Lattices of Interval-Valued Fuzzy Subgroups

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  • ABSTRACT

    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.


  • KEYWORD

    Interval-valued fuzzy set , Interval-valued fuzzy subgroup , Interval-valued 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 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 χA.

    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 (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 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 : XD(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 AI X.

    Definition 2.2 [13]. Let A,BD(I) X and let {Aα}α∈ΓD(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]. Let A, B, CD(I) X and let {Aα}α∈ΓD(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 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, yG, (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]. Let G be a group and let {Aα} α∈Γ ⊂ IVG(G). Then Aα ∈ 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) 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]. Let G be a group and let AG. 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, μI with λμ. Then the set A [λ,μ] = {xX : 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 Aα of interval-valued fuzzy subgroups to be the intersection ⋂Aα. However, the sup is defined as the interval-valued fuzzy subgroup generated by the union ⋃ Aα and denoted by ( ⋃ Aα). 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 {Aα}α be any subset of IVG(G). Then, by Result 2.A, ∈ IVG(G). Moreover, it is clear that Aα is the largest interval-valued fuzzy subgroup contained in Aα for each . So Aα = Aα. On the other hand, we can easily see that ( Aα) is the least intervalvalued fuzzy subgroup containing Aα for each . So Aα = ( Aα). 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 IVGf (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 IVGf (G)[resp. IVG[s, t](G)] is a sublattice of IVG(G). Moreover, IVGf (G)∩ IVG[s, t](G) is also a sublattice of IVG(G).

    Definition 3.2[11]. Let (X, ·) be a groupoid and let A,BD(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 xX, 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 zA[λ, μ] · B[λ, μ]. Then there exist x0, y0G such that z = x0y0. Thus AL(x0) ≥ λ, AU(x0) ≥ μ and AL(y0) ≥ λ, AU(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, y0G with z = x0y0 such that and Thus A L(x0) ≥ λ, A U(x0) ≥ μ and B L(y0) ≥ λ, B L(y0) ≥ μ. So x0A [λ, μ] and y0B [λ, μ], i.e., z = x0y0A [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, yG.

    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 λA L(e) and μA 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 AB ∈ 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 IVNf[s, t](G) = {A ∈ IVNG(G) : Im A is finite and A(e) = [s, t]}. Then IVNf[s, t](G) is a sublattice of IVGf (G)∩ IVG[s, t](G). Hence IVNf[s, t](G) is a sublattice of IVG(G).

    Proof. Let A, B ∈ IVNf[s, t](G). Then, by Result 3.C, AB ∈ IVG(G). Let zG. 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 AA B. By the similar arguments, we have BA B.

    Let C ∈ IVG(G) such that AC and BC. Let zG. Then Similarly, we haveThus A BC. So A B = AB.

    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 AB ∈ IVNf[s, t](G). From Proposition 3.8, it is clear that AB ∈ IVNG(G). Thus AB ∈ IVNf[s,t](G). Hence IVNf[s,t](G) is a sublattice of IVGf∩ 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, yG, 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, zL with xz[resp. xz], x∨(yz) = (xy) ∧ z[resp. x ∧ (yz) = (xy) ∨ z].

    In any lattice L, it is well-known [21, Lemma I.4.9] that for any x, y, zL if xz[resp. xz], then x ∨ (yz) ≤ (xy) ∧ z[resp. x ∧ (yz) ≥ (xy) ∨ 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 ∈ IVNf[s, t](G) such that AC. Then, by the modular inequality, (AB)∨CA∧(BC). Assume that A ∧ (BC) ⊄ (AB) ∨ C, i.e., there exists zG 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, y0G with z = x0y0 such that (B ∨ C)(z) = (B C)(z) (By the process of the proof of Theorem 3.9) Thus

    image

    On the other hand,

    image

    and

    image

    By (3.1), (3.2) and (3.3),

    A L(z) ∧ B L(x0) ∧ C L(y0) > A L(x0) ∧ B L(x0) ∧ C L(y0)

    and

    A U(z)∧B U(x0)∧C U(y0) > A U(x0)∧B U(x0)∧C U(y0).

    Then

    A L(z),B L(x0),C L(y0) > A L(x0) ∧ B L(x0) ∧ C L(y0)

    and

    A U(z),B U(x0),C U(y0) > A U(x0) ∧ B U(x0) ∧ CU(y0).

    Thus

    A L(x0) ∧ B L(x0))∧ C L(y0) = A L(x0)

    and

    A U(x0) ∧ B U(x0) ∧ C U(y0) = A U(x0).

    So

    A L(z) > A L(x0), A U(z) > A U(x0)

    and

    C L(y0) > A L(x0), C U(y0) > A U(x0).

    By Result 2.B,

    A L(x0 −1) = A L(x0) < A L(x0y0)

    and

    A U(x0 −1) = A U(x0) < A U(x0y0).

    By Result 3.D, A(x0) = A(x0 −1x0y0) = A(y0).

    Thus

    C L(y0) > A L(y0) and C U(y0) > A U(y0).

    This contradicts the fact that AC. So A ∧ (BC) ⊂ (AB) ∨ C. Hence A ∧ (BC) = (AB) ∨ C. Therefore IVNf[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) : [χA, χA] ⊂ B}, let B𝐵 and let xA. Then χA(x) = 1 ≤ BL(x) and χA(x) = 1 ≤ BU(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 ◁ G if 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)) = {[χA, χA] : AS(G)}. Then IVG(S(G)) forms a sublattice of IVGf (G) ∩ IVG[1,1](G) and hence of IVG(G).

    Proof. Let A,BS(G). Then it is clear that [χA, χA] ∩ [χB, χB] = [χAB, χAB] ∈ IVG(S(G)). By Lemma 3.12,

    < [χA, χA] ∪ [χB, χB] > = < [χAB, χAB] > = [χ<AB>, χ<AB>].

    Thus

    [χA, χA]∨[χB, χB] =< [χA, χA]∪[χB, χB] >∈ IVG(S(G)).

    Moreover, IVG(S(G)) ⊂ IVGf (G)∩ IVG[1,1](G).

    Hence IVG(S(G)) is a sublattice of IVGf (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) , IVNf[s, t](G) forms a modular sublattice of IVGf(G)∩ IVG[s, t](G). Therefore, for [s, t] = [1, 1], the sublattice IVNf[1, 1](G) is also modular. It is clear that IVNf[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)) = {[χN, χN] : NN(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).

    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 IVNf[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 IVGf (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.

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