2}))A}={([x]36,ρi)36×H[x]3636-{[1]36,[2]36,[19]36,[20]36,}and ρiH}.

    • S1S3 = {[0]36} × H then (36×D4)/(S1S3)=36×81×4=72.

      TS1S3¯(A)={([x]36,ρi)36×HT(([x]36,ρi))(S1S3)A}={([x]36,ρi)36×H({([x]36,ρi),([x+1]36,ρiμ)}({[0]36}×H))A}={([x]36,ρi)36×H[x]3636-{[1]36,[2]36,[3]36,[13]36,[14]36,[15]36,[19]36,[20]36,[25]36,[26]36,[27]36}and ρiH}.

    • S2S3 = ⟨[12]36⟩×{ρo, ρ2} then (36×D4)/(S2S3)=36×83×2=48.

      TS2S3¯(A)={([x]36,ρi)36×HT(([x]36,ρi))(S1S3)A}={([x]36,ρi)36×H({([x]36,ρi),([x+1]36,ρiμ)}([12]36×{ρo,ρ2}))A}={([x]36,ρi)36×H[x]3636-{[1]36,[2]36,[3]36,[13]36,[14]36,[15]36,[25]36,[26]36,[27]36}and ρiH}.

    From the above calculations, we obtain

    TN2()(A)¯={([x]36,ρi)36×HT(([x]36,ρi))2A,for 2}=TS1S2(A)¯TS1S3(A)¯TS2S3(A)¯={([x]36,ρi)36×H[x]3636-{[1]36,[2]36}and ρiH}.

  • (6) Determine TN3()(A)¯.


    For r = 3, the subsets ℬ3 ⊆ ℬ are {S1, S2, S3}. By the above explanation, we have T(x)ℬ3 = T(x)∩(∩S∈ℬ2S).


    • S1S2S3 = {[0]36} × {ρ0, ρ2} then (36×D4)/(S1S2S3)=36×81×2=144.

      TN3()(A)¯=TS1S2S3¯(A)={([x]36,ρi)36×H(([x]36,ρi))(S1S2S3)A}={([x]36,ρi)36×H({([x]36,ρi),([x+1]36,ρiμ)}({[0]36}×{ρ0,ρ2}))A}={([x]36,ρi)36×H[x]3636-{[1]36,[2]36,[3]36,[13]36,[14]36,[15]36,[19]36,[20]36,[25]36,[26]36,[27]36}and ρiH}.

In summary, we have TN3()(A)¯TN2()(A)¯TN1()(A)¯. The relationship between TNr()¯(A) and TNr()¯(A) is TN1()_(A)TN2()_(A)TN3()_(A)TN3()¯(A)TN2()¯(A)TN1()(A)¯.

Proposition 1

Let G be a group, (G, ) be a nearness approximation space and I : G*(G) be a set-valued mapping, where I(x) = {x} for each xG. Let A be a non-empty subset of G. Then,

INr()_(A)=Nr()(A)_and INr()¯(A)=Nr()(A)¯.
Proof

This is clear from Definitions 2 and 6.

Proposition 2

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued mapping. Let A be a non-empty subset of G′. Then,

  • (1) TNr()_(A)TNr()¯(A),

  • (2) TNr()_(G)=G=TNr()¯(G),

  • (3) If AB, then TNr()¯(A)TNr()¯(B) and TNr()¯(A)TNr()¯(B),

  • (4) TNr()_(A)(TNr()¯(AC))C,

  • (5) TNr()¯(A)(TNr()_(AC))C,

  • (6) TNr()¯(AB)TNr()¯(A)TNr()¯(B),

  • (7) TNr()¯(AB)TNr()¯(A)TNr()¯(B),

  • (8) TNr()¯(AB)TNr()¯(A)TNr()¯(B),

  • (9) TNr()¯(AB)=TNr()¯(A)TNr()¯(B).

Proof
  • (1) It is clear from Definition 6.

  • (2) It is clear from Definition 6.

  • (3) Let xTNr()¯(A). Subsequently, T(x)ℬrAB, for some ℬr ⊆ ℬ so that xTNr()¯(B). Therefore, TNr()¯(A)TNr()¯(B). Suppose yTNr()¯(A). Then, T(y)r¯A for some ℬr ⊆ ℬ. Since AB then T(y)ℬrB ≠ ∅ for some ℬr ⊆ ℬ such that yTNr()¯(B). Therefore, TNr()¯(A)TNr()¯(B).

  • (4) Let x(TNr()¯(AC))C. Then, x(TNr()¯(AC) so that T(x) ℬrAC = ∅ for every ℬr ⊆ ℬ, which implies that T(x) ℬrA for every ℬr ⊆ ℬ. Therefore, xTNr()¯(A) so that TNr()_(A)(TNr()¯(AC))C. The converse is not true in general. Let xTNr()¯(A). Then, T(x) ℬrA for some ℬr ⊆ ℬ but T(x)ℬrA for some ℬr ⊆ ℬ so that T(x)ℬrAC ≠ ∅, which implies xTNr()¯(AC). Thus, we have x(TNr()¯(AC))C so that TNr()_(A)(TNr()¯(AC))C.

  • (5) Let xTNr()¯(AC)C. Then xTNr()¯(AC) so that T(x)ℬrAC for every ℬr ⊆ ℬ which implies T(x)ℬrA ≠ ∅ for every ℬr ⊆ ℬ. Therefore, xTNr()¯(A) so that TNr()¯(A)(TNr()_(AC))C. The converse is not true in general. Let xTNr()¯(A) Then, T(x)ℬrA ≠ ∅ for some ℬr ⊆ ℬ but T(x)ℬrA = ∅ for some ℬr ⊆ ℬ such that T(x)ℬrAC, which implies xTNr()¯(AC). Therefore, xTNr()¯(AC)C so that TNr()¯(A)(TNr()_(AC))C.

  • (6) From part (3), we have TNr()¯(AB)TNr()¯(A) and TNr()¯(AB)TNr()¯(B). Then, TNr()¯(AB)TNr()¯(A)TNr()¯(B).

  • (7) From part (3), we have TNr()¯(AB)TNr()¯(A) and TNr()¯(AB)TNr()¯(B). Then, TNr()¯(AB)TNr()¯(A)TNr()¯(B).

  • (8) From part (3), we have TNr()¯(A)TNr()¯(AB) and TNr()¯(B)TNr()¯(AB). Then, TNr()¯(A)TNr()¯(B)TNr()¯(AB).

  • (9) Let xTNr()¯(AB). Then, T(x)ℬr ∩ (AB) ≠ ∅ for some ℬr ⊆ ℬ so that T(x)ℬrA ≠ ∅ or T(x)ℬrA ≠ ∅ which implies xTNr()¯(A) or xTNr()¯(B). Therefore, TNr()¯(AB)TNr()¯(A)TNr()¯(B). Conversely, from part (3), we have TNr()¯(A)TNr()¯(AB) and TNr()¯(B)TNr()¯(AB). Then, TNr()¯(A)TNr()¯(B)TNr()¯(AB).

The following examples show that the converse of Proposition 2 (6,7,8) is generally not true.

Example 8

Let ℤ18 be a group of integers modulo 18, ℬ = {S1, S2}, where S1 = ⟨[9]18⟩ = {[0]18, [9]18}, S2 = ⟨[6]18⟩ = {[0]18, [6]18, [12]18} be normal subgroups of ℤ18 and T : ℤ18* (ℤ18) be defined as T ([x]18) = {[x + 1]18}, for every [x]18 ∈ ℤ18. Suppose A, B ⊂ ℤ18, where

A=18-([1]18+[9]18[3]18+[9]18[2]18+[6]18)=18-{[1]18,[2]18,[3]18,[8]18,[10]18,[12]18,[14]18},

and

B=18-([1]18+[9]18[3]18+[9]18[3]18+[6]18)=18-{[1]18,[3]18,[4]18,[9]18,[10]18,[13]18,[15]18}.
  • (1) Determine TN1()¯(A) and TN1()¯(B).

    • • Cosets of S1 = ⟨[9]18⟩ contained in A are ⟨[9]18⟩, [4]18 + ⟨[9]18⟩, [6]18 + ⟨[9]18⟩, [7]18 + ⟨[9]18⟩.

      TS1_(A)={[x]1818[x+1]18+[9]18A}={[3]18,[5]18,[6]18,[8]18,[12]18,[14]18,[15]18,[17]18}.

    • • Cosets of S2 = ⟨[6]18⟩ contained in A are [5]18 + ⟨[6]18⟩.

      TS2_(A)={[x]1818[x+1]18+[6]18A}={[4]18,[10]18,[16]18}.

      From the above calculations, we have

      TN1()_(A)=TS1_(A)TS2_(A)={[3]18,[4]18,[5]18,[6]18,[8]18,[10]18,[12]18,[14]18,[15]18,[16]18,[17]18}.

    • • Cosets of S1 = ⟨[9]18⟩ contained in B are [2]18 + ⟨[9]18⟩, [5]18+⟨[9]18⟩, [7]18+⟨[9]18⟩, [8]18+⟨[9]18⟩.

      TS1_(B)={[x]1818[x+1]18+[9]18B}={[1]18,[4]18,[6]18,[7]18,[10]18,[13]18,[15]18,[16]18}.

    • • Cosets of S2 = ⟨[6]18⟩ contained in B are [0]18 + ⟨[6]18⟩, [2]18 + ⟨[6]18⟩, [5]18 + ⟨[6]18⟩.

      TS2_(B)={[x]1818[x+1]18+[6]18B}={[1]18,[4]18,[5]18,[7]18,[10]18,[11]18,[13]18,[16]18,[17]18}.

      From the above calculations, we have

      TN1()_(B)=TS1_(B)TS2_(B)={[1]18,[4]18,[5]18,[6]18,[7]18,[10]18,[11]18,[13]18,[15]18,[16]18,[17]18}.

      As a summary, we have TN1()¯(A)TN1()¯(B)={[4]18,[5]18,[6]18,[10]18,[15]18,[16]18,[17]18}.

  • (2) Determine TN1()¯(AB).


    AB = {[0]18, [5]18, [6]18, [7]18, [11]18, [16]18, [17]18}.


    • • Cosets of S1 = ⟨[9]18⟩ contained in AB are [7]18 + ⟨[9]18⟩.

      TS1_(AB)={[x]1818[x+1]18+[9]18AB}={[6]18,[15]18}.

    • • Cosets of S2 = ⟨[6]18⟩ contained in AB are [5]18 + ⟨[6]18⟩.

      TS2_(AB)={[x]1818[x+1]18+[6]18AB}={[4]18,[10]18,[16]18}.

      From the above calculations, we have

      TN1()_(AB)=TS1_(AB)TS2_(AB)={[4]18,[6]18,[10]18,[15]18,[16]18}.

    As a summary, we obtain TN1()¯(A)TN1()¯(B)TN1()¯(AB).

  • (3) Determine TN1()¯(A) and TN1()¯(B).

    • • Cosets of S1 = ⟨[9]18⟩ intersected with A are [0]18+⟨[9]18⟩, [2]18+⟨[9]18⟩, [4]18+⟨[9]18⟩, [5]18+ ⟨[9]18⟩, [6]18+⟨[9]18⟩, [7]18+⟨[9]18⟩, [8]18+⟨[9]18⟩.

      TS1¯(A)={[x]1818([x+1]18+[9]18)A}=18-{[0]18,[2]18,[9]18,[11]18}.

    • • Cosets of S2 = ⟨[6]18⟩ intersected with A are [0]18+⟨[6]18⟩, [1]18+⟨[6]18⟩, [3]18+⟨[6]18⟩, [4]18+ ⟨[6]18⟩, [5]18 + ⟨[6]18⟩.

      TS2¯(A)={[x]1818([x+1]18+[6]18)A}=18-{[1]18,[7]18,[13]18}.

      From the above calculations, we obtain

      TN1()¯(A)=TS1¯(A)TS2¯(A)=18.

    • • Cosets of S1 = ⟨[9]18⟩ intersected with B are [0]18+⟨[9]18⟩, [2]18+⟨[9]18⟩, [3]18+⟨[9]18⟩, [5]18+ ⟨[9]18⟩, [6]18+⟨[9]18⟩, [7]18+⟨[9]18⟩, [8]18+⟨[9]18⟩.

      TS1¯(B)={[x]1818([x+1]18+[9]18)B}=18-{[0]18,[3]18,[9]18,[12]18}.

    • • Cosets of S2 = ⟨[6]18⟩ intersected with B are [0]18+⟨[6]18⟩, [1]18+⟨[6]18⟩, [2]18+⟨[6]18⟩, [4]18+ ⟨[6]18⟩, [5]18 + ⟨[6]18⟩.

      TS2¯(B)={[x]1818([x+1]18+[6]18)B}=18-{[2]18,[8]18,[14]18}.

      From the above calculations, we obtain

      TN1()¯(B)=TS1¯(B)TS2¯(B)=18.

      As a summary, we have TN1()¯(A)TN1()¯(B)=18.

  • (4) Determine TN1()¯(AB).


    AB = {[0]18, [5]18, [6]18, [7]18, [11]18, [16]18, [17]18}.


    • • Cosets of S1 = ⟨[9]18⟩ intersected with AB are [0]18 + ⟨[9]18⟩, [2]18 + ⟨[9]18⟩, [5]18 + ⟨[9]18⟩, [6]18 + ⟨[9]18⟩, [7]18 + ⟨[9]18⟩, [8]18 + ⟨[9]18⟩.

      TS1¯(AB)={[x]1818([x+1]18+[9]18)(AB)}=18-{[0]18,[2]18,[3]18,[9]18,[11]18,[12]18}.

    • • Cosets of S2 = ⟨[6]18⟩ intersected with AB are [0]18 + ⟨[6]18⟩, [1]18 + ⟨[6]18⟩, [4]18 + ⟨[6]18⟩, [5]18 + ⟨[6]18⟩.

      TS2¯(AB)={[x]1818([x+1]18+[6]18)(AB)}=18-{[1]18,[2]18,[7]18,[8]18,[13]18,[14]18}.

      From the above calculations, we obtain

      TN1()¯(AB)=TS1¯(AB)TS2¯(AB)=18-{[2]18}.

    As a summary, we obtain TN1()¯(A)TN1()¯(B)TN1()¯(AB).

Example 9

Let ℤ18 be a group of integers modulo 18, ℬ = {S1, S2} where S1 = ⟨[9]18⟩ = {[0]18, [9]18}, S2 = ⟨[6]18⟩ = {[0]18, [6]18, [12]18} be normal subgroups of ℤ18 and T : ℤ18* (ℤ18) be defined as

T([x]18)={[x+1]18},

for every [x]18 ∈ ℤ18. Let A, B ⊂ ℤ18 where A = {[0]18, [5]18, [11]18}, B = {[9]18, [17]18} so AB = {[0]18, [5]18, [9]18, [11]18, [17]18}.

  • (1) Determine TN1()¯(A) and TN1()¯(B).

    • • There are no cosets of S1 = ⟨[9]18⟩ contained in A. TS1 (A) = {[x]18 ∈ ℤ18 | [x + 1]18 + ⟨[9]18⟩ ⊆ A} = ∅.

    • • There are no cosets of S2 = ⟨[6]18⟩ contained in A. TS2 (A) = {[x]18 ∈ ℤ18 | [x + 1]18 + ⟨[6]18⟩ ⊆ A} = ∅.

      From the above calculations, we obtain

      TN1()_(A)=TS1_(A)TS2_(A)=.

    • • There are no cosets of S1 = ⟨[9]18⟩ contained in B. TS1 (B) = {[x]18 ∈ ℤ18 | [x+1]18+⟨[9]18⟩ ⊆ B} = ∅.

    • • There are no cosets of S2 = ⟨[6]18⟩ contained in B. TS2 (B) = {[x]18 ∈ ℤ18 | [x+1]18+⟨[6]18⟩ ⊆ B} = ∅.

      From the above calculations, we obtain

      TN1()_(B)=TS1_(B)TS2_(B)=.

    As a summary, we have TN1()¯(A)TN1()¯(B)=

  • (2) Determine TN1()¯(AB).

    • • Cosets of S1 = ⟨[9]18⟩ contained in AB is ⟨[9]18⟩.

      TS1_(AB)={[x]1818[x+1]18+[9]18AB}={[8]18,[17]18}.

    • • Cosets of S2 = ⟨[6]18⟩ contained in AB is [5]18+ ⟨[6]18⟩.

      TS2_(AB)={[x]1818[x+1]18+[6]18AB}={[4]18,[10]18,[16]18}.

      From the above calculations, we obtain

      TN1()_(AB)=TS1_(AB)TS2_(AB)={[4]18,[8]18,[10]18,[16]18,[17]18}.

    As a summary, we have TN1()¯(AB)TN1()¯(A)TN1()¯(B).

Proposition 3

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued homomorphism. If A is a non-empty subset of G′ and e′A, then eTNr()¯(A) where e and e′ are neutral elements of G and G′, respectively.

Proof

Since T is a set-valued group homomorphism, it follows that for any aG, T(e) = T(aa−1) = T(a)T(a−1) = T(a)(T(a))−1. If xT(a) then e′ = xx−1T(a)(T(a))−1 = T(e). Because e′A and e′T(e) then e′T(e)ℬrA, for every ℬr ⊆ ℬ such that eTNr()¯(A).

Proposition 4

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued mapping. Then,

  • (1) TNr()¯(A)=rT¯(Ar),

  • (2) TN1()¯(A)rT¯(Ar),

where Aℬr = A (∩SrS).

Proof

(1) Similar to the proof of Theorem 4.5 in [26], we have

TNr()¯(A)=rTr¯(A)=rT¯(Ar).

(2) Similar to the proof of Theorem 4.5 in [26], we have TNr()¯(A)=rTr¯(A)rT¯(Ar).

The following corollary follows from Proposition 4.

Corollary 1

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued mapping. Let N be a normal subgroup of G′ and AG′, A ≠ ∅︀. Then,

  • (1) TNr()¯(A)=rTr¯(Ar),

  • (2) TNr()¯(A)rTr¯(Ar), where Aℬr = A (∩SrS).

Proof

(1) Similar to the proof of Theorem 4.5 in [26], we have Tr¯(Ar)=T¯(Arr)=T¯(Ar) so that with Proposition 4, we get TNr()¯(A)=rTr¯(Ar).

(2) Since AAℬr, it follows that TNr()(A) = ∪rTr (A) ⊆ ∪rTr (Aℬr).

Definition 7

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued mapping. Let A be a non-empty subset of G′. We define the sets

TNr()_(A)={xGT(x)rrA,for r,r},

and

TNr()¯(A)={xGT(x)rrA,for r,r},

where T(x)rr=T(x)(SrS)(RrR).

Proposition 5

Let G and G′ be two groups, (G′, ) be the nearness approximation space, and T : G*(G′) be a set-valued homomorphism. If A is a non-empty subset of G′, then

  • (1) TNr()¯(A)=TNr()¯(A),

  • (2) TNr()¯(A)TNr()¯(A).

Proof

(1) Let xTNr()¯(A). Then, T ( x ) r r A for some r, r . Since T ( x ) r T ( x ) r r A, we have xTNr()¯(A) so that TNr()¯(A)TNr()¯(A). Let xTNr()¯(A). Subsequently, T(x)rr = T(x)rA for some r so that xTNr()¯(A), which implies that TNr()¯(A)TNr()¯(A). Thus, TNr()¯(A)=TNr()¯(A).

(2) Let xTNr()¯(A). Then, T(x)rA ≠ ∅︀ for some r. Since T(x)rT(x)rr for every r, we obtain T(x)rrA such that xTNr()¯(A). Thus, TNr()¯(A)TNr()¯(A).

The following example shows that the converse of Proposition 5 (2) is not true in general.

Example 10

Let ℤ24 be a group of integers modulo 24, = {S1, S2} where S1 = ⟨[8]24⟩ = {[0]24, [8]24, [16]24}, S2 = ⟨[12]24⟩ = {[0]24, [12]24} be normal subgroups of ℤ24 and T : ℤ24*(ℤ24) be defined as

T([x]24)={(x+1)24},

for every [x]24 ∈ ℤ24. It is easy to see S1+S1 = S1, S2+S2 = S2 and S1 + S2 = ⟨[4]24⟩. Let A = {[4]24}. Then,

TS1+S1¯(A)=TS1¯(A)={[3]24,[11]24,[19]24},TS2+S2¯(A)=TS2¯(A)={[3]24,[15]24},TS1+S2¯(A)={[3]24,[7]24,[11]24,[15]24,[19]24,[23]24}.

We have

TN1()¯(A)=TS1¯(A)TS2¯(A)={[3]24,[11]24,[15]24,[19]24},TN1(+)¯(A)=TS1+S1¯(A)TS2+S2¯(A)TS1+S2¯(A)={[3]24,[7]24,[11]24,[15]24,[19]24,[23]24}.

Thus, we obtain TN1(+)(A)¯TN1()(A)¯.

Proposition 6

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued homomorphism. If A and B are two non-empty subsets of G′, then

TNr()¯(A)TNr()¯(B)TNr()¯(AB).
Proof

Let zTNr()¯(A)TNr()¯(B). Then, there exist xTNr()¯(A) and yTNr()¯(B) such that z = xy. Since xTNr()¯(A) and yTNr()¯(B) then T(x)rA ≠ ∅︀ for some r and T(y)rB for some r so there exist a, bG′ where aT(x)rA and bT(y)rB. Since aT(x)r, bT(y)r and T be a set-valued group homomorphism then abT(x)rT(y)r=T(x)T(y)rr=T(xy)rr so that abT(xy)rrAB which implies T(xy)rrABϕ. Therefore, z=xyTNr()¯(AB) so that TNr()¯(A)TNr()¯(B)TNr()¯(AB).

The following example shows that the converse of Proposition 6 is generally not true.

Example 11

Let ℤ24 be a group of integers modulo 24, = {S1, S2} where S1 = ⟨[8]24⟩ = {[0]24, [8]24, [16]24}, S2 = ⟨[12]24⟩ = {[0]24, [12]24}, S1 + S2 = ⟨[4]24⟩ = {[0]24, [4]24, [8]24, [12]24, [16]24, [20]24} be normal subgroups of ℤ24 and T : ℤ24*(ℤ24) be defined as

T([x]24)={8[x]24},

for every [x]24 ∈ ℤ24. Notice that T is a set-valued homomorphism. Let A = {[1]24, [5]24}, B = {[3]24}, A + B = {[4]24, [8]24}. We get

  • TS1¯(A)=TS2¯(A)=TS1¯(B)=TS2¯(B)=,

  • TS1+S2¯(A+B)=24,

  • TN1()¯(A)=TN1()¯(B)=,

  • TN1(+)¯(A+B)=24.

This shows that TN1(+)¯(A+B)TN1()¯(A)+TN1()¯(B).

Proposition 7

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued homomorphism. If A and B are two non-empty subsets of G′, then

TNr()_(A)TNr()_(B)TNr()_(AB).
Proof

Let zTNr()¯(A)TNr()¯(B). Then, there exist xTNr()¯(A) and yTNr()¯(B) so that z = xy. Since xTNr()¯(A) and yTNr()¯(B) then T(x)rA and T(y)rB for some r, r. Since T is a set-valued group homomorphism then T(x)rT(y)r=T(x)T(y)rr=T(xy)rrAB, which implies that z = xyTNr()¯(AB). Hence TNr()¯(A)TNr()¯(B)TNr()¯(AB). By Proposition 5(1), we have TNr()¯(A)TNr()¯(B)TNr()¯(AB).

Example 12

Let Q8 = {1, −1, i, −i, j, −j, k, −k} be a quaternion group, = {S1, S2} where S1 = {1, −1, i, −i}, S2 = {1, −1, j, −j} be the normal subgroups of Q8 and T : Q8*(Q8) be defined as

T(x)={x,-x},

for every xQ8. From Example 4.2, in [35], the mapping T is a set-valued homomorphism. Let A = {1, i}, B = {1, −1, −i}, A.B = {1, −1, i, −i} = S1. Then, we get

  • TS1 (A) = TS2 (A) = TS1 (B) = TS2 (B) = TS2 (A.B) = ∅︀,

  • TS1 (A.B) = {1, −1, i, −i}.

So that

  • TN1()¯(A)=TN1()¯(B)=,

  • TN1()¯(A.B)={1,1,i,i}.

This implies that TN1()¯(A.B)TN1()¯(A).TN1()¯(B).

Proposition 8

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued homomorphism. Let A be a subgroup of G′, TNr()¯(A)=TNr()¯(A) and TNr()¯(A). Subsequently, TNr()¯(A) is a subgroup of G.

Proof

Let x, yTNr()¯(A). Then, T(x)rA ≠ ∅︀ and T(y)rA for some r, r. So there exist aT(x)rA and bT(y)rA. Since A is a subgroup of G′, we have ab−1A. Since T is a set-valued homomorphism, we have ab-1T(x)r(T(y)r)-1=T(x)r(r)-1(T(y))-1=T(x)rrT(y-1)=T(x)T(y-1)rr=T(xy-1)rr so that ab-1T(xy-1)rrA which implies that xy-1TNr()¯(A)=TNr()¯(A). Therefore, TNr()¯(A) is a subgroup of G.

Proposition 9

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G* (G′) be a set-valued homomorphism. Let A be a normal subgroup of G′, TNr()¯(A)=TNr()¯(A) and TNr()¯(A). Subsequently, TNr()¯(A) is a normal subgroup of G.

Proof

This is sufficient to demonstrate that TNr()¯(A) is normal. Let xTNr()¯(A) and gG. Then, T(x)rA ≠ ∅︀ for some r and T(g) is a non-empty set, we can select g′ ∈ T(g). Thus, aT(x)rA. exists and because A is a normal subgroup of G′, we have aga−1A. Since T is a set-valued homomorphism, we have aga−1T(x)rT(g)(T(x)r)−1 = T(x)rT(g)r−1T(x)−1=T(x) rT(g)r−1T(x−1)=T(x)T(g)T(x−1)rr−1=T(xgx−1) r such that aga−1T(xgx−1)rA which implies that xgx-1TNr()¯(A). Therefore, TNr()¯(A) is a normal subgroup of G.

Example 13

Let ℤ24 be a group of integers modulo 24, = {S1, S2}, where S1 = ⟨[8]24⟩ = {[0]24, [8]24, [16]24}, S2 = ⟨[12]24⟩ = {[0]24, [12]24} be normal subgroups of ℤ24 and T : ℤ24*(ℤ24) be defined as

T([x]24)={3[x]24},

for every [x]24 ∈ ℤ24. Observe that T is a set-valued homomorphism, S1 + S1 = S1, S2 + S2 = S2, and S1 + S2 = ⟨[4]24⟩. Let A ⊂ ℤ24 where A = 2ℤ24. Then, A is a normal subgroup of ℤ24. We get

TS1¯(A)=TS2¯(A)=TS1+S1¯(A)=TS2+S2¯(A)=TS1+S2¯(A)=224.

so that TN1()¯(A)=TN1(+)¯(A)=224 is a normal subgroup of ℤ24.

Proposition 10

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued homomorphism. Let A be a subgroup of G′ and TNr()¯(A) ≠ ∅. Then, TNr()¯(A) is a subgroup of G.

Proof

Let x, yTNr()¯(A). Then, T(x)rA and T(y)rA for some r, r. Since A is a subgroup of G′, we have (T(y)r)-1=T(y-1)rA. Since T is a set-valued homomorphism, we have T(x)rT(y-1)r=T(x)T(y-1)rr=T(xy-1)rrA which implies that xy−1TNr(B)¯(A). From Proposition 5(1), we have xy−1TNr()¯(A). Therefore, TNr()¯(A) is a subgroup of G.

Proposition 11

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued homomorphism. Let A be a normal subgroup of G′ and TNr()¯(A) ≠ ∅. Then, TNr()¯(A) is a normal subgroup of G.

Proof

This suffices for demonstrating that TNr()¯(A) is normal. Let xTNr()¯(A) and gG. Then, T(x)rA for some r and T(g) is a non-empty set. Since A is a subgroup of G′, we have (T(x)r)−1 = T(x−1)rA. Since A is a normal subgroup of G′ and T is a set-valued homomorphism, we have T(x)rT(g)T(x−1)r = T(x)T(g)T(x−1)r = T(xgx−1)rA which implies that xgx−1TNr()¯(A). Therefore, TNr()¯(A) is a normal subgroup of G.

Proposition 12

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued mapping. Let A be a non-empty subset of G′. Then,

  • (1) If || = k, then TNk()¯(A)=T¯(A) and TNk()¯(A)=T¯(A).

  • (2) If 1 ≤ r < k where || = k, then Tr¯(A)TNr()¯(A) and Tr¯(A)TNr()¯(A).

Proof

The proof results from Definitions 5 and 6

From Proposition 12(1), we observe that the generalized approximations in a group based on set-valued mapping is a special type of near generalized approximation.

Proposition 13

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued mapping and ij, 1 ≤ i < j ≤ ||. Let A be a non-empty subset of G′. Then

  • (1) TNr(i)¯(A)TNr(j)¯(A),

  • (2) TNr(i)¯(A)TNr(j)¯(A).

Proof

(1) From Definition 6, we have TNr(k)¯(A)=rkTr¯(A), for 1 ≤ k ≤ ||. Since 1 ≤ i < j ≤ ||, we have ij then TNr(i)¯(A)TNr(j)¯(A).

(2) From Definition 6, we obtain TNr(k)¯(A)=rkTr¯(A) for 1 ≤ k ≤ ||. Since 1 ≤ i < j ≤ ||, we have ij then TNr(i)¯(A)TNr(j)¯(A).

Proposition 14

Let G and G′ be two groups, (G′, ), be a nearness approximation space, and T : G*(G′) be a set-valued mapping. Let A be a non-empty subset of G′. Then,

TNi()_(A)TNj()_(A)TNj()¯(A)TNi()¯(A),

where 1 ≤ i < j ≤ ||.

Proof

Let xTNi()¯(A). Subsequently, T(x)iA for a certain i. Since i < j then ij so that T(x)j = T(x)∩Sj ST(x)∩Si S = T(x)iA which yields that xTNj()¯(A). Therefore, TNi()¯(A)TNj()¯(A). From Proposition 2 (1), we have TNj()_(A)TNj()¯(A). Let xTNj()¯(A). Then, T(x)jA ≠ ∅︀ for some j. Since i < j then ij so that T(x)j = T(x)∩SjST(x)∩SiS = T(x)i. Thus, we have T(x)ℬiA ≠ ∅︀, which yields xTNi()¯(A). Therefore, TNi()_(A)TNj()_(A)TNj()¯(A)TNi()¯(A).

In this study, we showed that near set theory is applicable to group theory. We demonstrated the concepts of near-generalized approximations in a group based on set-valued mapping, an extension of generalized approximations in a group based on set-valued mapping and near approximations in a group. We demonstrate that near-generalized lower and upper approximations of a (normal) subgroup in a group based on set-valued mapping can form a (normal) subgroup of a group. Furthermore, we showed that the generalized approximations in a group based on a set-valued mapping are a special type of near-generalized approximations in a group based on a set-valued mapping. Finally, we found that if the r index is larger, then the near-generalized lower approximations in a group based on set-valued mapping are greater (in terms of the inclusion sets), but the near-generalized upper approximations in a group based on set-valued mapping are smaller sets. Near-generalized lower approximations in a group based on set-valued mapping are subsets of near-generalized upper approximations in a group based on set-valued mapping. As an extension of this study, we are interested in applying this structure in a ring.

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DianWinda Setyawati is a doctoral student in the Department of Mathematics at Institut Teknologi Sepuluh Nopember, Indonesia. Since 2003, she has worked as a lecturer in the Department of Mathematics at Institut Teknologi Sepuluh Nopember, Indonesia. Her research areas include group theory, ring theory, and rough set theory.

Subiono is a professor in the Department of Mathematics at Institut Teknologi Sepuluh Nopember, Indonesia. He has worked as a lecturer in the Department of Mathematics at Institut Teknologi Sepuluh Nopember, Indonesia, since 1984. His research areas are min-maxpluss algebra and applied algebra.

Bijan Davvaz received his B.Sc. degree in Applied Mathematics from Shiraz University in 1988 and his M.Sc. degree in Pure Mathematics from Tehran University in 1990. In 1998, he received his Ph.D. in mathematics at Tarbiat Modares University. He is a member of the Editorial Boards of 24 mathematical journals. He has also served as the Head of the Department of Mathematics (1998–2002), Chairman of the Faculty of Science (2004–2006), and Vice-President for Research (2006–2008) at Yazd University, Iran. His areas of interest include algebra (groups, rings, and modules), algebraic hyperstructures, rough sets, and fuzzy logic. He has published 12 books on mathematics. He is currently a distinguished professor of Mathematics at Yazd University.

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(2): 125-140

Published online June 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.2.125

Copyright © The Korean Institute of Intelligent Systems.

Near-Generalized Approximations in Groups Based on a Set-Valued Mapping

Dian Winda Setyawati1, Subiono1, and Bijan Davvaz2

1Department of Mathematics, Institut Teknologi Sepuluh Nopember, Kampus ITS, Sukolilo-Surabaya, Indonesia
2Department of Mathematical Sciences, Yazd University, Yazd, Iran

Correspondence to:Subiono (subiono2008@matematika.its.ac.id)

Received: June 5, 2023; Revised: August 24, 2023; Accepted: June 5, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A normal subgroup of a group can partition a group into equivalence classes. Therefore, approximations can be constructed within a group. The near approximations in a group are extensions of the approximations in a group. A set-valued mapping T from group G to the set of all non-empty subsets of group G′ can establish generalized approximations in group G based on the set-valued mapping T. In this study, we introduce the notion of near-generalized approximations in a group based on set-valued mapping, an extension of the concept of generalized approximations in a group based on set-valued mapping and near approximations in a group. We then present some properties of nearby subgroups in a group based on set-valued mapping. Furthermore, we compare these types of near-generalized and generalized approximations in a group based on set-valued mapping.

Keywords: Normal subgroup, Group, Approximations, Near approximations, Generalized approximations, Near-generalized approximations

1. Introduction

In the 1980s, Pawlak [1] introduced rough set theory as a mathematical tool for dealing with the problems of vagueness and uncertainty in decision-making. The theory has been used in various fields, including machine learning, data mining, intelligent systems, pattern recognition, decision analysis, inductive reasoning, mereology, etc. [210]. The rough set theory is based on the concept of equivalence relation that is reflexive, symmetric, and transitive. The foundation of this theory is an equivalence class, which is used to construct the lower and upper approximations of a set. The lower approximation of set A is the union of the equivalent classes, subsets of set A, and upper approximation of set A is the union of the equivalent classes that intersect with set A. The concept of lower and upper approximations of a set with respect to the normal subgroups of a group has been extensively studied by many scientists [1114]. In 2004, Davvaz [15] introduced the concept of the lower and upper approximations of a set with respect to the ideals of a ring. Furthermore, in 2006, Davvaz and Mahdavipour [16] introduced the concept of lower and the upper approximations of a set with respect to the submodules of an R-module. Scientists studied the rough set of the ideals of a ring and the submodules of an R-module [1522]. In 2022, Salama et al. [23] constructed the lower and upper approximations of a set using dominance relation that is reflexive, antisymmetric, and transitive.

The algebraic structure of rough sets is generalized to approximation mapping. It is generalized to approximation mappings with respect to an equivalence relation of algebraic structures, such as generalized approximation mapping with respect to the normal subgroups of a group [24], the ideals of a ring [25, 26] and the submodules of an R-module [27]. In 2022, Setyawati and Subiono [24] discussed the relationship between the lower and upper approximations of different sets with different normal subgroups of a group based on set-valued mapping.

The near set was introduced by Peters [28], in which the objects were perceived to be close to each other, with a similarity of description to some degree. The basic idea behind near sets is the description and classification of objects, that is, the members of a universal set based on the perceptual information system. Peters and Wasilewski [29] introduced an approach to the foundations of information science considered in the context of near sets. The algebraic properties of near sets are described in [30]. In 2019, Bağirmaz [31] introduced and derived the properties of near approximations in a group. Furthermore, he compared the near approximations and the approximations in a group. The near approximations in a group are an extension of the rough approximations in a group. Near approximations in a group used two or more normal subgroups of a group, whereas approximations in a group used the normal subgroup of a group. Davvaz et al. [32,33] extended the ideas presented by Bağirmaz [31] by using the ideals of a ring and the submodules of an R-module. In 2023, Setyawati et al. [34] applied the concept of near approximations to the Cayley graphs.

In this study, we introduce the notion of near-generalized approximations in a group based on set-valued mapping (an extension of the concept of generalized approximations in a group based on set-valued mapping [24] and near approximations in a group [31]) and investigate their properties. We then present some properties of nearby subgroups in a group based on set-valued mapping. Furthermore, we compare these types of near-generalized and generalized approximations in a group based on set-valued mapping.

The remainder of this paper is organized as follows. After its introduction, Section 2 reviews fundamental definitions of and near approximations in a group. Section 3 reviews fundamental definitions of approximation in a group based on set-valued mapping. Section 4 introduces the notion of near-generalized approximations in a group based on set-valued mapping and proves their properties. Section 5 presents the properties of lower and upper near subgroups in a group based on set-valued mapping. Section 6 compares the near-generalized and generalized approximations in a group based on a set-valued mapping.

2. Approximations and Near Approximations in a Group

In this section, we define near approximations in a group, which is an extended definition of rough approximations in a group. Let K be a finite set of objects and Q be an equivalence relation on K. A pair (K, Q) is called the approximation space. If S is a normal subgroup of G, then pair (G, S) is an approximation space.

Definition 1 [11]

Let G be a group, S a normal subgroup of G, and A a non-empty subset of G. We define the sets

AprS_(A)={xGxSA},

and

AprS¯(A)={xGxSA},

called lower and upper approximations, respectively. of set A with respect to the normal subgroup S.

Example 1

Let D6 = {ρ0, ρ1, ρ2, ρ3, ρ4, ρ5, μ, ρ1μ, ρ2μ, ρ3μ, ρ4μ, ρ5μ} be a dihedral group with order 12 and S = {ρ0, ρ2, ρ4} be a normal subgroup of D6 where ρ0 is a neutral element of D6. Let A = {ρ1, ρ3, ρ5, μρ2}. Then, we obtain AprS(A) = {ρ1, ρ3,ρ5} and AprS¯(A)={ρ1,ρ3,ρ5,μ,μρ2,μρ4}.

We denote as the set of all normal subgroups in the group G and ℬ as a subset of . A pair (G, ) is called a nearness approximation space. If (G, ) is a nearness approximation space, then

xr=x(SrS),

where ℬr ⊆ ℬ.

Definition 2 [31]

Let (G, ) be a nearness approximation space and A be a non-empty subset of G. We define the sets

Nr()_(A)={xGxrA,for r},

and

Nr()¯(A)={xGxrA,for r},

called near lower and near upper approximations of set A with respect to the normal subgroups of G, respectively. Clearly, Nr()¯(A)=rAprr¯(A) and Nr()¯(A)=rAprr¯(A).

Consider the following example. In this example [31], three different near approximations of the subset A of G defined by families N1(ℬ)(A), N2(ℬ)(A), N3(ℬ)(A).

Example 2

Let ℤ12 be a group of integers modulo 12 and ℬ = {S1, S2, S3} where S1 = {[0]12, [6]12}, S2 = {[0]12, [4]12, [8]12}, S3 = {[0]12, [3]12, [6]12, [9]12} be normal subgroups of ℤ18. Let A = {[0]12, [1]12, [3]12, [4]12, [7]12, [8]12, [10]12}. Then, we get

  • N1()¯(A)=[0]12,[1]12,[4]12,[7]12,[8]12,[10]12,

  • N2()¯(A)=A,

  • N3()¯(A)=A,

  • N1()¯(A)=12,

  • N2()¯(A)={[0]12,[1]12,[2]12,[3]12,[4]12,[6]12,[7]12,[8]12,[9]12,[10]12},

  • N3()¯(A)=A.

From the calculations above, we can conclude that N1()_(A)N2()_(A)=N3()_(A)=N3()¯(A)N2()¯(A)N1()¯(A).

3. Approximations in a Group based on a Set-Valued Mapping

In this study, if N is a non-empty set, then *(N) denotes the set of all non-empty subsets of N. The following definition is a generalization of Definition 1.

Definition 3 [35]

Let M and N be two non-empty sets and T : M*(N) be a set-valued mapping. Let A be a non-empty subset of N. We define the sets

T_(A)={xMT(x)A},

and

T¯(A)={xMT(x)A},

which are called lower and upper approximations of A under T, respectively.

Example 3

Let M = {p, q, r, s, t}, N = {a, b, c, d, e, f, g, h, i, j, k}, and A = {a, b, c, g, h, k}. Let the set-valued mapping T : M*(N) be defined as T(p) = {a, c, h}, T(q) = {b}, T(r) = {e, f, g}, T(s) = {d, i, j}, T(t) = {j, k}. Then, T(A) = {p, q} and (A) = {p, q, r, t}.

In [35], the set-valued homomorphism between the two groups is defined.

Definition 4

Let G and G′ be the two groups. A mapping T : G*(G′) is called a set-valued homomorphism if

  • (1) T(xy) = T(x)T(y),

  • (2) (T(x))−1 = {a−1 | aT(x)} = T (x−1),

for all x, yG.

Example 4 shows that each group homomorphism is a set-valued homomorphism.

Example 4

Let G and G′ be two groups and f : GG′ be a group homomorphism. Subsequently, the set-valued mapping T : G*(G′) defined by T(x) = {f(x)} is a set-valued homomorphism because for every x, yG we have

  • (1) T(xy) = {f(xy)} = {f(x)f(y)} = {f(x)} {f(y)} = T(x)T(y),

  • (2) (T(x))−1 = {(f(x))−1} = {f (x−1)} = T(x−1).

Example 5 [35]

The set ℝ* of non-zero real numbers is a group. Suppose that D={[100a]|a*}. The set-valued mapping T : ℝ**(D) defined by T(a)={[100a],[100-a]} is a set-valued homomorphism.

Furthermore, we introduced the generalized lower and upper approximations of a set with respect to the normal subgroup of a group [24]. This definition is a generalization of Definition 3. The following definition is similar to that in [25].

Definition 5

Let G and G′ be two groups and T : G* (G′) be a set-valued mapping. Let S be a normal subgroup of G′ and A be a non-empty subset of G′. Then,

TS_(A)={xGT(x)SA},

and

TS¯(A)={xG(T(x)S)A},

called the generalized lower and upper approximations of A under T with respect to a normal subgroup S of G′, respectively.

Example 6

Let D6 = {ρ0, ρ1, ρ2, ρ3, ρ4, ρ5, μ, ρ1μ, ρ2μ, ρ3μ, ρ4μ, ρ5μ} be a dihedral group with order 12, S = {ρ0, ρ2, ρ4} be a normal subgroup of D6 where ρ0 is a neutral element of D6. Let T : D6* (D6) be defined as

T(x)={{x}if x{ρ0,ρ1,ρ2,ρ3,ρ4,ρ5},{ρi,μ}if x=ρiμ{μ,ρ1μ,ρ2μ,ρ3μ,ρ4μ,ρ5μ}.

Let A = {ρ1, ρ3, ρ5, μρ2}. Then, we get

  • TS(A) = {ρ1, ρ3, ρ5},

  • TS¯(A)={ρ1,ρ3,ρ5,μ,ρμ,ρ2μ,ρ3μ,ρ4μ,ρ5μ}.

From the above calculations, we conclude that TS_(A)TS¯(A).

4. Near-Generalized Approximations in a Group based on a Set-Valued Mapping

Definitions 2 and 5 motivate to form Definition 6. We denote as the set of all normal subgroups in a group G′ and ℬ as a subset of . A pair (G′, ) is defined as nearness approximation space. If (G′, ) is a nearness approximation space, then

T(x)r=T(x)(SrS),

where ℬr ⊆ ℬ.

Definition 6

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued mapping. Let A be a non-empty subset of G′. We define the sets

TNr()_(A)={xGT(x)rA,for r},

and

TNr()¯(A)={xGT(x)rA,for r},

which are called the near-generalized lower and upper approximations of A under T with respect to the normal subgroups ℬ of G′, respectively. Clearly, TNr()¯(A)=rTr¯(A) and TNr()¯(A)=rTr¯(A).

Example 7

Let D4 = {ρ0, ρ1, ρ2, ρ3, μ, ρ1μ, ρ2μ, ρ3μ} be a dihedral group with order 8, HD4 and H = {ρ0, ρ1, ρ2, ρ3} be a group where ρ0 is a neutral element of D4 and H. Let T : ℤ36 × H*(ℤ36 × D4) be defined by

T(([x]36,ρi))={([x]36,ρi),([x+1]36,ρiμ)},

for every ([x]18, ρi) ∈ ℤ36 × H. Let N = {ρ0, ρ2, μ, ρ2μ}, H be normal subgroups of D4 and ℬ = {S1, S2, S3}, where S1 = ⟨[9]36⟩ × H, S2 = ⟨[6]36⟩ × N, S3 = ⟨[4]36⟩ × H be normal subgroups of ℤ36 × D4. Let A ⊂ ℤ36 × D4 where

A={([a]36,b)[a]36Z36-{[1]36,[2]36,[3]36,[4]36,[13]36,[14]36,[15]36,[16]36,[19]36,[20]36,[21]36,[25]36,[26]36,[27]36,[28]36}andb{ρ0,ρ2,ρ3,μ,ρ2μ,ρ3μ}}.

We determine TNr()¯(A) and TNr()¯(A).

  • • The first set of components of A is ℤ36 – {[1]36, [2]36, [3]36, [4]36, [13]36, [14]36, [15]36, [16]36, [19]36, [20]36, [21]36, [25]36, [26]36, [27]36, [28]36}. There are no cosets of ⟨[4]36⟩ contained in the first set of components of A. Cosets of ⟨[6]36⟩ contained in the first set of components of A are [0]18 + ⟨[6]36⟩, [5]18 + ⟨[6]36⟩. There are no cosets of ⟨[9]36⟩ contained in the first set of components of A. Cosets of ⟨[12]36⟩ contained in the first set of components of A are [0]36+⟨[12]36⟩, [5]36+⟨[12]36⟩, [6]36+ ⟨[12]36⟩, [10]36 + ⟨[12]36⟩, [11]36 + ⟨[12]36⟩. Cosets of ⟨[18]36⟩ contained in the first set of components of A are [0]36 + ⟨[18]36⟩, [5]36 + ⟨[18]36⟩, [6]36 + ⟨[18]36⟩, [11]36 + ⟨[18]36⟩, [12]36 + ⟨[18]36⟩, [17]36 + ⟨[18]36⟩.

  • • The second set of components of A is {ρ0, ρ2, ρ3, μ, ρ2μ, ρ3μ}. Cosets of {ρ0, ρ2} contained in the second set of components of A are {ρ0, ρ2}, {μ, ρ2μ}. Cosets of N contained in the second set of components of A are N. There are no cosets of H contained in the second set of components of A.

  • (1) Determine TN1()¯(A).


    For r = 1, the subsets ℬ1 ⊆ ℬ are {S1}, {S2}, {S3}.

    • S1 = ⟨[9]36⟩×H then (36×D4)/S1=36×84×4=18.

      TS1_(A)={([x]36,ρi)36×HT(([x]36,ρi))S1A}={([x]36,ρi)36×H{([x]36,ρi),([x+1]36,ρiμ)}([9]36×H)A}=.

    • S2 = ⟨[6]36⟩×N then (36×D4)/S2=36×86×4=12.

      TS2_(A)={([x]36,ρi)36×HT(([x]36,ρi))S2A}={([x]36,ρi)36×H{([x]36,ρi),([x+1]36,ρiμ)}([6]36×N)A}={([x]36,ρi)36×H[x]36{[5]36,[11]36,[17]36,[23]36,[29]36,[35]36}and ρi{ρ0,ρ2}}.

    • S3 = ⟨[4]36⟩×H then (36×D4)/S3=36×89×4=8.

      TS3_(A)={([x]36,ρi)36×HT(([x]36,ρi))S3A}={([x]36,ρi)36×H{([x]36,ρi),([x+1]36,ρiμ)}([4]36×H)A}=.

    From the above calculations, we obtain

    TN1()_(A)={([x]36,ρi)36×HT(([x]36,ρi))1A,for 1}=TS1_(A)TS2_(A)TS3_(A)={([x]36,ρi)36×H[x]36{[5]36,[11]36,[17]36,[23]36,[29]36,[35]36}and ρi{ρ0,ρ2}}.

  • (2) Determine TN2()¯(A).


    For r = 2, the subsets ℬ2 ⊆ℬ are {S1, S2}, {S1, S3}, {S2, S3}. By the above explanation, we have T(x)ℬ2 = T(x)(∩S∈ℬ2S.)

    • S1S2 = ⟨[18]36⟩×{ρ0, ρ2} then (36×D4)/(S1S2)=36×82×2=72.

      TS1S2_(A)={([x]36,ρi)36×HT(([x]36,ρi))(S1S2)A}={([x]36,ρi)36×H{([x]36,ρi),([x+1]36,ρiμ)}([18]36×{ρ0,ρ2})A}={([x]36,ρi)36×H[x]36{[5]36,[11]36,[17]36,[23]36,[29]36,[35]36}and ρi{ρ0,ρ2}}.

    • S1S3 = {[0]36} × H then (36×D4)/(S1S3)=36×81×4=72.

      TS1S3_(A)={([x]36,ρi)36×HT(([x]36,ρi))(S1S3)A}={([x]36,ρi)36×H{([x]36,ρi),([x+1]36,ρiμ)}({[0]36}×H)A}=.

    • S2S3 = ⟨[12]36⟩×{ρo, ρ2} then (36×D4)/(S2S3)=36×83×2=48.

      TS2S3_(A)={([x]36,ρi)36×HT(([x]36,ρi))(S2S3)A}={([x]36,ρi)36×H{([x]36,ρi),([x+1]36,ρiμ)}([12]36×{ρ0,ρ2})A}={([x]36,ρi)36×H[x]36{[5]36,[10]36,[11]36,[17]36,[22]36,[23]36,[29]36,[34]36,[35]36}and ρi{ρ0,ρ2}}.

    From the above calculations, we obtain

    TN2()_(A)={([x]36,ρi)36×HT(([x]36,ρi))2A,for 2}=TS1S2_(A)TS1S3_(A)TS2S3_(A)={([x]36,ρi)36×H[x]36{[5]36,[10]36,[11]36,[17]36,[22]36,[23]36,[29]36,[34]36,[35]36}and ρi{ρ0,ρ2}}.

  • (3) Determine TN3()¯(A).


    For r = 3, the subsets ℬ3 ⊆ ℬ then {S1, S2, S3}. By the above explanation, we have T(x)ℬ3 = T(x) (∩S∈ℬ3S).

    • S1S2S3 = {[0]36} × {ρo, ρ2} then (36×D4)/(S1S2S3)=36×81×2=144.

      TN3()_(A)=TS1S2S3_(A)={([x]36,ρi)36×HT(([x]36,ρi))(S1S2S3)A}={([x]36,ρi)36×H{([x]36,ρi),([x+1]36,ρiμ)}({[0]36×{ρ0,ρ2})A}={([x]36,ρi)36×H[x]36{[5]36,[6]36,[7]36,[8]36,[9]36,[10]36,[11]36,[17]36,[22]36,[23]36,[29]36,[30]36,[31]36,[32]36,[33]36,[34]36,[35]36}and ρi{ρ0,ρ2}}.

    As a summary, we have TN1()¯(A)TN2()¯(A)TN3()¯(A).

    • • The first set of components of A is ℤ36 – {[1]36, [2]36, [3]36, [4]36, [13]36, [14]36, [15]36, [16]36, [19]36, [20]36, [21]36, [25]36, [26]36, [27]36, [28]36}.


      Every coset of ⟨[4]36⟩ intersects with the first set of components of A.


      Every coset of ⟨[6]36⟩ intersects with the first set of components of A.


      Every coset of ⟨[9]36⟩ intersects with the first set of components of A.


      Every coset of ⟨[12]36⟩ intersects with the first set of components of A except [1]36 +⟨[12]36⟩, [2]36 + ⟨[12]36⟩, [3]36 + ⟨[12]36⟩, [4]36 + ⟨[12]36⟩.


      Every coset of ⟨[18]36⟩ always intersects with the first set of components of A except [1]36 +⟨[18]36⟩, [2]36 + ⟨[18]36⟩, [3]36 + ⟨[18]36⟩.

    • • The second set of components of A is {ρ0, ρ2, ρ3, μ, ρ2μ, ρ3μ}.


      Every coset of {ρ0, ρ2} intersects with the second set of components of A.


      Every coset of N intersects with the second set of components of A.


      Every coset of H intersects with the second set of components of A.

  • (4) Determine TN1()(A)¯.


    For r = 1, the subsets ℬ1 ⊆ ℬ are {S1}, {S2}, {S3}.

    • S1 = ⟨[9]36⟩×H then (36×D4)/S1=36×84×4=18.

      TS1¯(A)={([x]36,ρi)36×HT(([x]36,ρi))S1A}={([x]36,ρi)36×H({([x]36,ρi),([x+1]36,ρiμ)}([9]36×H))A}=36×H.

    From the above calculations, we obtain

    TN1()(A)¯={([x]36,ρi)36×HT(([x]36,ρi)1A,for 1}=TS1(A)¯TS2(A)¯TS3(A)¯=36×H.

  • (5) Determine TN2()(A)¯.


    For r = 2, the subsets ℬ2 ⊆ ℬ are {S1, S2}, {S1, S3}, {S2, S3}. By the above explanation, we have T(x)ℬ2 = T(x) (∩S∈ℬ2S).


    • S1S2 = ⟨[18]36⟩ × {ρo, ρ2} then (36×D4)/(S1S2)=36×82×2=72.

      TS1S2¯(A)={([x]36,ρi)36×HT(([x]36,ρi))(S1S2)A}={([x]36,ρi)36×H({([x]36,ρi),([x+1]36,ρiμ)}([18]36×{ρ0,ρ2}))A}={([x]36,ρi)36×H[x]3636-{[1]36,[2]36,[19]36,[20]36,}and ρiH}.

    • S1S3 = {[0]36} × H then (36×D4)/(S1S3)=36×81×4=72.

      TS1S3¯(A)={([x]36,ρi)36×HT(([x]36,ρi))(S1S3)A}={([x]36,ρi)36×H({([x]36,ρi),([x+1]36,ρiμ)}({[0]36}×H))A}={([x]36,ρi)36×H[x]3636-{[1]36,[2]36,[3]36,[13]36,[14]36,[15]36,[19]36,[20]36,[25]36,[26]36,[27]36}and ρiH}.

    • S2S3 = ⟨[12]36⟩×{ρo, ρ2} then (36×D4)/(S2S3)=36×83×2=48.

      TS2S3¯(A)={([x]36,ρi)36×HT(([x]36,ρi))(S1S3)A}={([x]36,ρi)36×H({([x]36,ρi),([x+1]36,ρiμ)}([12]36×{ρo,ρ2}))A}={([x]36,ρi)36×H[x]3636-{[1]36,[2]36,[3]36,[13]36,[14]36,[15]36,[25]36,[26]36,[27]36}and ρiH}.

    From the above calculations, we obtain

    TN2()(A)¯={([x]36,ρi)36×HT(([x]36,ρi))2A,for 2}=TS1S2(A)¯TS1S3(A)¯TS2S3(A)¯={([x]36,ρi)36×H[x]3636-{[1]36,[2]36}and ρiH}.

  • (6) Determine TN3()(A)¯.


    For r = 3, the subsets ℬ3 ⊆ ℬ are {S1, S2, S3}. By the above explanation, we have T(x)ℬ3 = T(x)∩(∩S∈ℬ2S).


    • S1S2S3 = {[0]36} × {ρ0, ρ2} then (36×D4)/(S1S2S3)=36×81×2=144.

      TN3()(A)¯=TS1S2S3¯(A)={([x]36,ρi)36×H(([x]36,ρi))(S1S2S3)A}={([x]36,ρi)36×H({([x]36,ρi),([x+1]36,ρiμ)}({[0]36}×{ρ0,ρ2}))A}={([x]36,ρi)36×H[x]3636-{[1]36,[2]36,[3]36,[13]36,[14]36,[15]36,[19]36,[20]36,[25]36,[26]36,[27]36}and ρiH}.

In summary, we have TN3()(A)¯TN2()(A)¯TN1()(A)¯. The relationship between TNr()¯(A) and TNr()¯(A) is TN1()_(A)TN2()_(A)TN3()_(A)TN3()¯(A)TN2()¯(A)TN1()(A)¯.

Proposition 1

Let G be a group, (G, ) be a nearness approximation space and I : G*(G) be a set-valued mapping, where I(x) = {x} for each xG. Let A be a non-empty subset of G. Then,

INr()_(A)=Nr()(A)_and INr()¯(A)=Nr()(A)¯.
Proof

This is clear from Definitions 2 and 6.

Proposition 2

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued mapping. Let A be a non-empty subset of G′. Then,

  • (1) TNr()_(A)TNr()¯(A),

  • (2) TNr()_(G)=G=TNr()¯(G),

  • (3) If AB, then TNr()¯(A)TNr()¯(B) and TNr()¯(A)TNr()¯(B),

  • (4) TNr()_(A)(TNr()¯(AC))C,

  • (5) TNr()¯(A)(TNr()_(AC))C,

  • (6) TNr()¯(AB)TNr()¯(A)TNr()¯(B),

  • (7) TNr()¯(AB)TNr()¯(A)TNr()¯(B),

  • (8) TNr()¯(AB)TNr()¯(A)TNr()¯(B),

  • (9) TNr()¯(AB)=TNr()¯(A)TNr()¯(B).

Proof
  • (1) It is clear from Definition 6.

  • (2) It is clear from Definition 6.

  • (3) Let xTNr()¯(A). Subsequently, T(x)ℬrAB, for some ℬr ⊆ ℬ so that xTNr()¯(B). Therefore, TNr()¯(A)TNr()¯(B). Suppose yTNr()¯(A). Then, T(y)r¯A for some ℬr ⊆ ℬ. Since AB then T(y)ℬrB ≠ ∅ for some ℬr ⊆ ℬ such that yTNr()¯(B). Therefore, TNr()¯(A)TNr()¯(B).

  • (4) Let x(TNr()¯(AC))C. Then, x(TNr()¯(AC) so that T(x) ℬrAC = ∅ for every ℬr ⊆ ℬ, which implies that T(x) ℬrA for every ℬr ⊆ ℬ. Therefore, xTNr()¯(A) so that TNr()_(A)(TNr()¯(AC))C. The converse is not true in general. Let xTNr()¯(A). Then, T(x) ℬrA for some ℬr ⊆ ℬ but T(x)ℬrA for some ℬr ⊆ ℬ so that T(x)ℬrAC ≠ ∅, which implies xTNr()¯(AC). Thus, we have x(TNr()¯(AC))C so that TNr()_(A)(TNr()¯(AC))C.

  • (5) Let xTNr()¯(AC)C. Then xTNr()¯(AC) so that T(x)ℬrAC for every ℬr ⊆ ℬ which implies T(x)ℬrA ≠ ∅ for every ℬr ⊆ ℬ. Therefore, xTNr()¯(A) so that TNr()¯(A)(TNr()_(AC))C. The converse is not true in general. Let xTNr()¯(A) Then, T(x)ℬrA ≠ ∅ for some ℬr ⊆ ℬ but T(x)ℬrA = ∅ for some ℬr ⊆ ℬ such that T(x)ℬrAC, which implies xTNr()¯(AC). Therefore, xTNr()¯(AC)C so that TNr()¯(A)(TNr()_(AC))C.

  • (6) From part (3), we have TNr()¯(AB)TNr()¯(A) and TNr()¯(AB)TNr()¯(B). Then, TNr()¯(AB)TNr()¯(A)TNr()¯(B).

  • (7) From part (3), we have TNr()¯(AB)TNr()¯(A) and TNr()¯(AB)TNr()¯(B). Then, TNr()¯(AB)TNr()¯(A)TNr()¯(B).

  • (8) From part (3), we have TNr()¯(A)TNr()¯(AB) and TNr()¯(B)TNr()¯(AB). Then, TNr()¯(A)TNr()¯(B)TNr()¯(AB).

  • (9) Let xTNr()¯(AB). Then, T(x)ℬr ∩ (AB) ≠ ∅ for some ℬr ⊆ ℬ so that T(x)ℬrA ≠ ∅ or T(x)ℬrA ≠ ∅ which implies xTNr()¯(A) or xTNr()¯(B). Therefore, TNr()¯(AB)TNr()¯(A)TNr()¯(B). Conversely, from part (3), we have TNr()¯(A)TNr()¯(AB) and TNr()¯(B)TNr()¯(AB). Then, TNr()¯(A)TNr()¯(B)TNr()¯(AB).

The following examples show that the converse of Proposition 2 (6,7,8) is generally not true.

Example 8

Let ℤ18 be a group of integers modulo 18, ℬ = {S1, S2}, where S1 = ⟨[9]18⟩ = {[0]18, [9]18}, S2 = ⟨[6]18⟩ = {[0]18, [6]18, [12]18} be normal subgroups of ℤ18 and T : ℤ18* (ℤ18) be defined as T ([x]18) = {[x + 1]18}, for every [x]18 ∈ ℤ18. Suppose A, B ⊂ ℤ18, where

A=18-([1]18+[9]18[3]18+[9]18[2]18+[6]18)=18-{[1]18,[2]18,[3]18,[8]18,[10]18,[12]18,[14]18},

and

B=18-([1]18+[9]18[3]18+[9]18[3]18+[6]18)=18-{[1]18,[3]18,[4]18,[9]18,[10]18,[13]18,[15]18}.
  • (1) Determine TN1()¯(A) and TN1()¯(B).

    • • Cosets of S1 = ⟨[9]18⟩ contained in A are ⟨[9]18⟩, [4]18 + ⟨[9]18⟩, [6]18 + ⟨[9]18⟩, [7]18 + ⟨[9]18⟩.

      TS1_(A)={[x]1818[x+1]18+[9]18A}={[3]18,[5]18,[6]18,[8]18,[12]18,[14]18,[15]18,[17]18}.

    • • Cosets of S2 = ⟨[6]18⟩ contained in A are [5]18 + ⟨[6]18⟩.

      TS2_(A)={[x]1818[x+1]18+[6]18A}={[4]18,[10]18,[16]18}.

      From the above calculations, we have

      TN1()_(A)=TS1_(A)TS2_(A)={[3]18,[4]18,[5]18,[6]18,[8]18,[10]18,[12]18,[14]18,[15]18,[16]18,[17]18}.

    • • Cosets of S1 = ⟨[9]18⟩ contained in B are [2]18 + ⟨[9]18⟩, [5]18+⟨[9]18⟩, [7]18+⟨[9]18⟩, [8]18+⟨[9]18⟩.

      TS1_(B)={[x]1818[x+1]18+[9]18B}={[1]18,[4]18,[6]18,[7]18,[10]18,[13]18,[15]18,[16]18}.

    • • Cosets of S2 = ⟨[6]18⟩ contained in B are [0]18 + ⟨[6]18⟩, [2]18 + ⟨[6]18⟩, [5]18 + ⟨[6]18⟩.

      TS2_(B)={[x]1818[x+1]18+[6]18B}={[1]18,[4]18,[5]18,[7]18,[10]18,[11]18,[13]18,[16]18,[17]18}.

      From the above calculations, we have

      TN1()_(B)=TS1_(B)TS2_(B)={[1]18,[4]18,[5]18,[6]18,[7]18,[10]18,[11]18,[13]18,[15]18,[16]18,[17]18}.

      As a summary, we have TN1()¯(A)TN1()¯(B)={[4]18,[5]18,[6]18,[10]18,[15]18,[16]18,[17]18}.

  • (2) Determine TN1()¯(AB).


    AB = {[0]18, [5]18, [6]18, [7]18, [11]18, [16]18, [17]18}.


    • • Cosets of S1 = ⟨[9]18⟩ contained in AB are [7]18 + ⟨[9]18⟩.

      TS1_(AB)={[x]1818[x+1]18+[9]18AB}={[6]18,[15]18}.

    • • Cosets of S2 = ⟨[6]18⟩ contained in AB are [5]18 + ⟨[6]18⟩.

      TS2_(AB)={[x]1818[x+1]18+[6]18AB}={[4]18,[10]18,[16]18}.

      From the above calculations, we have

      TN1()_(AB)=TS1_(AB)TS2_(AB)={[4]18,[6]18,[10]18,[15]18,[16]18}.

    As a summary, we obtain TN1()¯(A)TN1()¯(B)TN1()¯(AB).

  • (3) Determine TN1()¯(A) and TN1()¯(B).

    • • Cosets of S1 = ⟨[9]18⟩ intersected with A are [0]18+⟨[9]18⟩, [2]18+⟨[9]18⟩, [4]18+⟨[9]18⟩, [5]18+ ⟨[9]18⟩, [6]18+⟨[9]18⟩, [7]18+⟨[9]18⟩, [8]18+⟨[9]18⟩.

      TS1¯(A)={[x]1818([x+1]18+[9]18)A}=18-{[0]18,[2]18,[9]18,[11]18}.

    • • Cosets of S2 = ⟨[6]18⟩ intersected with A are [0]18+⟨[6]18⟩, [1]18+⟨[6]18⟩, [3]18+⟨[6]18⟩, [4]18+ ⟨[6]18⟩, [5]18 + ⟨[6]18⟩.

      TS2¯(A)={[x]1818([x+1]18+[6]18)A}=18-{[1]18,[7]18,[13]18}.

      From the above calculations, we obtain

      TN1()¯(A)=TS1¯(A)TS2¯(A)=18.

    • • Cosets of S1 = ⟨[9]18⟩ intersected with B are [0]18+⟨[9]18⟩, [2]18+⟨[9]18⟩, [3]18+⟨[9]18⟩, [5]18+ ⟨[9]18⟩, [6]18+⟨[9]18⟩, [7]18+⟨[9]18⟩, [8]18+⟨[9]18⟩.

      TS1¯(B)={[x]1818([x+1]18+[9]18)B}=18-{[0]18,[3]18,[9]18,[12]18}.

    • • Cosets of S2 = ⟨[6]18⟩ intersected with B are [0]18+⟨[6]18⟩, [1]18+⟨[6]18⟩, [2]18+⟨[6]18⟩, [4]18+ ⟨[6]18⟩, [5]18 + ⟨[6]18⟩.

      TS2¯(B)={[x]1818([x+1]18+[6]18)B}=18-{[2]18,[8]18,[14]18}.

      From the above calculations, we obtain

      TN1()¯(B)=TS1¯(B)TS2¯(B)=18.

      As a summary, we have TN1()¯(A)TN1()¯(B)=18.

  • (4) Determine TN1()¯(AB).


    AB = {[0]18, [5]18, [6]18, [7]18, [11]18, [16]18, [17]18}.


    • • Cosets of S1 = ⟨[9]18⟩ intersected with AB are [0]18 + ⟨[9]18⟩, [2]18 + ⟨[9]18⟩, [5]18 + ⟨[9]18⟩, [6]18 + ⟨[9]18⟩, [7]18 + ⟨[9]18⟩, [8]18 + ⟨[9]18⟩.

      TS1¯(AB)={[x]1818([x+1]18+[9]18)(AB)}=18-{[0]18,[2]18,[3]18,[9]18,[11]18,[12]18}.

    • • Cosets of S2 = ⟨[6]18⟩ intersected with AB are [0]18 + ⟨[6]18⟩, [1]18 + ⟨[6]18⟩, [4]18 + ⟨[6]18⟩, [5]18 + ⟨[6]18⟩.

      TS2¯(AB)={[x]1818([x+1]18+[6]18)(AB)}=18-{[1]18,[2]18,[7]18,[8]18,[13]18,[14]18}.

      From the above calculations, we obtain

      TN1()¯(AB)=TS1¯(AB)TS2¯(AB)=18-{[2]18}.

    As a summary, we obtain TN1()¯(A)TN1()¯(B)TN1()¯(AB).

Example 9

Let ℤ18 be a group of integers modulo 18, ℬ = {S1, S2} where S1 = ⟨[9]18⟩ = {[0]18, [9]18}, S2 = ⟨[6]18⟩ = {[0]18, [6]18, [12]18} be normal subgroups of ℤ18 and T : ℤ18* (ℤ18) be defined as

T([x]18)={[x+1]18},

for every [x]18 ∈ ℤ18. Let A, B ⊂ ℤ18 where A = {[0]18, [5]18, [11]18}, B = {[9]18, [17]18} so AB = {[0]18, [5]18, [9]18, [11]18, [17]18}.

  • (1) Determine TN1()¯(A) and TN1()¯(B).

    • • There are no cosets of S1 = ⟨[9]18⟩ contained in A. TS1 (A) = {[x]18 ∈ ℤ18 | [x + 1]18 + ⟨[9]18⟩ ⊆ A} = ∅.

    • • There are no cosets of S2 = ⟨[6]18⟩ contained in A. TS2 (A) = {[x]18 ∈ ℤ18 | [x + 1]18 + ⟨[6]18⟩ ⊆ A} = ∅.

      From the above calculations, we obtain

      TN1()_(A)=TS1_(A)TS2_(A)=.

    • • There are no cosets of S1 = ⟨[9]18⟩ contained in B. TS1 (B) = {[x]18 ∈ ℤ18 | [x+1]18+⟨[9]18⟩ ⊆ B} = ∅.

    • • There are no cosets of S2 = ⟨[6]18⟩ contained in B. TS2 (B) = {[x]18 ∈ ℤ18 | [x+1]18+⟨[6]18⟩ ⊆ B} = ∅.

      From the above calculations, we obtain

      TN1()_(B)=TS1_(B)TS2_(B)=.

    As a summary, we have TN1()¯(A)TN1()¯(B)=

  • (2) Determine TN1()¯(AB).

    • • Cosets of S1 = ⟨[9]18⟩ contained in AB is ⟨[9]18⟩.

      TS1_(AB)={[x]1818[x+1]18+[9]18AB}={[8]18,[17]18}.

    • • Cosets of S2 = ⟨[6]18⟩ contained in AB is [5]18+ ⟨[6]18⟩.

      TS2_(AB)={[x]1818[x+1]18+[6]18AB}={[4]18,[10]18,[16]18}.

      From the above calculations, we obtain

      TN1()_(AB)=TS1_(AB)TS2_(AB)={[4]18,[8]18,[10]18,[16]18,[17]18}.

    As a summary, we have TN1()¯(AB)TN1()¯(A)TN1()¯(B).

Proposition 3

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued homomorphism. If A is a non-empty subset of G′ and e′A, then eTNr()¯(A) where e and e′ are neutral elements of G and G′, respectively.

Proof

Since T is a set-valued group homomorphism, it follows that for any aG, T(e) = T(aa−1) = T(a)T(a−1) = T(a)(T(a))−1. If xT(a) then e′ = xx−1T(a)(T(a))−1 = T(e). Because e′A and e′T(e) then e′T(e)ℬrA, for every ℬr ⊆ ℬ such that eTNr()¯(A).

Proposition 4

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued mapping. Then,

  • (1) TNr()¯(A)=rT¯(Ar),

  • (2) TN1()¯(A)rT¯(Ar),

where Aℬr = A (∩SrS).

Proof

(1) Similar to the proof of Theorem 4.5 in [26], we have

TNr()¯(A)=rTr¯(A)=rT¯(Ar).

(2) Similar to the proof of Theorem 4.5 in [26], we have TNr()¯(A)=rTr¯(A)rT¯(Ar).

The following corollary follows from Proposition 4.

Corollary 1

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued mapping. Let N be a normal subgroup of G′ and AG′, A ≠ ∅︀. Then,

  • (1) TNr()¯(A)=rTr¯(Ar),

  • (2) TNr()¯(A)rTr¯(Ar), where Aℬr = A (∩SrS).

Proof

(1) Similar to the proof of Theorem 4.5 in [26], we have Tr¯(Ar)=T¯(Arr)=T¯(Ar) so that with Proposition 4, we get TNr()¯(A)=rTr¯(Ar).

(2) Since AAℬr, it follows that TNr()(A) = ∪rTr (A) ⊆ ∪rTr (Aℬr).

Definition 7

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued mapping. Let A be a non-empty subset of G′. We define the sets

TNr()_(A)={xGT(x)rrA,for r,r},

and

TNr()¯(A)={xGT(x)rrA,for r,r},

where T(x)rr=T(x)(SrS)(RrR).

Proposition 5

Let G and G′ be two groups, (G′, ) be the nearness approximation space, and T : G*(G′) be a set-valued homomorphism. If A is a non-empty subset of G′, then

  • (1) TNr()¯(A)=TNr()¯(A),

  • (2) TNr()¯(A)TNr()¯(A).

Proof

(1) Let xTNr()¯(A). Then, T ( x ) r r A for some r, r . Since T ( x ) r T ( x ) r r A, we have xTNr()¯(A) so that TNr()¯(A)TNr()¯(A). Let xTNr()¯(A). Subsequently, T(x)rr = T(x)rA for some r so that xTNr()¯(A), which implies that TNr()¯(A)TNr()¯(A). Thus, TNr()¯(A)=TNr()¯(A).

(2) Let xTNr()¯(A). Then, T(x)rA ≠ ∅︀ for some r. Since T(x)rT(x)rr for every r, we obtain T(x)rrA such that xTNr()¯(A). Thus, TNr()¯(A)TNr()¯(A).

The following example shows that the converse of Proposition 5 (2) is not true in general.

Example 10

Let ℤ24 be a group of integers modulo 24, = {S1, S2} where S1 = ⟨[8]24⟩ = {[0]24, [8]24, [16]24}, S2 = ⟨[12]24⟩ = {[0]24, [12]24} be normal subgroups of ℤ24 and T : ℤ24*(ℤ24) be defined as

T([x]24)={(x+1)24},

for every [x]24 ∈ ℤ24. It is easy to see S1+S1 = S1, S2+S2 = S2 and S1 + S2 = ⟨[4]24⟩. Let A = {[4]24}. Then,

TS1+S1¯(A)=TS1¯(A)={[3]24,[11]24,[19]24},TS2+S2¯(A)=TS2¯(A)={[3]24,[15]24},TS1+S2¯(A)={[3]24,[7]24,[11]24,[15]24,[19]24,[23]24}.

We have

TN1()¯(A)=TS1¯(A)TS2¯(A)={[3]24,[11]24,[15]24,[19]24},TN1(+)¯(A)=TS1+S1¯(A)TS2+S2¯(A)TS1+S2¯(A)={[3]24,[7]24,[11]24,[15]24,[19]24,[23]24}.

Thus, we obtain TN1(+)(A)¯TN1()(A)¯.

Proposition 6

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued homomorphism. If A and B are two non-empty subsets of G′, then

TNr()¯(A)TNr()¯(B)TNr()¯(AB).
Proof

Let zTNr()¯(A)TNr()¯(B). Then, there exist xTNr()¯(A) and yTNr()¯(B) such that z = xy. Since xTNr()¯(A) and yTNr()¯(B) then T(x)rA ≠ ∅︀ for some r and T(y)rB for some r so there exist a, bG′ where aT(x)rA and bT(y)rB. Since aT(x)r, bT(y)r and T be a set-valued group homomorphism then abT(x)rT(y)r=T(x)T(y)rr=T(xy)rr so that abT(xy)rrAB which implies T(xy)rrABϕ. Therefore, z=xyTNr()¯(AB) so that TNr()¯(A)TNr()¯(B)TNr()¯(AB).

The following example shows that the converse of Proposition 6 is generally not true.

Example 11

Let ℤ24 be a group of integers modulo 24, = {S1, S2} where S1 = ⟨[8]24⟩ = {[0]24, [8]24, [16]24}, S2 = ⟨[12]24⟩ = {[0]24, [12]24}, S1 + S2 = ⟨[4]24⟩ = {[0]24, [4]24, [8]24, [12]24, [16]24, [20]24} be normal subgroups of ℤ24 and T : ℤ24*(ℤ24) be defined as

T([x]24)={8[x]24},

for every [x]24 ∈ ℤ24. Notice that T is a set-valued homomorphism. Let A = {[1]24, [5]24}, B = {[3]24}, A + B = {[4]24, [8]24}. We get

  • TS1¯(A)=TS2¯(A)=TS1¯(B)=TS2¯(B)=,

  • TS1+S2¯(A+B)=24,

  • TN1()¯(A)=TN1()¯(B)=,

  • TN1(+)¯(A+B)=24.

This shows that TN1(+)¯(A+B)TN1()¯(A)+TN1()¯(B).

Proposition 7

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued homomorphism. If A and B are two non-empty subsets of G′, then

TNr()_(A)TNr()_(B)TNr()_(AB).
Proof

Let zTNr()¯(A)TNr()¯(B). Then, there exist xTNr()¯(A) and yTNr()¯(B) so that z = xy. Since xTNr()¯(A) and yTNr()¯(B) then T(x)rA and T(y)rB for some r, r. Since T is a set-valued group homomorphism then T(x)rT(y)r=T(x)T(y)rr=T(xy)rrAB, which implies that z = xyTNr()¯(AB). Hence TNr()¯(A)TNr()¯(B)TNr()¯(AB). By Proposition 5(1), we have TNr()¯(A)TNr()¯(B)TNr()¯(AB).

Example 12

Let Q8 = {1, −1, i, −i, j, −j, k, −k} be a quaternion group, = {S1, S2} where S1 = {1, −1, i, −i}, S2 = {1, −1, j, −j} be the normal subgroups of Q8 and T : Q8*(Q8) be defined as

T(x)={x,-x},

for every xQ8. From Example 4.2, in [35], the mapping T is a set-valued homomorphism. Let A = {1, i}, B = {1, −1, −i}, A.B = {1, −1, i, −i} = S1. Then, we get

  • TS1 (A) = TS2 (A) = TS1 (B) = TS2 (B) = TS2 (A.B) = ∅︀,

  • TS1 (A.B) = {1, −1, i, −i}.

So that

  • TN1()¯(A)=TN1()¯(B)=,

  • TN1()¯(A.B)={1,1,i,i}.

This implies that TN1()¯(A.B)TN1()¯(A).TN1()¯(B).

5. The Lower and Upper Near Subgroups in a Group based on a Set-Valued Mapping

Proposition 8

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued homomorphism. Let A be a subgroup of G′, TNr()¯(A)=TNr()¯(A) and TNr()¯(A). Subsequently, TNr()¯(A) is a subgroup of G.

Proof

Let x, yTNr()¯(A). Then, T(x)rA ≠ ∅︀ and T(y)rA for some r, r. So there exist aT(x)rA and bT(y)rA. Since A is a subgroup of G′, we have ab−1A. Since T is a set-valued homomorphism, we have ab-1T(x)r(T(y)r)-1=T(x)r(r)-1(T(y))-1=T(x)rrT(y-1)=T(x)T(y-1)rr=T(xy-1)rr so that ab-1T(xy-1)rrA which implies that xy-1TNr()¯(A)=TNr()¯(A). Therefore, TNr()¯(A) is a subgroup of G.

Proposition 9

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G* (G′) be a set-valued homomorphism. Let A be a normal subgroup of G′, TNr()¯(A)=TNr()¯(A) and TNr()¯(A). Subsequently, TNr()¯(A) is a normal subgroup of G.

Proof

This is sufficient to demonstrate that TNr()¯(A) is normal. Let xTNr()¯(A) and gG. Then, T(x)rA ≠ ∅︀ for some r and T(g) is a non-empty set, we can select g′ ∈ T(g). Thus, aT(x)rA. exists and because A is a normal subgroup of G′, we have aga−1A. Since T is a set-valued homomorphism, we have aga−1T(x)rT(g)(T(x)r)−1 = T(x)rT(g)r−1T(x)−1=T(x) rT(g)r−1T(x−1)=T(x)T(g)T(x−1)rr−1=T(xgx−1) r such that aga−1T(xgx−1)rA which implies that xgx-1TNr()¯(A). Therefore, TNr()¯(A) is a normal subgroup of G.

Example 13

Let ℤ24 be a group of integers modulo 24, = {S1, S2}, where S1 = ⟨[8]24⟩ = {[0]24, [8]24, [16]24}, S2 = ⟨[12]24⟩ = {[0]24, [12]24} be normal subgroups of ℤ24 and T : ℤ24*(ℤ24) be defined as

T([x]24)={3[x]24},

for every [x]24 ∈ ℤ24. Observe that T is a set-valued homomorphism, S1 + S1 = S1, S2 + S2 = S2, and S1 + S2 = ⟨[4]24⟩. Let A ⊂ ℤ24 where A = 2ℤ24. Then, A is a normal subgroup of ℤ24. We get

TS1¯(A)=TS2¯(A)=TS1+S1¯(A)=TS2+S2¯(A)=TS1+S2¯(A)=224.

so that TN1()¯(A)=TN1(+)¯(A)=224 is a normal subgroup of ℤ24.

Proposition 10

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued homomorphism. Let A be a subgroup of G′ and TNr()¯(A) ≠ ∅. Then, TNr()¯(A) is a subgroup of G.

Proof

Let x, yTNr()¯(A). Then, T(x)rA and T(y)rA for some r, r. Since A is a subgroup of G′, we have (T(y)r)-1=T(y-1)rA. Since T is a set-valued homomorphism, we have T(x)rT(y-1)r=T(x)T(y-1)rr=T(xy-1)rrA which implies that xy−1TNr(B)¯(A). From Proposition 5(1), we have xy−1TNr()¯(A). Therefore, TNr()¯(A) is a subgroup of G.

Proposition 11

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued homomorphism. Let A be a normal subgroup of G′ and TNr()¯(A) ≠ ∅. Then, TNr()¯(A) is a normal subgroup of G.

Proof

This suffices for demonstrating that TNr()¯(A) is normal. Let xTNr()¯(A) and gG. Then, T(x)rA for some r and T(g) is a non-empty set. Since A is a subgroup of G′, we have (T(x)r)−1 = T(x−1)rA. Since A is a normal subgroup of G′ and T is a set-valued homomorphism, we have T(x)rT(g)T(x−1)r = T(x)T(g)T(x−1)r = T(xgx−1)rA which implies that xgx−1TNr()¯(A). Therefore, TNr()¯(A) is a normal subgroup of G.

6. Comparison between Near-Generalized and Generalized Approximations in a Group based on a Set-Valued Mapping

Proposition 12

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued mapping. Let A be a non-empty subset of G′. Then,

  • (1) If || = k, then TNk()¯(A)=T¯(A) and TNk()¯(A)=T¯(A).

  • (2) If 1 ≤ r < k where || = k, then Tr¯(A)TNr()¯(A) and Tr¯(A)TNr()¯(A).

Proof

The proof results from Definitions 5 and 6

From Proposition 12(1), we observe that the generalized approximations in a group based on set-valued mapping is a special type of near generalized approximation.

Proposition 13

Let G and G′ be two groups, (G′, ) be a nearness approximation space, and T : G*(G′) be a set-valued mapping and ij, 1 ≤ i < j ≤ ||. Let A be a non-empty subset of G′. Then

  • (1) TNr(i)¯(A)TNr(j)¯(A),

  • (2) TNr(i)¯(A)TNr(j)¯(A).

Proof

(1) From Definition 6, we have TNr(k)¯(A)=rkTr¯(A), for 1 ≤ k ≤ ||. Since 1 ≤ i < j ≤ ||, we have ij then TNr(i)¯(A)TNr(j)¯(A).

(2) From Definition 6, we obtain TNr(k)¯(A)=rkTr¯(A) for 1 ≤ k ≤ ||. Since 1 ≤ i < j ≤ ||, we have ij then TNr(i)¯(A)TNr(j)¯(A).

Proposition 14

Let G and G′ be two groups, (G′, ), be a nearness approximation space, and T : G*(G′) be a set-valued mapping. Let A be a non-empty subset of G′. Then,

TNi()_(A)TNj()_(A)TNj()¯(A)TNi()¯(A),

where 1 ≤ i < j ≤ ||.

Proof

Let xTNi()¯(A). Subsequently, T(x)iA for a certain i. Since i < j then ij so that T(x)j = T(x)∩Sj ST(x)∩Si S = T(x)iA which yields that xTNj()¯(A). Therefore, TNi()¯(A)TNj()¯(A). From Proposition 2 (1), we have TNj()_(A)TNj()¯(A). Let xTNj()¯(A). Then, T(x)jA ≠ ∅︀ for some j. Since i < j then ij so that T(x)j = T(x)∩SjST(x)∩SiS = T(x)i. Thus, we have T(x)ℬiA ≠ ∅︀, which yields xTNi()¯(A). Therefore, TNi()_(A)TNj()_(A)TNj()¯(A)TNi()¯(A).

7. Conclusion

In this study, we showed that near set theory is applicable to group theory. We demonstrated the concepts of near-generalized approximations in a group based on set-valued mapping, an extension of generalized approximations in a group based on set-valued mapping and near approximations in a group. We demonstrate that near-generalized lower and upper approximations of a (normal) subgroup in a group based on set-valued mapping can form a (normal) subgroup of a group. Furthermore, we showed that the generalized approximations in a group based on a set-valued mapping are a special type of near-generalized approximations in a group based on a set-valued mapping. Finally, we found that if the r index is larger, then the near-generalized lower approximations in a group based on set-valued mapping are greater (in terms of the inclusion sets), but the near-generalized upper approximations in a group based on set-valued mapping are smaller sets. Near-generalized lower approximations in a group based on set-valued mapping are subsets of near-generalized upper approximations in a group based on set-valued mapping. As an extension of this study, we are interested in applying this structure in a ring.

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