Lattices of IntervalValued Fuzzy Subgroups
 Author: Lee Jeong Gon, Hur Kul, Lim Pyung Ki
 Organization: Lee Jeong Gon; Hur Kul; Lim Pyung Ki
 Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 14, Issue2, p154~161, 25 June 2014

ABSTRACT
We discuss some interesting sublattices of intervalvalued fuzzy subgroups. In our main result, we consider the set of all intervalvalued fuzzy normal subgroups with finite range that attain the same value at the identity element of the group. We then prove that this set forms a modular sublattice of the lattice of intervalvalued fuzzy subgroups. In fact, this is an intervalvalued fuzzy version of a wellknown result from classical lattice theory. Finally, we employ a lattice diagram to exhibit the interrelationship among these sublattices.

KEYWORD
Intervalvalued fuzzy set , Intervalvalued fuzzy subgroup , Intervalvalued fuzzy normal subgroup , Level subset , Modular lattice

1. Introduction
In 1965, Zadeh [1] introduced the concept of a fuzzy set, and later generalized this to the notion of an intervalvalued fuzzy set [2]. Since then, there has been tremendous interest in this subject because of the diverse range of applications, from engineering and computer science to social behavior studies. In particular, Gorzalczany [3] developed an inference method using intervalvalued fuzzy sets.
In 1995, Biswas [4] studied intervalvalued fuzzy subgroups. Subsequently, a number of researchers applied intervalvalued fuzzy sets to algebra [511], and Lee et al. [12] furthered the investigation of intervalvalued fuzzy subgroups in the sense of a lattice.
Later, in 1999, Mondal and Samanta [13] applied intervalvalued fuzzy sets to topology, and Jun et al. [14] studied intervalvalued fuzzy strong semiopenness and intervalvalued fuzzy strong semicontinuity. Furthermore, Min [1517] investigated intervalvalued fuzzy almost Mcontinuity, the characterization of intervalvalued fuzzy msemicontinuity and intervalvalued fuzzy m
β continuity, and then Min and Yoo [18] researched intervalvalued fuzzy mα continuity. In particular, Choi et al. [19] introduced the concept of an intervalvalued smooth topology, and described some relevant properties.In this paper, we discuss some interesting sublattices of the lattice of intervalvalued fuzzy subgroups of a group.
In the main result of our paper, we consider the set of all intervalvalued fuzzy normal subgroups with finite range that attain the same value at the identity element of the group. We prove that this set forms a modular sublattice of the lattice of intervalvalued fuzzy subgroups. In fact, this is an intervalvalued fuzzy version of a wellknown result from classical lattice theory. Finally, we use a lattice diagram to exhibit the interrelationship among these sublattices.
2. Preliminaries
In this section, we list some basic concepts and wellknown results which are needed in the later sections. Throughout this paper, we will denote the unit interval [0, 1] as
I . For any ordinary subsetA on a setX , we will denote the characteristic function ofA asχ _{A}.Let
D (I ) be the set of all closed subintervals of the unit interval [0, 1]. The elements ofD (I ) are generally denoted by capital lettersM ,N , ···, and note thatM = [M ^{ L},M ^{ U}], whereM ^{ L} andM ^{ U} are the lower and the upper end points respectively. Especially, we denote0 = [0, 0],1 = [1, 1], anda = [a ,a ] for everya (0, 1). We also note that (i) (∀M,N D(I)) (M = N M L= N L,M U = N U), (ii) (∀M,N D(I)) (M = N M L N L,M U N U). For everyM D (I ), thecomplement ofM , denoted byM ^{ C}, is defined byM ^{ C} = 1 −M = [1 −M ^{ U}, 1 −M ^{ L}](See [13]).Definition 2.1 [2,3]. A mappingA :X →D (I ) is called an intervalvalued fuzzy set (IVFS) inX , denoted byA = [A ^{ L},A ^{ U}], ifA ^{ L},A ^{ L}I ^{ X} such thatA ^{ L}A ^{ U}, i.e.,A ^{ L}(x )A ^{ U}(x ) for eachx X , whereA ^{ L}(x )[respA ^{ U}(x )] is called thelower [respupper ]end point of x to A . For any [a, b ]D(I) , the intervalvalued fuzzyA inX defined byA(x) = [A ^{ L}(x ),A ^{ U}(x )] = [a, b ] for eachx X is denoted by and ifa =b , then the IVFS is denoted by simply . In particular, and denote theintervalvalued fuzzy empty set and theintervalvalued fuzzy whole set inX , respectively.We will denote the set of all IVFSs in X as
D (I )^{ X}. It is clear that setA = [A,A ] ∈D (I )^{ X} for eachA ∈I ^{ X}.Definition 2.2 [13] . LetA,B ∈D (I )^{ X} and let {A _{α}}_{α∈Γ} ⊂D (I )^{ X}. Then(i) A ⊂ B iff A L ≤ B L and A U ≤ B U. (ii) A = B iff A ⊂ B and B ⊂ A. (iii) A C = [1 − A U, 1 − A L]. (iv) A ∪ B = [A L ∨ B L , A U ∨ B U].(iv)' Aα = [ ,]. (v) A ∩ B = [A L ∧ B L, A U ∧ B U]. (v)' Aα = [,].
Result 2.A[13, Theorem 1]. LetA, B, C ∈D (I )^{ X} and let {A _{α}}_{α∈Γ} ⊂D (I )^{ X}. Then(a) ⊂ A ⊂ . (b) A ∪ B = B ∪ A , A ∩ B = B ∩ A. (c) A∪(B∪C) = (A∪B)∪C , A∩(B∩C) = (A∩B)∩C. (d) A, B ⊂ A ∪ B , A ∩ B ⊂ A, B. (e) A ∩ ( Aα) = ( A ∩ Aα). (f) A ∪ ( Aα ) = (A ∪ Aα). (g) () c = , () c = . (h) (A c) c = A. (i) (Aα) c = A cα , ( Aα) c = Acα.
Definition 2.3 [8] . Let (X , ·) be a groupoid and letA D (I )^{ X}. ThenA is called an intervalvalued fuzzy subgroupoid (IVGP) inX if A L(xy) ≥ A L(x) ∧ A L(y) and A U(xy) ≥ A U(x) ∧ A U(y), ∀x, y X. It is clear that , IVGP(X ).Definition 2.4 [4]. LetA be an IVFs in a groupG . ThenA is called an intervalvalued fuzzy subgroup (IVG) inG if it satisfies the conditions : For anyx ,y ∈G , (i) A L(xy) ≥ A L(x) ∧ A L(y) and A U(xy) ≥ A U(x) ∧ A U(y). (ii) A L(x −1) ≥ A L(x) and A U(x −1) ≥ A U(x). We will denote the set of all IVGs of G as IVG(G).Result 2.A[8, Proposition 4.3]. LetG be a group and let {A _{α}} _{α∈Γ} ⊂ IVG(G ). ThenA _{α} ∈ IVG(G ).Result 2.B [4, Proposition 3.1]. LetA be an IVG in a groupG . Then (a) A(x −1) = A(x), ∀x ∈ G. (b) A L(e) ≥ A L(x) and A U(e) ≥ A U(x), ∀x ∈G, where e is the identity of G.Result 2.C [8, Proposition 4.2]. LetG be a group and letA ⊂G . ThenA is a subgroup ofG if and only if [χ^{ A} ,χ^{ A} ] ∈ IVG(G ).Definition 2.5 [8]. LetA be an IVFS in a setX and letλ ,μ ∈I withλ ≤μ . Then the setA ^{ [λ,μ]} = {x ∈X :A ^{ L}(x ) ≥λ andA ^{ U}(x ) ≥μ } is called a [λ, μ ]level subset ofA .3. Lattices of IntervalValued Fuzzy Subgroups
In this section, we study the lattice structure of the set of intervalvalued fuzzy subgroups of a given group. From Definitions 2.1 and 2.2, we can see that for a set
X ,D (I )^{X} forms a complete lattice under the usual ordering of intervalvalued fuzzy inclusion ⊂, where the inf and the sup are the intersection and the union of intervalvalued fuzzy sets, respectively. To construct the lattice of intervalvalued fuzzy subgroups, we define the inf of a familyA _{α} of intervalvalued fuzzy subgroups to be the intersection ⋂A _{α}. However, the sup is defined as the intervalvalued fuzzy subgroup generated by the union ⋃A _{α} and denoted by ( ⋃A _{α}). Thus we have the following result.Proposition 3.1. LetG be a group. Then IVG(G ) forms a complete lattice under the usual ordering of intervalvalued fuzzy set inclusion ⊂.Proof. Let {A _{α}}_{α} be any subset of IVG(G ). Then, by Result 2.A, ∈ IVG(G ). Moreover, it is clear thatA _{α} is the largest intervalvalued fuzzy subgroup contained inA _{α} for each . SoA _{α} =A _{α}. On the other hand, we can easily see that (A _{α}) is the least intervalvalued fuzzy subgroup containingA _{α} for each . SoA _{α} = (A _{α}). Hence IVG(G ) is a complete lattice.Next we construct certain sublattice of the lattice IVG(
G ). In fact, these sublattices reflect certain peculiarities of the intervalvalued fuzzy setting. For a groupG , let IVG_{f} (G ) = {A ∈ IVG(G ) : ImA is finite } and let IVG_{[s, t]}(G ) = {A ∈ IVG(G ) : A(e ) = [s, t ]}, wheree is the identity ofG . Then it is clear that IVG_{f} (G )[resp. IVG_{[s, t]}(G )] is a sublattice of IVG(G ). Moreover, IVG_{f} (G )∩ IVG_{[s, t]}(G ) is also a sublattice of IVG(G ).Definition 3.2 [11] . Let (X, ·) be a groupoid and letA ,B ∈D (I )^{ X}. Then theintervalvalued fuzzy product of A and B, denoted byA B , is an IVFS inX defined as follows : For eachx ∈X , Now to obtain our main results, we start with following two lemmas.Lemma 3.3. LetG be a group and let A,B ∈ IVG(G ). Then for each [λ, μ ] ∈D (I ),A ^{ [λ, μ]} ·B ^{ [λ, μ]} ⊂ (A B )^{ [λ, μ]}.Proof. Letz ∈A ^{[λ, μ]} ·B ^{[λ, μ]}. Then there existx_{0} ,y_{0} ∈G such thatz =x_{0}y_{0} . ThusA ^{L}(x_{0} ) ≥λ ,A ^{U}(x_{0} ) ≥μ andA ^{L}(y_{0} ) ≥λ ,A ^{U}(y_{0} ) ≥μ . So and Thus . HenceThe following is the converse of Lemma 3.2.
Lemma 3.4. LetG be a group and letA ,B ∈ IVG(G ). If ImA and ImB are finite, then for each ,Proof. Let Then and Since ImA and ImB are finite, there existx_{0} ,y_{0} ∈G withz =x_{0}y_{0} such that and ThusA ^{ L}(x _{0}) ≥λ ,A ^{ U}(x _{0}) ≥μ andB ^{ L}(y _{0}) ≥λ ,B ^{ L}(y _{0}) ≥μ . Sox _{0} ∈A ^{ [λ, μ]} andy _{0} ∈B ^{ [λ, μ]}, i.e.,z =x _{0}y _{0} ∈A ^{ [λ, μ]} · B^{ [λ, μ]}. Hence (A B )^{ [λ, μ]} ⊂A ^{ [λ, μ]} · B^{ [λ, μ]}. This completes the proof.The following is the immediate result of Lemmas 3.3 and 3.4.
Proposition 3.5. LetG be a group and letA ,B ∈ IVG(G ). If ImA and ImB are finite, then for each [λ, μ ] ∈D (I ), (A B) [λ, μ] = A [λ, μ] · B [λ, μ].Definition 3.6 [8]. LetG be a group and letA ∈ IVG(G ). ThenA is called intervalvalued fuzzy normal subgroup (IVNG) ofG ifA (xy ) =A (yx ) for anyx ,y ∈G .We will denote the set of all IVNGs of
G as IFNG(G ). It is clear that ifG is abelian, then every IVG ofG is an IVNG ofG .Result 3.A [6, Proposition 2.13]. LetG be a group, letA ∈ IFNG(G ) and let such thatλ ≤A ^{ L}(e ) andμ ≤A ^{ U}(e ). ThenA ^{ [λ, μ]} ◁G , whereA ^{ [λ, μ]} ◁G means thatA ^{ [λ, μ]} is a normal subgroup ofG .Result 3.B [6, Proposition 2.17]. LetG be a group and letA ∈ IVG(G ). IfA ^{ [λ, μ]} ◁G for each [λ, μ ] ∈ ImA , ThenA ∈ IVNG(G ).The following is the immediate result of Results 3.A and 3.B.
Theorem 3.7. LetG be a group and letA ∈ IVG(G ). ThenA ∈ IVNG(G ) if and only if for each [λ, μ ] ∈ ImA ,A ^{ [λ, μ]} ◁G .Result 3.C[8, Proposition 5.3]. LetG be a group and letA ∈ IVNG(G ). IfB ∈ IVG(G ), thenB A ∈IVG(G ).The following is the immediate result of Result 2.A and Definition 3.6.
Proposition 3.8. LetG be a group and letA ,B ∈ IVNG(G ). ThenA ∩B ∈ IVNG(G ).It is wellknown that the set of all normal subgroups of a group forms a sublattice of the lattice of its subgroups. As an intervalvalued fuzzy analog of this classical result we obtain the following result.
Theorem 3.9. LetG be a group and let IVN_{f[s, t]}(G ) = {A ∈ IVNG(G ) : ImA is finite andA (e ) = [s, t ]}. Then IVN_{f[s, t]}(G ) is a sublattice of IVG_{f} (G )∩ IVG_{[s, t]}(G ). Hence IVN_{f[s, t]}(G ) is a sublattice of IVG(G ).Proof. LetA ,B ∈ IVN_{f[s, t]}(G ). Then, by Result 3.C,A B ∈ IVG(G ). Letz ∈G . Then [Since A(e) = (s, t) = B(e)] = A L(z). [By Result 2.B] Similarly, we have (A B )^{ U}(z ) ≥A ^{ U}(z ). ThusA ⊂A B . By the similar arguments, we haveB ⊂A B .Let
C ∈ IVG(G ) such thatA ⊂C andB ⊂C . Letz ∈G . Then Similarly, we haveThusA B ⊂C . SoA B =A ∨B .Now let [
λ, μ ] ∈D (I ). SinceA ,B ∈ IVNG(G ),A ^{ [λ, μ]}◁G andB ^{ [λ, μ]}◁G . ThenA ^{ (λ,μ)}B ^{ [λ, μ]}◁G . By Proposition 3.5, (A B )^{ [λ, μ]} ◁G . Thus, by Theorem 3.7,A B ∈ IVNG(G ). SoA ∨B ∈ IVN_{f[s, t]}(G ). From Proposition 3.8, it is clear thatA ∧B ∈ IVNG(G ). ThusA ∧B ∈ IVN_{f[s,t]}(G ). Hence IVN_{f[s,t]}(G ) is a sublattice of IVG_{f}∩ IVG_{[s,t]}(G ), and therefore of IVG(G ). This complete the proof.The relationship of different sublattice of the lattice of intervalvalued fuzzy subgroup discussed herein can be visualized by the lattice diagram in Figure 1.
It is also wellknown[20, Theorem I.11] that the sublattice of normal subgroups of a group is modular. As an intervalvalued fuzzy version to the classical theoretic result, we prove that IVN(_{[s, t]}(
G ) forms a modular lattice.Result 3.D [11, Lemma 3.2]. LetG be a group and letA ∈ IVG(G ). If for anyx, y ∈G ,A^{ L} (x ) <A^{ L} (y ) andA^{ U} (x ) <A^{ U} (y ), thenA (xy ) =A (x ) =A (yx ).Definition 3.10 [20,21]. A lattice (L ,∧,∨) is said to bemodular if for anyx, y, z ∈L withx ≤z [resp.x ≥z ],x ∨(y ∧z ) = (x ∨y ) ∧ z[resp.x ∧ (y ∨z ) = (x ∧y ) ∨z ].In any lattice
L , it is wellknown [21, Lemma I.4.9] that for anyx, y, z ∈L ifx ≤z [resp.x ≥z ], thenx ∨ (y ∧z ) ≤ (x ∨y ) ∧z [resp.x ∧ (y ∨z ) ≥ (x ∧y ) ∨z ]. The inequality is called themodular inequality .Theorem 3.11. The lattice IVN_{ f[s, t]} (G ) is modular.Proof. LetA,B,C ∈ IVN_{f[s, t]} (G ) such thatA ⊃C . Then, by the modular inequality, (A ∧B )∨C ⊂A ∧(B ∨C ). Assume thatA ∧ (B ∨C ) ⊄ (A ∧B ) ∨C , i.e., there existsz ∈G such that [A ∧ (B ∨ C)] L(z) > [(A ∧ B) ∨ C] L(z) and [A ∧ (B ∨ C)] U(z) > [(A ∧ B) ∨ C] U(z). Since ImB and ImC are finite, there existx_{0} ,y_{0} ∈G withz =x_{0}y_{0} such that (B ∨ C)(z) = (B C)(z) (By the process of the proof of Theorem 3.9) ThusOn the other hand,
and
By (3.1), (3.2) and (3.3),
A^{ L} (z ) ∧B^{ L} (x _{0}) ∧C^{ L} (y _{0}) >A^{ L} (x _{0}) ∧B^{ L} (x _{0}) ∧C^{ L} (y _{0})and
A^{ U} (z )∧B^{ U} (x _{0})∧C^{ U} (y _{0}) >A^{ U} (x _{0})∧B^{ U} (x _{0})∧C^{ U} (y _{0}).Then
A^{ L} (z ),B^{ L} (x _{0}),C^{ L} (y _{0}) >A^{ L} (x _{0}) ∧B^{ L} (x _{0}) ∧C^{ L} (y _{0})and
A^{ U} (z ),B^{ U} (x _{0}),C^{ U} (y _{0}) >A^{ U} (x _{0}) ∧B^{ U} (x _{0}) ∧C^{U} (y _{0}).Thus
A^{ L} (x _{0}) ∧B^{ L} (x _{0}))∧C^{ L} (y _{0}) =A^{ L} (x _{0})and
A^{ U} (x _{0}) ∧B^{ U} (x _{0}) ∧C^{ U} (y _{0}) =A^{ U} (x _{0}).So
A^{ L} (z ) >A^{ L} (x _{0}),A^{ U} (z ) >A^{ U} (x _{0})and
C^{ L} (y _{0}) >A^{ L} (x _{0}),C^{ U} (y _{0}) >A^{ U} (x _{0}).By Result 2.B,
A^{ L} (x _{0}^{ −1}) =A^{ L} (x _{0}) <A^{ L} (x _{0}y _{0})and
A^{ U} (x _{0}^{ −1}) =A^{ U} (x _{0}) <A^{ U} (x _{0}y _{0}).By Result 3.D,
A (x _{0}) =A (x _{0}^{ −1}x _{0}y _{0}) =A (y _{0}).Thus
C^{ L} (y _{0}) >A^{ L} (y _{0}) andC^{ U} (y _{0}) >A^{ U} (y _{0}).This contradicts the fact that
A ⊃C . SoA ∧ (B ∨C ) ⊂ (A ∧B ) ∨C . HenceA ∧ (B ∨C ) = (A ∧B ) ∨C . Therefore IVN_{f[s,t]} (G ) is modular. This completes the proof.We discuss some interesting facts concerning a special class of intervalvalued fuzzy subgroups that attain the value [1, 1] at the identity element of
G .Lemma 3.12. Let A be a subset of a groupG . Then where <A > is the subgroup generated byA .Proof. Let𝐵 = {B ∈ IVG(G ) : [χ_{A} ,χ_{A} ] ⊂B }, letB ∈𝐵 and letx ∈A . Then χA(x) = 1 ≤ BL(x) and χA(x) = 1 ≤ BU(x). ThusB (x ) = [1, 1]. SinceB ∈ IVG(G ),B = for any composite of elements ofA . So [χ _{},χ _{}] ⊂B . Hence [χ _{},χ _{}>] ⊂ ⋂𝐵 . By Result 2.C, [χ _{},χ _{}] ∈ IVG(G ). Moreover, [χ _{},χ _{}] ∈𝐵 . Therefore [χ _{},χ _{}] = ⋂𝐵 =< [χ _{},χ _{}] >.The following can be easily seen.
Lemma 3.13. LetA andB subgroups of a groupG . Then (a) A ◁ G if and only if [χA, χA] ∈IVN(G). (b) [χA,χA] [χB, χB] = [χA·B, χA·B].Proposition 3.14. LetS (G ) be the set of all subgroup of a groupG and let IVG(S (G )) = {[χ_{A}, χ_{A}] :A ∈S (G )}. Then IVG(S (G )) forms a sublattice of IVG_{f} (G ) ∩ IVG_{[1,1]}(G ) and hence of IVG(G ).Proof. LetA,B ∈S (G ). Then it is clear that [χ_{A} ,χ_{A} ] ∩ [χ_{B} ,χ_{B} ] = [χ _{A∩B},χ _{A∩B}] ∈ IVG(S (G )). By Lemma 3.12,< [
χ_{A} ,χ_{A} ] ∪ [χ_{B} ,χ_{B} ] > = < [χ _{A∪B},χ _{A∪B}] > = [χ _{<A∪B>},χ _{<A∪B>}].Thus
[
χ_{A} ,χ_{A} ]∨[χ_{B} ,χ_{B} ] =< [χ_{A} ,χ_{A} ]∪[χ_{B} ,χ_{B} ] >∈ IVG(S (G )).Moreover, IVG(
S (G )) ⊂ IVG_{f} (G )∩ IVG_{[1,1]}(G ).Hence IVG(
S (G )) is a sublattice of IVG_{f} (G )∩IVG_{[1,1]}(G ).Proposition 3.14 allows us to consider the lattice of subgroups
S (G ) ofG a groupG as a sublattice of the lattice of all intervalvalued fuzzy subgroups IVG(G ) ofG .Now, in view of Theorems 3.9 and 3.11, for each fixed [
s, t ] ∈D (I ) , IVN_{f[s, t]}(G ) forms a modular sublattice of IVG_{f}(G )∩ IVG_{[s, t]}(G ). Therefore, for [s, t ] = [1, 1], the sublattice IVN_{f[1, 1]}(G ) is also modular. It is clear that IVNf[1, 1](G) ∩ IVG(S(G)) = IVN(N(G)); whereN (G ) denotes the set of all normal subgroups ofG and IVN(N (G )) = {[χ_{N} ,χ_{N} ] :N ∈N (G )}. Moreover, IVG(N (G )) is also modular.The lattice structure of these sublattices can be visualized by the diagram in Figure 2,
By using Lemmas 3.12 and 3.13, we obtain a wellknown classical result.
Corollary 3.15. LetG be a group. ThenN (G ) forms a modular sublattice ofS (G ).4. Conclusion
Lee et al. [11] studied intervalvalued fuzzy subgroup in the sense of a lattice. Cheong and Hur [5], Lee et al. [10], Jang et al. [6], Kang and Hur [8] investigated intervalvalued fuzzy ideals/(generalized) biideals, subgroup and ring, respectively.
In this paper, we mainly study sublattices of the lattice of intervalvalued fuzzy subgroups of a group. In particular, we prove that the lattice IVN_{f[s, t]}(G) is modular lattice (See Theorem 3.11). Finally, for subgroup
S (G ) of a groupG , IVG(S (G )) forms a sublattice of IVG_{f} (G )∩ IVG_{[1,1]}(G ) and hence of IVG(G ) (See Proposition 3.14).In the future, we will investigate sublattices of the lattice of intervalvalued fuzzy subrings of a ring.

[Figure 1.]

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[Figure 2.]