We discuss some interesting sublattices of interval-valued fuzzy subgroups. In our main result, we consider the set of all interval-valued 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 interval-valued fuzzy subgroups. In fact, this is an interval-valued fuzzy version of a well-known result from classical lattice theory. Finally, we employ a lattice diagram to exhibit the interrelationship among these sublattices. In 1965, Zadeh [1] introduced the concept of a fuzzy set, and later generalized this to the notion of an interval-valued 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 interval-valued fuzzy sets. In 1995, Biswas [4] studied interval-valued fuzzy subgroups. Subsequently, a number of researchers applied interval-valued fuzzy sets to algebra [5 - 11] , and Lee et al. [12] furthered the investigation of interval-valued fuzzy subgroups in the sense of a lattice. Later, in 1999, Mondal and Samanta [13] applied interval-valued fuzzy sets to topology, and Jun et al. [14] studied interval-valued fuzzy strong semi-openness and interval-valued fuzzy strong semicontinuity. Furthermore, Min [15 - 17] investigated interval-valued fuzzy almost M-continuity, the characterization of interval-valued fuzzy m-semicontinuity and intervalvalued fuzzy m β -continuity, and then Min and Yoo [18] researched interval-valued fuzzy m α -continuity. In particular, Choi et al. [19] introduced the concept of an interval-valued smooth topology, and described some relevant properties. In this paper, we discuss some interesting sublattices of the lattice of interval-valued fuzzy subgroups of a group. In the main result of our paper, we consider the set of all interval-valued 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 interval-valued fuzzy subgroups. In fact, this is an interval-valued fuzzy version of a well-known result from classical lattice theory. Finally, we use a lattice diagram to exhibit the interrelationship among these sublattices. In this section, we list some basic concepts and well-known results which are needed in the later sections. Throughout this paper, we will denote the unit interval [0, 1] as I . For any ordinary subset A on a set X , we will denote the characteristic function of A as χ _A . Let D ( I ) be the set of all closed subintervals of the unit interval [0, 1]. The elements of D ( I ) are generally denoted by capital letters M , N , ···, and note that M = [ M ^L , M ^U ], where M ^L and M ^U are the lower and the upper end points respectively. Especially, we denote 0 = [0, 0], 1 = [1, 1], and a = [ a , a ] for every a (0, 1). We also note that For every M D ( I ), the complement of M , denoted by M ^C , is defined by M ^C = 1 − M = [1 − M ^U , 1 − M ^L ](See [13] ). Definition 2.1 [2,3]. A mapping A : X → D ( I ) is called an interval-valued fuzzy set (IVFS) in X , denoted by A = [ A ^L , A ^U ], if A ^L , A ^L I ^X such that A ^L A ^U , i.e., A ^L ( x ) A ^U ( x ) for each x X , where A ^L ( x )[resp A ^U ( x )] is called the lower [resp upper ] end point of x to A . For any [ a, b ] D(I) , the interval-valued fuzzy A in X defined by A(x) = [ A ^L ( x ), A ^U ( x )] = [ a, b ] for each x X is denoted by and if a = b , then the IVFS is denoted by simply . In particular, and denote the interval-valued fuzzy empty set and the interval-valued fuzzy whole set in X , respectively. We will denote the set of all IVFSs in X as D ( I ) ^X . It is clear that set A = [ A,A ] ∈ D ( I ) ^X for each A ∈ I ^X . Definition 2.2 [13] . Let A,B ∈ D ( I ) ^X and let { A _α } _α∈Γ ⊂ D ( I ) ^X . Then Result 2.A[13, Theorem 1]. Let A, B, C ∈ D ( I ) ^X and let { A _α } _α∈Γ ⊂ D ( I ) ^X . Then Definition 2.3 [8] . Let ( X , ·) be a groupoid and let A D ( I ) ^X . Then A is called an interval-valued fuzzy subgroupoid (IVGP) in X 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]. Let A be an IVFs in a group G . Then A is called an interval-valued fuzzy subgroup (IVG) in G if it satisfies the conditions : For any x , y ∈ G , Result 2.A[8, Proposition 4.3]. Let G be a group and let { A _α } _α∈Γ ⊂ IVG( G ). Then A _α ∈ IVG( G ). Result 2.B [4, Proposition 3.1]. Let A be an IVG in a group G . Then Result 2.C [8, Proposition 4.2]. Let G be a group and let A ⊂ G . Then A is a subgroup of G if and only if [ χ ^A , χ ^A ] ∈ IVG( G ). Definition 2.5 [8]. Let A be an IVFS in a set X and let λ , μ ∈ I with λ ≤ μ . Then the set A ^[λ,μ] = { x ∈ X : A ^L ( x ) ≥ λ and A ^U ( x ) ≥ μ } is called a [ λ, μ ]- level subset of A . 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 interval-valued fuzzy inclusion ⊂, where the inf and the sup are the intersection and the union of interval-valued fuzzy sets, respectively. To construct the lattice of interval-valued fuzzy subgroups, we define the inf of a family A _α of interval-valued fuzzy subgroups to be the intersection ⋂ A _α . However, the sup is defined as the interval-valued fuzzy subgroup generated by the union ⋃ A _α and denoted by ( ⋃ A _α ). Thus we have the following result. Proposition 3.1. Let G be a group. Then IVG( G ) forms a complete lattice under the usual ordering of interval-valued 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 that A _α is the largest interval-valued fuzzy subgroup contained in A _α for each . So A _α = A _α . On the other hand, we can easily see that ( A _α ) is the least intervalvalued fuzzy subgroup containing A _α for each . So A _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 group G , let IVG _f ( G ) = { A ∈ IVG( G ) : Im A is finite } and let IVG _{[s, t]} ( G ) = { A ∈ IVG( G ) : A( e ) = [ s, t ]}, where e is the identity of G . 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 let A , B ∈ D ( I ) ^X . Then the interval-valued fuzzy product of A and B, denoted by A B , is an IVFS in X defined as follows : For each x ∈ X , Now to obtain our main results, we start with following two lemmas. Lemma 3.3. Let G 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. Let z ∈ A ^{[λ, μ]} · B ^{[λ, μ]} . Then there exist x₀ , y₀ ∈ G such that z = x₀y₀ . Thus A ^L ( x₀ ) ≥ λ , A ^U ( x₀ ) ≥ μ and A ^L ( y₀ ) ≥ λ , A ^U ( y₀ ) ≥ μ . So and Thus . Hence The following is the converse of Lemma 3.2. Lemma 3.4. Let G be a group and let A , B ∈ IVG( G ). If Im A and Im B are finite, then for each , Proof. Let Then and Since Im A and Im B are finite, there exist x₀ , y₀ ∈ G with z = x₀y₀ such that and Thus A ^L ( x ₀ ) ≥ λ , A ^U (x ₀ ) ≥ μ and B ^L ( y ₀ ) ≥ λ , B ^L ( y ₀ ) ≥ μ . So x ₀ ∈ A ^{[λ, μ]} and y 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. Let G be a group and let A , B ∈ IVG( G ). If Im A and Im B are finite, then for each [ λ, μ ] ∈ D ( I ), (A B) ^{[λ, μ]} = A ^{[λ, μ]} · B ^{[λ, μ]}. Definition 3.6 [8]. Let G be a group and let A ∈ IVG( G ). Then A is called interval-valued fuzzy normal subgroup (IVNG) of G if A ( xy ) = A ( yx ) for any x , y ∈ G . We will denote the set of all IVNGs of G as IFNG( G ). It is clear that if G is abelian, then every IVG of G is an IVNG of G . Result 3.A [6, Proposition 2.13]. Let G be a group, let A ∈ IFNG( G ) and let such that λ ≤ A ^L ( e ) and μ ≤ A ^U ( e ). Then A ^{[λ, μ]} ◁ G , where A ^{[λ, μ]} ◁ G means that A ^{[λ, μ]} is a normal subgroup of G . Result 3.B [6, Proposition 2.17]. Let G be a group and let A ∈ IVG( G ). If A ^{[λ, μ]} ◁ G for each [ λ, μ ] ∈ Im A , Then A ∈ IVNG( G ). The following is the immediate result of Results 3.A and 3.B. Theorem 3.7. Let G be a group and let A ∈ IVG( G ). Then A ∈ IVNG( G ) if and only if for each [ λ, μ ] ∈ Im A , A ^{[λ, μ]} ◁ G . Result 3.C[8, Proposition 5.3]. Let G be a group and let A ∈ IVNG( G ). If B ∈ IVG( G ), then B A ∈IVG( G ). The following is the immediate result of Result 2.A and Definition 3.6. Proposition 3.8. Let G be a group and let A , B ∈ IVNG( G ). Then A ∩ B ∈ IVNG( G ). It is well-known that the set of all normal subgroups of a group forms a sublattice of the lattice of its subgroups. As an interval-valued fuzzy analog of this classical result we obtain the following result. Theorem 3.9. Let G be a group and let IVN _{f[s, t]} ( G ) = { A ∈ IVNG( G ) : Im A is finite and A ( 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. Let A , B ∈ IVN _{f[s, t]} ( G ). Then, by Result 3.C, A B ∈ IVG( G ). Let z ∈ 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 ). Thus A ⊂ A B . By the similar arguments, we have B ⊂ A B . Let C ∈ IVG( G ) such that A ⊂ C and B ⊂ C . Let z ∈ G . Then Similarly, we have Thus A B ⊂ C . So A B = A ∨ B . Now let [ λ, μ ] ∈ D ( I ). Since A , B ∈ IVNG( G ), A ^{[λ, μ]} ◁ G and B ^{[λ, μ]} ◁ G . Then A ^(λ,μ) B ^{[λ, μ]} ◁ G . By Proposition 3.5, ( A B ) ^{[λ, μ]} ◁ G . Thus, by Theorem 3.7, A B ∈ IVNG( G ). So A ∨ B ∈ IVN _{f[s, t]} ( G ). From Proposition 3.8, it is clear that A ∧ B ∈ IVNG( G ). Thus A ∧ 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 interval-valued fuzzy subgroup discussed herein can be visualized by the lattice diagram in Figure 1 . It is also well-known [20 , Theorem I. 11] that the sublattice of normal subgroups of a group is modular. As an interval-valued 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]. Let G be a group and let A ∈ IVG( G ). If for any x, y ∈ G , A ^L ( x ) < A ^L ( y ) and A ^U ( x ) < A ^U ( y ), then A ( xy ) = A ( x ) = A ( yx ). Definition 3.10 [20,21]. A lattice ( L ,∧,∨) is said to be modular if for any x, y, z ∈ L with x ≤ z [resp. x ≥ z ], x ∨( y ∧ z ) = ( x ∨ y ) ∧ z[resp. x ∧ ( y ∨ z ) = ( x ∧ y ) ∨ z ]. In any lattice L , it is well-known [ 21 , Lemma I.4.9] that for any x, y, z ∈ L if x ≤ z [resp. x ≥ z ], then x ∨ ( y ∧ z ) ≤ ( x ∨ y ) ∧ z [resp. x ∧ ( y ∨ z ) ≥ ( x ∧ y ) ∨ z ]. The inequality is called the modular inequality . Theorem 3.11. The lattice IVN _{f[s, t]} ( G ) is modular. Proof. Let A,B,C ∈ IVN _{f[s, t]} ( G ) such that A ⊃ C . Then, by the modular inequality, ( A ∧ B )∨ C ⊂ A ∧( B ∨ C ). Assume that A ∧ ( B ∨ C ) ⊄ ( A ∧ B ) ∨ C , i.e., there exists z ∈ 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 Im B and Im C are finite, there exist x₀ , y₀ ∈ G with z = x₀y₀ such that (B ∨ C)(z) = (B C)(z) (By the process of the proof of Theorem 3.9) Thus On 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 ) and C ^U ( y ₀ ) > A ^U ( y ₀ ). This contradicts the fact that A ⊃ C . So A ∧ ( B ∨ C ) ⊂ ( A ∧ B ) ∨ C . Hence A ∧ ( 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 interval-valued fuzzy subgroups that attain the value [1, 1] at the identity element of G . Lemma 3.12. Let A be a subset of a group G . Then where < A > is the subgroup generated by A . Proof. Let 𝐵 = { B ∈ IVG( G ) : [ χ_A , χ_A ] ⊂ B }, let B ∈ 𝐵 and let x ∈ A . Then χ_A(x) = 1 ≤ B^L(x) and χ_A(x) = 1 ≤ B^U(x). Thus B ( x ) = [1, 1]. Since B ∈ IVG( G ), B = for any composite of elements of A . 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. Let A and B subgroups of a group G . Then

• (a)A◁Gif and only if [χA, χA] ∈IVN(G).
• (b) [χA,χA][χB,χB] = [χA·B,χA·B].

Proposition 3.14. Let S ( G ) be the set of all subgroup of a group G 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. Let A,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 ) of G a group G as a sublattice of the lattice of all intervalvalued fuzzy subgroups IVG( G ) of G .

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 IVN_{f[1, 1]}(G) ∩ IVG(S(G)) = IVN(N(G)); where N ( G ) denotes the set of all normal subgroups of G 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 well-known classical result. Corollary 3.15. Let G be a group. Then N ( G ) forms a modular sublattice of S ( G ). Lee et al. [11] studied interval-valued 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 interval-valued fuzzy ideals/(generalized) bi-ideals, subgroup and ring, respectively. In this paper, we mainly study sublattices of the lattice of interval-valued 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 group G , 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 interval-valued fuzzy subrings of a ring. Jeong Gon Lee received the Ph.D degree in The Department of Mathematics Education from Korea National University of Education. He is currently Assistant Professor inWonkwang University, Korea. His research interests are Measure Theory, Operator Theory, Mathematics Education, Category Theory, Hyperspace, and Topology. At present he has worked as one of ”Managing Editors” in Annals of Fuzzy Mathematics and Informatics (AFMI). E-mail: jukolee@wku.ac.kr Kul Hur received the Ph.D degree in The Department of Mathematics from Yonsei University. He was a Professor inWonkwang University. His research interests are Category Theory, Hyperspace and Topology. He retired from Wonkwang University on February 2012. At present he has worked as one of ”Editors-in-Chief” in Annals of Fuzzy Mathematics and Informatics (AFMI). E-mail: kulhur@wonkwang.ac.kr Pyung Ki Lim received the Ph.D degree in The Department of Mathematics from Chonnam National University, Korea. He is currently Professor in Wonkwang University. His research interests are Category Theory, Hyperspace and Topology. E-mail: pklim@wonkwang.ac.kr