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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 mβ-continuity, and then Min and Yoo [18] researched interval-valued fuzzy mα-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.
2. Preliminaries
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
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(0, 1). We also note that
(i) (∀M,ND(I)) (M=NML=NL,MU=NU),
(ii) (∀M,ND(I)) (M=NMLNL,MUNU).
For every M
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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}
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I^{ X} such that A^{ L}
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A^{ U}, i.e., A^{ L}(x)
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A^{ U}(x) for each x
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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]
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D(I), the interval-valued fuzzy A in X defined by A(x) = [A^{ L}(x),A^{ U}(x)] = [a, b] for each x
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X is denoted by
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and if a = b, then the IVFS
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is denoted by simply
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. In particular,
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and
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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
(i)A⊂BiffAL≤BLandAU≤BU.
(ii)A=BiffA⊂BandB⊂A.
(iii)AC= [1 −AU, 1 −AL].
(iv)A∪B= [AL∨BL,AU∨BU].
(iv)'Aα= [,].
(v)A∩B= [AL∧BL,AU∧BU].
(v)'Aα= [,].
Result 2.A[13, Theorem 1]. Let A, 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) (Ac)c=A.
(i) (Aα)c=Acα, (Aα)c=Acα.
Definition 2.3[8]. Let (X, ·) be a groupoid and let A
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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, yX. It is clear that
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,
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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,
(i)AL(xy) ≥AL(x) ∧AL(y) andAU(xy) ≥AU(x) ∧AU(y).
(ii)AL(x−1) ≥AL(x) andAU(x−1) ≥AU(x).
We will denote the set of all IVGs of G as IVG(G).
Result 2.A[8, Proposition 4.3]. Let G be a group and let {A_{α}} _{α∈Γ} ⊂ IVG(G). Then
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A_{α} ∈ IVG(G).Result 2.B [4, Proposition 3.1]. Let A be an IVG in a group G. Then
(a)A(x−1) =A(x), ∀x∈G.
(b)AL(e) ≥AL(x) andAU(e) ≥AU(x), ∀x∈G, whereeis the identity ofG.
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.
3. Lattices of Interval-Valued 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 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}
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be any subset of IVG(G). Then, by Result 2.A,
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∈ IVG(G). Moreover, it is clear that
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A_{α} is the largest interval-valued fuzzy subgroup contained in A_{α} for each
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. So
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A_{α} =
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A_{α}. On the other hand, we can easily see that (
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A_{α}) is the least intervalvalued fuzzy subgroup containing A_{α} for each
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. So
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A_{α} = (
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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
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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
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B)^{ [λ, μ]}.Proof. Let z ∈ A^{[λ, μ]} · B^{[λ, μ]}. Then there exist x_{0}, y_{0} ∈ G such that z = x_{0}y_{0}. Thus A^{L}(x_{0}) ≥ λ, A^{U}(x_{0}) ≥ μ and A^{L}(y_{0}) ≥ λ, A^{U}(y_{0}) ≥ μ. So and Thus
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. Hence
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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
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,
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Proof. Let
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Then and Since Im A and Im B are finite, there exist x_{0}, y_{0} ∈ G with z = x_{0}y_{0} 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
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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), (AB)^{ [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
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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
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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
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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
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B)^{ U}(z) ≥ A^{ U}(z). Thus A ⊂ A
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B. By the similar arguments, we have B ⊂ A
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B.Let C ∈ IVG(G) such that A ⊂ C and B⊂ C. Let z ∈ G. Then Similarly, we have
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Thus A
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B ⊂ C. So A
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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)}
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B^{ [λ, μ]}◁G. By Proposition 3.5, (A
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B)^{ [λ, μ]} ◁ G. Thus, by Theorem 3.7, A
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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 x_{0}, y_{0} ∈ G with z = x_{0}y_{0} such that (B ∨ C)(z) = (BC)(z)(By the process of the proof of Theorem 3.9) Thus
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On the other hand,
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and
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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})andA^{ U}(z)∧B^{ U}(x_{0})∧C^{ U}(y_{0}) > A^{ U}(x_{0})∧B^{ U}(x_{0})∧C^{ U}(y_{0}).ThenA^{ L}(z),B^{ L}(x_{0}),C^{ L}(y_{0}) > A^{ L}(x_{0}) ∧ B^{ L}(x_{0}) ∧ C^{ L}(y_{0})andA^{ U}(z),B^{ U}(x_{0}),C^{ U}(y_{0}) > A^{ U}(x_{0}) ∧ B^{ U}(x_{0}) ∧ C^{U}(y_{0}).ThusA^{ L}(x_{0}) ∧ B^{ L}(x_{0}))∧ C^{ L}(y_{0}) = A^{ L}(x_{0})andA^{ U}(x_{0}) ∧ B^{ U}(x_{0}) ∧ C^{ U}(y_{0}) = A^{ U}(x_{0}).SoA^{ L}(z) > A^{ L}(x_{0}), A^{ U}(z) > A^{ U}(x_{0})andC^{ 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})andA^{ 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}).ThusC^{ 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 =
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).
4. Conclusion
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.
No potential conflict of interest relevant to this article was reported.
Acknowledgements
This work was supported by the research grant of theWonkwang University in 2014.
BIO
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.krKul 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.krPyung 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
Zadeh L. A.
1965
“Fuzzy sets, ”
Information and Control
http://dx.doi.org/10.1016/S0019-9958(65)90241-X
8
(3)
338 -
353
DOI : 10.1016/S0019-9958(65)90241-X
Zadeh L. A.
1975
“The concept of a linguistic variable and its application to approximate reasoning−I,”
Information Sciences
http://dx.doi.org/10.1016/0020-0255(75)90036-5
8
(3)
199 -
249
DOI : 10.1016/0020-0255(75)90036-5
Gorzaczany M. B.
1987
“A method of inference in approximate reasoning based on interval-valued fuzzy sets,”
Fuzzy Sets and Systems
http://dx.doi.org/10.1016/0165-0114(87)90148-5
21
(1)
1 -
17
DOI : 10.1016/0165-0114(87)90148-5
Biswas R.
1994
“Rosenfeld’s fuzzy subgroups with intervalvalued membership functions,”
Fuzzy Sets and Systems
http://dx.doi.org/10.1016/0165-0114(94)90148-1
63
(1)
87 -
90
DOI : 10.1016/0165-0114(94)90148-1
Cheong M.
,
Hur K.
2011
“Interval-valued fuzzy ideals and Bi-ideals of a semigroup,”
International Journal of Fuzzy Logic and Intelligent Systems
http://dx.doi.org/10.5391/IJFIS.2011.11.4.259
11
(4)
259 -
266
DOI : 10.5391/IJFIS.2011.11.4.259
Jang S. U.
,
Hur K.
,
Lim P. K.
2012
“Interval-valued fuzzy normal subgroups,”
International Journal of Fuzzy Logic and Intelligent Systems
http://dx.doi.org/10.5391/IJFIS.2012.12.3.205
12
(3)
205 -
214
DOI : 10.5391/IJFIS.2012.12.3.205
Kang H. W.
2011
“Interval-valued fuzzy subgroups and homomorphisms,”
Honam Mathematical Journal
http://dx.doi.org/10.5831/HMJ.2011.33.4.499
33
(4)
499 -
518
DOI : 10.5831/HMJ.2011.33.4.499
Kang H. W.
,
Hur K.
2010
“Interval-valued fuzzy subgroups and rings,”
Honam Mathematical Journal
http://dx.doi.org/10.5831/HMJ.2010.32.4.593
32
(4)
593 -
617
DOI : 10.5831/HMJ.2010.32.4.593
Kim S. M.
,
Hur K.
,
Cheong M. S.
,
Chae G. B.
2012
“Interval-valued fuzzy quasi-ideals in a semigroups,”
International Journal of Fuzzy Logic and Intelligent Systems
http://dx.doi.org/10.5391/IJFIS.2012.12.3.215
12
(3)
215 -
225
DOI : 10.5391/IJFIS.2012.12.3.215
Lee K. C.
,
Kang H. W.
,
Hur K.
2011
“Interval-valued fuzzy generalized bi-ideals of a semigroup,”
Honam Mathematical Journal
http://dx.doi.org/10.5831/HMJ.2011.33.4.603
33
(4)
603 -
616
DOI : 10.5831/HMJ.2011.33.4.603
Lee J. G.
,
Hur K.
,
Lim P. K.
2013
“Interval-valued fuzzy subgroups,”
Honam Mathematical Journal
http://dx.doi.org/10.5831/HMJ.2013.35.4.565
35
(4)
565 -
582
DOI : 10.5831/HMJ.2013.35.4.565
Lee K. C.
,
Choi G. H.
,
Hur K.
2011
“The lattice of interval-valued intuitionistic fuzzy relations,”
Journal of Korean Institute of Intelligent Systems
http://dx.doi.org/10.5391/JKIIS.2011.21.1.145
21
(1)
145 -
152
DOI : 10.5391/JKIIS.2011.21.1.145
Jun Y. B.
,
Bae J. J.
,
Cho S. H.
,
Kim C. S.
2006
“ Interval-valued fuzzy strong semi-openness and interval-valued fuzzy strong semi-continuity,”
Honam Mathematical Journal
28
(3)
417 -
431
Min W. K.
2010
“Interval-valued fuzzy almost M-continuous mapping on interval-valued fuzzy topological spaces,”
International Journal of Fuzzy Logic and Intelligent Systems
http://dx.doi.org/10.5391/IJFIS.2010.10.2.142
10
(2)
142 -
145
DOI : 10.5391/IJFIS.2010.10.2.142
Min W. K.
2009
“Characterizations for interval-valued fuzzy m-semicontinuous mappings on interval-valued fuzzy minimal spaces,”
Journal of Korean Institute of Intelligent Systems
http://dx.doi.org/10.5391/JKIIS.2009.19.6.848
19
(6)
848 -
851
DOI : 10.5391/JKIIS.2009.19.6.848
Min W. K.
2011
“Interval-valued fuzzy mβ-continuous mappings on interval-valued fuzzy minimal spaces,”
International Journal of Fuzzy Logic and Intelligent Systems
http://dx.doi.org/10.5391/IJFIS.2011.11.1.033
11
(1)
33 -
37
DOI : 10.5391/IJFIS.2011.11.1.033
Min W. K.
,
Yoo Y. H.
2010
“Interval-valued fuzzy mα-continuous mappings on interval-valued fuzzy minimal spaces,”
International Journal of Fuzzy Logic and Intelligent Systems
http://dx.doi.org/10.5391/IJFIS.2010.10.1.054
10
(1)
54 -
58
DOI : 10.5391/IJFIS.2010.10.1.054
Choi J. Y.
,
Kim S. R.
,
Hur K.
2010
“Interval-valued smooth topological spaces,”
Honam Mathematical Journal
http://dx.doi.org/10.5831/HMJ.2010.32.4.711
32
(4)
711 -
738
DOI : 10.5831/HMJ.2010.32.4.711
@article{ E1FLA5_2014_v14n2_154}
,title={Lattices of Interval-Valued Fuzzy Subgroups}
,volume={2}
, url={http://dx.doi.org/10.5391/IJFIS.2014.14.2.154}, DOI={10.5391/IJFIS.2014.14.2.154}
, number= {2}
, journal={International Journal of Fuzzy Logic and Intelligent Systems}
, publisher={Korean Institute of Intelligent Systems}
, author={Lee, Jeong Gon
and
Hur, Kul
and
Lim, Pyung Ki}
, year={2014}
, month={Jun}
TY - JOUR
T2 - International Journal of Fuzzy Logic and Intelligent Systems
AU - Lee, Jeong Gon
AU - Hur, Kul
AU - Lim, Pyung Ki
SN - 1598-2645
TI - Lattices of Interval-Valued Fuzzy Subgroups
VL - 14
PB - Korean Institute of Intelligent Systems
DO - 10.5391/IJFIS.2014.14.2.154
PY - 2014
UR - http://dx.doi.org/10.5391/IJFIS.2014.14.2.154
ER -
Lee, J. G.
,
Hur, K.
,
&
Lim, P. K.
( 2014).
Lattices of Interval-Valued Fuzzy Subgroups.
International Journal of Fuzzy Logic and Intelligent Systems,
14
(2)
Korean Institute of Intelligent Systems.
doi:10.5391/IJFIS.2014.14.2.154
Lee, JG
,
Hur, K
,
&
Lim, PK
2014,
Lattices of Interval-Valued Fuzzy Subgroups,
International Journal of Fuzzy Logic and Intelligent Systems,
vol. 2,
no. 2,
Retrieved from http://dx.doi.org/10.5391/IJFIS.2014.14.2.154
[1]
JG Lee
,
K Hur
,
and
PK Lim
,
“Lattices of Interval-Valued Fuzzy Subgroups”,
International Journal of Fuzzy Logic and Intelligent Systems,
vol. 2,
no. 2,
Jun
2014.
Lee, Jeong Gon
and
,
Hur, Kul
and
,
Lim, Pyung Ki
and
,
“Lattices of Interval-Valued Fuzzy Subgroups”
International Journal of Fuzzy Logic and Intelligent Systems,
2.
2
2014:
Lee, JG
,
Hur, K
,
Lim, PK
Lattices of Interval-Valued Fuzzy Subgroups.
International Journal of Fuzzy Logic and Intelligent Systems
[Internet].
2014.
Jun ;
2
(2)
Available from http://dx.doi.org/10.5391/IJFIS.2014.14.2.154
Lee, Jeong Gon
,
Hur, Kul
,
and
Lim, Pyung Ki
,
“Lattices of Interval-Valued Fuzzy Subgroups.”
International Journal of Fuzzy Logic and Intelligent Systems
2
no.2
()
Jun,
2014):
http://dx.doi.org/10.5391/IJFIS.2014.14.2.154