Interval-Valued Fuzzy Congruences on a Semigroup

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  • ABSTRACT

    We introduce the concept of interval-valued fuzzy congruences on a semigroup S and we obtain some important results: First, for any interval-valued fuzzy congruence R on a group G, the interval-valued congruence class Re is an interval-valued fuzzy normal subgroup of G. Second, for any interval-valued fuzzy congruence R on a groupoid S, we show that a binary operation * an S/R is well-defined and also we obtain some results related to additional conditions for S. Also we improve that for any two interval-valued fuzzy congruences R and Q on a semigroup S such that RQ, there exists a unique semigroup homomorphism g : S/R → S/G.


  • KEYWORD

    Interval-valued fuzzy set , Interval-valued fuzzy (normal) subgroup , Interval-valued fuzzy congruence

  • 1. Introduction

    As a generalization of fuzzy sets introduced by Zadeh [1], Zadeh [2] also suggested the concept of interval-valued fuzzy sets. After that time, Biswas [3] applied it to group theory, and Gorzalczany [4] introduced a method of inference in approximate reasoning by using interval-valued fuzzy sets. Moreover, Mondal and Samanta [5] introduced the concept of interval-valued fuzzy topology and investigated some of it’s properties. In particular, Roy and Biswas [6] introduced the notion of interval-valued fuzzy relations and studied some of it’s properties. Recently, Jun et al. [7] investigated strong semi-openness and strong semicontinuity in interval-valued fuzzy topology. Moreover, Min [8] studied characterizations for interval-valued fuzzy m-semicontinuous mappings, Min and Kim [9,10] investigated intervalvalued fuzzy m*-continuity and m*-open mappings. Hur et al. [11] studied interval-valued fuzzy relations in the sense of a lattice theory. Also, Choi et al. [12] introduced the concept of interval-valued smooth topological spaces and investigated some of it’s properties.

    On the other hand, Cheong and Hur [13], and Lee et al. [14] studied interval-valued fuzzy ideals/(generalized)bi-ideals in a semigroup. In particular, Kim and Hur [15] investigated interval-valued fuzzy quasi-ideals in a semigroup. Kang [16], Kang and Hur [17] applied the notion of interval-valued fuzzy sets to algebra. Jang et al. [18] investigated interval-valued fuzzy normal subgroups.

    In this paper, we introduce the concept of interval-valued fuzzy congruences on a semigroup S and we obtain some important results:

    (i) For any interval-valued fuzzy congruence R on a group G, the interval-valued congruence class Re is an interval-valued fuzzy normal subgroup of G (Proposition 3.11).

    (ii) For any interval-valued fuzzy congruence R on a groupoid S, we show that a binary operation * an S/R is well-defined (Proposition 3.20) and also we obtain some results related to additional conditions for S (Theorem 3.21, Corollaries 3.21-1, 3.21-2, and 3.21-3). Also we improve that for any two intervalvalued fuzzy congruences R and Q on a semigroup S such that RQ, there exists a unique semigroup homomorphism g : S/R → S/G (Theorem 4.3).

    2. Preliminaries

    In this section, we list some concepts and well-known results which are needed in later sections.

    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 = [ML, MU], where ML and MU are the lower and the upper end points respectively. Especially, we denoted , 0 = [0, 0], 1 = [1, 1], and a = [a, a] for every a ∈ (0, 1), We also note that

    (i) (∀M, ND(I)) (M = NML = NL, MU = NU),

    (ii) (∀M, ND(I)) (M = NMLNL, MUNU),

    For every MD(I), the complement of M, denoted by MC, is defined by MC = 1 ? M = [1 ? MU, 1 ? ML]([7, 14]).

    Definition 2.1 [4, 10, 14]. A mapping A : XD(I) is called an interval-valued fuzzy set(IVFS) in X, denoted by A = [AL, AU], if AL, ALIX such that ALAU, i.e., AL(x) ≤ AU(x) for each xX, where AL(x)[resp AU(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) = [AL(x), AU(x)] = [a, b] for each xX is denoted by

    image

    and if a = b, then the IVFS

    image

    is denoted by simply a. In particular,

    image

    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 AIX.

    Definition 2.2 [14]. Let A, BD(I)X and let {Aαα∈ГD(I)X. Then

    (i) A ⊂ B iff AL ≤ BL and AU ≤ BU.

    (ii) A = B iff A ⊂ B and B ⊂ A.

    (iii) AC = [1 ? AU, 1 ? AL].

    (iv) A ∪ B = [AL ? BL, AU ? BU].

    image

    (v) A ∩ B = [AL ? BL, AU ? BU].

    image

    Result 2.A [14, Theorem 1]. Let A, B, C D(I)X and let {Aα}α∈ГD(I)X. Then

    image

    (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.

    image

    (h) (Ac)c = A.

    image

    Definition 2.3 [8]. Let X be a set. Then a mapping R = [RL , RU] : XXD(I) is called an interval-valued fuzzy relation(IVFR) on X.

    We will denote the set of all IVFRs on X as IVR(X).

    Definition 2.4 [8]. Let R ∈ IVR(X). Then the inverse of R, R?1 is defined by R?1(x,y) = R(y,x), for each x, yX.

    Definition 2.5 [11]. Let X be a set and let R, Q ∈ IVR(X). Then the composition of R and Q, QR, is defined as follows : For any x, yX,

    image

    and

    image

    Result 2.B [11, Proposition 3.4]. Let X be a set and let R, R1, R2, R3, Q1, Q2 ∈ IVR(X). Then

    (a) (R1 ○ R2) ○ R3 = R1 ○ (R2 ○ R3).

    (b) If R1 ⊂ R2 and Q1 ⊂ Q2, then R1 ○ Q1 ⊂ R2 ○ Q2.

    In particular, if Q1 ⊂ Q2, then R1 ○ Q1 ⊂ R1 ○ Q2.

    (c) R1(R2 ∪ R3) = (R1 ○ R2) ∪ (R1 ○ R3),

    R1(R2 ∩ R3) = (R1 ○ R2) ∩ (R1 ○ R3).

    image

    Definition 2.6 [11]. An IVFR R on a set X is called an interval-valued fuzzy equivalence relation(IV FER) on X if it satisfies the following conditions :

    (1) it is interval-valued fuzzy reflexiv, i.e., R(x, x) = [1, 1], for each xX,

    (2) it is interval-valued fuzzy symmetric, i.e., R?1 = R,

    (3) it is interval-valued fuzzy transitive, i.e., RRR.

    We will denote the set of all IVFERS on X as IVE(X).

    From Definition 2.6, we can easily see that the following hold.

    Remark 2.7 (a) If R is an fuzzy equivalence relation on a set X, then [R, R] ∈ IVE(X).

    (b) If R ∈ IVE(X), then RL and RU are fuzzy equivalence relation on X.

    (c) Let R be an ordinary relation on a set X. Then R is an equivalence relation on X if and only if [ΧR, ΧR] ∈ IVE(X).

    Result 2.C [11, Proposition 3.9]. Let X be a set and let Q, R ∈ IVE(X). If QR = RQ, then RQ ∈ IVE(X).

    Let R be an IVFER on a set X and let aX. We define a mapping Ra : XD(I) as follows : For each aX,

    Ra(x) = R(a, x).

    Then clearly RaD(I)X. In this case, Ra is called the interval-valued fuzzy equivalence class of R containing aX. The set {Ra : aX} is called the interval-valued fuzzy quotient set of X by R and denoted by X/R.

    Result 2.D [11, Proposition 3.10]. Let R be an IVFER on a set X. Then the following hold :

    (a) Ra = Rb if and only if R(a, b) = [1, 1], for any a, bX.

    (b) R(a, b) = [0, 0] if and only if RaRb =

    image

    for any a, bX.

    image

    (d) There exits the surjection π : XX/R defined by π(x) = Rx for each xX.

    Definition 2.8 [11]. Let X be a set, let R ∈ IVR(X) and let {Rα}α∈Г be the family of all IVFERs on X containing R. Then ∩α∈Г Rα is called the IVFER generated by R and denoted by Re.

    It is easily seen that Re is the smallest IVFER containing R.

    Definition 2.9 [11]. Let X be a set and let R ∈ IVR(X). Then the interval-valued fuzzy transitive closure of R, denoted R, is defined as followings :

    image

    ,where Rn = RR ○ … ○ R(n factors).

    Definition 2.10 [11]. We define two mappings △, ▽ : XD(I) as follows : For any x, yX,

    image

    and

    ▽(x, y) = [1, 1].

    It is clear that △, ▽ ∈ IVE(X) and R is an interval-valued fuzzy reflexive relation on X if and only if △ ⊂ R.

    Result 2.E [11, Proposition 4.7]. If R is an IVFR on a set X, then

    Re = [R ∪ R?1 ∪ △]∞.

    Definition 2.11 [17]. Let (X, ·) be a groupoid and let A, BD(I)X. Then the interval-valued fuzzy product of A and B, AB is defined as follows : For each aX,

    image

    and

    image

    Definition 2.12 [17]. Let (X, ·) be a groupoid and let AD(I)X. Then A is called an iinterval-valued fuzzy subgroupoid (IVGP) of X if for any x, yX,

    AL ≥ AL(x) ∧ AL(y)

    and

    AU ≥ AU(x) ∧ AU(y).

    We will denote the set of all IVGPs of X as IVGP(X). Then it is clear that

    image

    Definition 2.13 [17]. Let G be a group and let A ∈ IVGP(G). Then A is an iinterval-valued fuzzy subgroup (IVG) of G if for each xG,

    A(x?1) ≥ A(x),

    i.e.,

    AL(x?1) ≥ AL(x) and AU(x?1) ≥ AU(x).

    We will denote the set of all IVGs of G as IVG(G).

    Definition 2.14 [17]. Let G be a group and let A ∈ IVG(G). Then A is said to be normal if A(xy) = A(yx), for any x, yG.

    We will denote the set of all interval-valued fuzzy normal subgroups of G as IVNG(G). In particular, we will denote the set {N ∈ IVNG(G) : N(e) = [1, 1]} as IVN(G).

    Result 2.F [17, Proposition 5.2]. Let G be a group and let AD(I)G. If B ∈ IVNG(G), then AB = BA.

    Definition 2.15 [18]. Let G be a group, let A ∈ IVG(G) and let xG. We define two mappings

    Ax : G → D(I)

    and

    xA : G → D(I)

    as follows, respectively : For each gG,

    Ax(g) = A(gx?1) and xA(g) = A(x?1g).

    Then Ax[resp: xA] is called the interval-valued fuzzy right[resp. left] coset of G determined by x and A.

    It is obvious that if A ∈ IVNG(G), then the interval-valued fuzzy left coset coincides with the interval-valued fuzzy right coset of A on G. In this case, we will call interval-valued fuzzy coset instead of interval-valued fuzzy left coset or intervalvalued fuzzy right coset.

    3. Interval-Valued Fuzzy Congruences

    Definition 3.1 [19]. A relation R on a groupoid S is said to be:

    (1) left compatible if (a, b) ∈ R implies (xa, xb) ∈ R, for any a, bS,

    (2) right compatible if (a, b) ∈ R implies (ax, bx) ∈ R, for any a, bS,

    (3) compatible if (a, b) ∈ R and (s, d) ∈ R imply (ab, cd) ∈ R, for any a, b, c, dS,

    (4) a left[resp. right] congruence on S if it is a left[resp. right] compatible equivalence relation.

    (5) a congruence on S if it is both a left and a right congruence on S.

    It is well-known [19, Proposition I.5.1] that a relation R on a groupoid S is congruence if and only if it is both a left and a right congruence on S. We will denote the set of all ordinary congruences on S as C(S).

    Now we will introduce the concept of interval-valued fuzzy compatible relation on a groupoid.

    Definition 3.2 An IVFR R on a groupoid S is said to be :

    (1) interval-valued fuzzy left compatible if for any x, y, zG,

    RL(x, y) ≤ RL(zx, zy) and RU(x, y) ≤ RU(zx, zy),

    (2) interval-valued fuzzy right compatible if for any x, y, zG,

    RL(x, y) ≤ RL(xz, yz) and RU(x, y) ≤ RU(xz, yz),

    (3) interval-valued fuzzy compatible if for any x, y, z, tG,

    RL(x, y) ∧ RL(z, t) ≤ RL(xz, yz)

    and

    RU(x, y ∧ RU(z, t) ≤ RU(xz, yz).

    Example 3.3 Let S = e, a, b be the groupoid with multiplication table :

    image

    (a) Let R1 : S × SD(I) be the mapping defined as the matrix :

    image

    where [λij , μij] ∈ D(I) such that [λ1i , μ1i](i = 1, 2, 3),

    [λ21 , μ21] and [λ31 , μ31] are arbitrary, and

    [λ23 , μ23] = [λ32 , μ32], [λ22 , μ22] = [λ33 , μ33],

    [λ11 , μ11] ≤ [λ22 , μ22],

    [λ12 , μ12] ≤ [λ23 , μ23] ∧ [λ22 , μ22],

    [λ13 , μ13] ≤ [λ23 , μ23] ∧ [λ22 , μ22],

    [λ21 , μ21] ≤ [λ23 , μ23] ∧ [λ22 , μ22],

    [λ31 , μ31] ≤ [λ23 , μ23] ∧ [λ22 , μ22].

    Then we can see that R1 is an interval-valued fuzzy left compatible relation on S.

    (b) Let R2 : S × SD(I) be the mapping defined as the matrix :

    image

    where [λij , μij] ∈ D(I) such that [λij , μij](i, j = 1, 2, 3) is arbitrary and

    [λ11 , μ11] ≤ [λ21 , μ21], [λ12 , μ12] ≤ [λ31 , μ31],

    [λ13 , μ13] ≤ [λ31 , μ31], [λ21 , μ21] ≤ [λ31 , μ31],

    [λ32 , μ32] ≤ [λ22 , μ22],

    [λ33 , μ33] ≤ [λ23 , μ23] = [λ22 , μ22].

    Then we can see that R2 is an interval-valued fuzzy right compatible relation on S.

    (c) Let R3 : S × SD(I) be the mapping defined as the matrix :

    image

    where [λij , μij] ∈ D(I) such that

    λ11 ∧ λ12 ≤ λ12, μ11 ∧ μ12 ≤ μ12, λ11 ∧ λ13 ≤ λ13,

    μ11 ∧ μ13 ≤ μ13, λ12 ∧ λ13 ≤ λ12, μ12 ∧ μ13 ≤ μ12,

    λ21 ∧ λ22 ≤ λ32, μ21 ∧ μ22 ≤ μ32, λ21 ∧ λ23 ≤ λ33,

    μ21 ∧ μ23 ≤ μ33, λ22 ∧ λ23 ≤ λ32, μ22 ∧ μ23 ≤ μ32,

    λ31 ∧ λ32 ≤ λ22, μ31 ∧ μ32 ≤ μ22, λ31 ∧ λ33 ≤ λ23,

    μ31 ∧ μ33 ≤ μ23, λ32 ∧ λ33 ≤ λ22, μ32 ∧ μ33 ≤ μ22,

    Then we can see that R3 is an interval-valued fuzzy compatible relation on S.

    Lemma 3.4 Let R be a relation on a groupoid S. Then R is left compatible if and only if [ΧR, ΧR] is interval-valued fuzzy left compatible.

    Proof. (⇒) : Suppose R is left compatible. Let a, b, xS.

    Case(1) Suppose (a, b) ∈ R. Then ΧR(a, b) = 1. Since R is left compatible, (xa, xb) ∈ R, for each xS. Thus ΧR(xa, xb) = 1 = ΧR(a, b).

    Case(2) Suppose ¬(a, b) ∈ R. Then, for each xS, it holds that ΧR(a, b) = 0 ≤ ΧR(xa, xb). Thus, in either cases, [ΧR, ΧR].

    (?) : Suppose [ΧR, ΧR] is interval-valued fuzzy compatible. Let a, b, xS and (a, b) ∈ R. Then, by hypothesis, ΧR(xa, xb) ≥ ΧR(a, b) = 1. Thus ΧR(xa, xb) = 1. So (xa, xb) ∈ R. Hence R is left compatible.

    Lemma 3.5 [The dual of Lemma 3.4]. Let R be a relation on a groupoid S. Then R is right compatible if and only if [ΧR, ΧR] is interval-valued fuzzy right compatible.

    Definition 3.6 An IVFER R on a groupoid S is called an :

    (1) interval-valued fuzzy left congruence (IVLC) if it is intervalvalued fuzzy left compatible,

    (2) interval-valued fuzzy right congruence (IVRC) if it is interval-valued fuzzy right compatible,

    (3) interval-valued fuzzy congruence (IVC) if it is intervalvalued fuzzy compatible.

    We will denote the set of all IVCs[resp. IVLCs and IVRCs] on S as IVC(S) [resp: IVLC(S) and IVRC(S)].

    Example 3.7 Let S = e, a, b be the groupoid defined in Example 3.3. Let R1 : S × SD(I) be the mapping defined as the matrix :

    image

    Then it can easily be checked that R ∈ IVE(S). Moreover we can see that R ∈ IVC(S).

    Proposition 3.8 Let S be a groupoid and let R ∈ IVE(S). Then R ∈ IVC(S) if and only if it is both an IVLC and an IVRC.

    Proof. (⇒) : Suppose R ∈ IVC(S) and let x, y, zS. Then

    RL(x, y) = RL(x, y) ∧ RL(z, z) ≤ RL(xz, yz)

    and

    RU(x, y) = RU(x, y) ∧ RU(z, z) ≤ RU(xz, yz).

    Also,

    RL(x, y) = RL(z, z) ∧ RL(x, y) ≤ RL(zx, zy)

    and

    RU(x, y) = RU(z, z) ∧ RU(x, y) ≤ RU(zx, zy).

    Thus R is both an IVLC and an IVRC.

    (?) : Suppose R is both an IVLC and an IVRC. and let x, y, z, tS. Then

    image

    By the similar arguments, we have that

    RU(x, y) ∧ RU(z, t) ≤ RU(xz, yt)

    So R is interval-valued fuzzy compatible. Hence R ∈ IVC(S).

    The following is the immediate result of Remark 2.7(c), Lemmas 3.4 and 3.5, and Proposition 3.5.

    Theorem 3.9 Let R be a relation on a groupoid S. Then R ∈ C(S) if and only if [ΧR, ΧR] ∈ IVC(S).

    For any interval-valued fuzzy left[resp. right] compatible relation R, it is obvious that if G is a group, then R(x, y) = R(tx, ty)[resp: R(x, y) = R(xt, yt)], for any x, y, tG. Thus we have following result.

    Lemma 3.10 Let R be an IVC on a group G. Then

    R(xay, xby) = R(xa, xb) = R(ay, by) = R(a, b),

    for any a, b, x, yG.

    Example 3.11 Let V be the Klein 4-group with multiplication table :

    image

    Let R : V × VD(I) be the mapping defined as the matrix :

    image

    Then we can see that R ∈ IVC(V). Furthermore, it is easily checked that Lemma 3.10 holds : For any s, t, x, yV,

    R(xsy, xty) = R(xs, xt) = R(sy, ty) = R(s, t)

    The following is the immediate result of Proposition 3.8 and Lemma 3.10.

    Theorem 3.12 Let R be an IVFR on a group G. Then R ∈ IVC(G) if and only if it is interval-valued fuzzy left(right) compatible equivalence relation.

    Lemma 3.13 Let P and Q be interval-valued fuzzy compatible relations on a groupoid S. Then QP is also an interval-valued fuzzy compatible relation on S.

    Proof. Let a, b, xS. Then

    image

    By the similar arguments, we have that

    (Q○P)U(ax, bx) ≥ PU(a, c) ∧ QU(c, b) for each c ∈ S:

    Thus

    image

    and

    image

    So QP is interval-valued fuzzy right compatible. Similarly, we can see that QP is interval-valued fuzzy left compatible. Hence QP is interval-valued fuzzy compatible.

    Theorem 3.14 Let P and Q be IVC on a groupoid S. Then the following are equivalent :

    (a) Q○P ∈ IVC(S).

    (b) Q○P ∈ IVE(S).

    (c) Q○P is interval-valued fuzzy symmetric.

    (d) Q○P = P○Q

    Proof. It is obvious that (a) ⇒ (b) ⇒ (c).

    (c) ⇒ (d) : Suppose the condition (c) holds and let a, bS. Then

    image

    Similarly, we have that

    (Q○P)U(a, b) = (P○Q)U(a, b).

    Hence QP = PQ

    (d) ⇒ (a) : Suppose the condition (d) holds. Then , by Result 2.C, QP ∈ IVE(S). Since P and Q are interval-valued fuzzy compatible, by Lemma 3.13, QP is interval-valued fuzzy compatible. So QP ∈ IVC(S). This completes the proof.

    Proposition 3.15 Let S be a groupoid and let Q, P ∈ IVC(S). If QP = PQ, then PQ ∈ IVC(S).

    Proof. By Result 2.C, it is clear that PQ ∈ IVE(S). Let x, y, tS. Then, since P and Q are interval-valued fuzzy right compatible,

    image

    Similarly, we have that

    (P○Q)U(x, y) ≤ (P○Q)U(xt, yt).

    By the similar arguments, we have that

    (P○Q)L(x, y) ≤ (P○Q)L(tx, ty)

    and

    (P○Q)U(x, y) ≤ (P○Q)U(tx, ty).

    So PQ is interval-valued fuzzy left and right compatible.

    Hence PQ ∈ IVC(S).

    Let R be an IVC on a groupoid S and let aS. Then RaD(I)S is called an interval-valued fuzzy congruence class of R containing aS and we will denote the set of all interval-valued fuzzy congruence classes of R as S/R.

    Proposition 3.16 If R is an IVC on a groupoid S, then RaRbRab, for any a, bS.

    Proof. Let xS. If x is not expressible as x = yz, then clearly (RaRb)(x) = [0, 0]. Thus RaRbRab. Suppose x is expressible as x = yz. Then

    image

    Similarly, we have that

    (RaRb)U(x) ≤ (Rab)U(x).

    Thus RaRbRab. This completes the proof.

    Proposition 3.17 Let G be a group with the identity e and let R ∈ IVC(G). We define the mapping AR : GD(I) as follows : For each aG,

    AR(a) = R(a, e) = Re(a).

    Then AR = Re ∈ IVNG(G).

    Proof. From the definition of AR, it is obvious that ARD(I)G. Let a, bG. Then

    image

    Similarly, we have that

    image

    On the other hand,

    image

    Moreover,

    image

    So AR ∈ IVG(G) such that AR(e) = [1, 1].

    Finally,

    image

    Hence AR ∈ IVNG(G). This completes the proof.

    The following is the immediate result of Proposition 3.17 and Result 2.F. Proposition 3.18 Let G be a group with the identity e. If P,Q ∈ IVNG(G), then PeQe = QePe.

    Proposition 3.19 Let G be a group with the identity e. If R ∈ IVC(G), then any interval-valued fuzzy congruence class Rx of xG by R is an interval-valued fuzzy coset of Re. Conversely, each interval-valued fuzzy coset of Re is an interval-valued fuzzy congruence class by R.

    Proof. Suppose R ∈ IVC(G) and let x.gG. Then Rx(g) = R(x, g). Since R is interval-valued fuzzy left compatible, by Lemma 3.10, R(x, g) = R(e, x?1 g). Thus

    Rx(g) = R(e, x?1 g) = Re(?1g) = (xRe)(g).

    So Rx = xRe. Hence Rx is an interval-valued fuzzy coset of Re.

    Conversely, let A be any interval-valued fuzzy coset of Re. Then there exists an xG such that A = xRe. Let gG.

    Then

    A(g) = (xRe)(g) = Re(x?1g) = R(e, x?1 g).

    Since R is interval-valued fuzzy left compatible,

    R(e, x?1 g) = R(x, g) = Rx(g).

    So A = Rx. Hence A is an interval-valued fuzzy congruence class of x by R.

    Proposition 3.20 Let R be an IVC on a groupoid S. We define the binary operation * on S/R as follows : For any a, bS,

    Ra * Rb = Rab.

    Then * is well-defined.

    Proof. Suppose Ra = Rx and Rb = Ry, where a, b, x, yS. Then, by Result 2.D(a),

    R(a, x) = R(b, y) = [1, 1].

    Thus

    image

    Similarly, we have that

    RU(ab, xy) ≥ 1.

    Thus R(ab, xy) = [1, 1]. By Result 2.D(a), Rab = Rxy. So Ra * Rb = Rx * Ry. Hence * is well-defined.

    From Proposition 3.20 and the definition of semigroup, we obtain the following result.

    Theorem 3.21 Let R be an IVC on a semigroup S. Then (S/R, *) is a semigroup.

    A semigroup S is called an inverse semigroup [7] if each aS has a unique inverse, i.e., there exists a unique a?1S such that aa?1a = a and a?1 = a?1aa?1.

    Corollary 3.21-1 Let R be an IVC on an inverse semigroup S. Then (S/R, *) is an inverse semigroup. Proof. By Theorem 3.21, (S/R, *) is a semigroup. Let aS. Since S is an inverse semigroup, there exists a unique a?1S such that aa?1a = a and a?1 = a?1aa?1. Moreover, it is clear that (Ra)?1 = Ra?1. Then (Ra)?1 * Ra * (Ra)?1 = Ra?1 * Ra * Ra?1 = Ra?1aa?1 = Ra?1 and Ra * (Ra)?1 * Ra = Ra * Ra?1 * Ra = Raa?1a = Ra.

    So Ra?1 is an inverse of Ra for each aS.

    An element a of a semigroup S is said to be regular if aaSa, i.e., there exists an xS such that a = axa. The semigroup S is said to be regular if for each aS, a is a regular element. Corresponding to a regular element a, there exists at least one aS such that a = aaa and a = aaa. Such an a is called an inverse of a.

    Corollary 3.21-2 Let R be an IVC on a regular semigroup S.

    Then (S/R, *) is a regular semigroup.

    Proof. By Theorem 3.21, (S/R, *) is a semigroup. Let aS. Since S is a regular semigroup, there exists an xS such that a = axa. It is obvious that RxS/R. Moreover, Ra * Rx * Ra = Raxa = Ra. So Ra is an regular element of S/R. Hence S/R is a regular semigroup.

    Corollary 3.21-3 Let R be an IVC on a group G. Then (G/R, *) is a group.

    Proof. By Theorem 3.21, (G/R, *) is a semigroup. Let xG. Then

    Rx * Re = Rxe = Rx = Rex = Re * Rx.

    Thus Re is the identity in G/R with respect to *. Moreover,

    Rx * Rx?1 = Rxx?1 = Re = Rx?1x = Rx?1 * Rx.

    So Rx?1 is the inverse of Rx with respect to *. Hence G/R is a group.

    Proposition 3.22 Let G be a group and let R ∈ IVC(G). We define the mapping π : G/RD(I) as follows : For each xG,

    π(Rx) = [(Rx)L(e), Rx)U(e)].

    Then π ∈ IVG(G/R).

    Proof. From the definition of π, it is clear that π = [πL, πU] ∈ D(I)G/R. Let x, yG. Then

    image

    Similarly, we have that

    πU(Rx * Ry) ≥ πU(Rx) ∧ πU(Ry).

    By the process of the proof of Corollary 3.21-1, (Rx)?1 = Rx?1. Thus

    π((Rx)?1) = π(Rx?1) = R(x?1, e) = R(e, x) = π(Rx).

    So π((Rx)?1) = π(Rx) for each xG. Hence π ∈ IVG(G/R).

    Proposition 3.23 If R is an IVC on an inverse semigroup S. Then R(x?1, y?1) = R(x, y) for any x, yS. Proof. By Corollary 3.21-1, S/R is an inverse semigroup with (Rx)?1 = Rx?1 for each xS. Let x, yS. Then

    image

    Hence R(x?1, y?1) = R(x, y).

    The following is the immediate result of Proposition 3.22

    Corollary 3.23 Let R be an IVC on a group G. Then

    R(x?1, y?1) = R(x, y)

    for any x, yG.

    Proposition 3.24 Let R be an IVC on a semigroup S. Then

    R?1([1, 1]) = {(a, b) ∈ S × S : R(a, b) = [1; 1]}

    is a congruence on S. Proof. It is clear that R?1([1, 1]) is reflexive and symmetric. Let (a, b), (b, c) ∈ R?1([1, 1]). Then R(a, b) = R(b, c) = [1, 1]. Thus

    image

    Similarly, we have that RU(a, c) ≥ 1. So R(a, c) = [1, 1], i.e., (a, c) ∈ R?1([1, 1]). Hence R?1([1, 1]) is an equivalence relation on S.

    Now let (a, b) ∈ R?1([1, 1]) and let xS. Since R is an IVC on S,

    RL(ax, bx) ≥ RL(a, b) = 1 and RU(ax, bx) ≥ RU(a, b) = 1.

    Then R(ax, bx) = [1, 1]. Thus (ax, bx) ∈ R?1([1, 1]). Similarly, (xa, xb) ∈ R?1([1, 1]). So R?1([1, 1]) is compatible. Hence R?1([1, 1]) is a congruence on S.

    Let S be a semigroup. Then S1 denotes the monoid defined as follows :

    image

    Definition 3.25 Let S be a semigroup and let R ∈ IVR(S). Then we define a mapping R* : S × SD(I) as follows : For any c, dS,

    image

    and

    image

    It is obvious that R* ∈ IVR(S).

    Proposition 3.26 Let S be a semigroup and let R, P,Q ∈ IVR(S). Then :

    (a) R ⊂ R*.

    (b) (R*)?1 = (R?1)*.

    (c) If P ⊂ Q, then P* ⊂ Q*.

    (d) (P*)* = P*.

    (e) (P ∪ Q)* = P* ∪ Q*.

    (f) R = R* if and only if R is left and right compatible.

    Proof. From Definition 3.25, the proofs of (a), (b) and (c) are clear.

    (d) By (a) and (c), it is clear that R* ⊂ (R*)*. Let c, dS. Then

    image

    By the similar arguments, we have that

    ((R*)*)U(c, d) ≤ (R*)U(c, d).

    Thus (R*)* ⊂ R*. So (R*)* = R*.

    (e) By (c), R* ⊂ (PQ)* and Q* ⊂ (PQ)*. Thus P* ∪ Q* ⊂ (PQ)*. Let c, dS. Then

    image

    Similarly, we have that

    ((P ∪ Q)*)U(c, d) ≤ (P*)U(a, b) ∧ (Q)*)U(c, d).

    Thus (PQ)* ⊂ P* ∪ Q*. So (PQ)* = P* ∪ Q*.

    (f) (⇒) : Suppose R = R* and let c, d, eS. Then

    image

    Similarly, we have that

    RU(ec, ed) ≥ RU(c, d).

    By the similar arguments, we have that

    RL(ce, de) ≥ RL(c, d) and RU(ce, de) ≥ RU(c, d).

    (?) : Suppose R is interval-valued fuzzy left and right compatible. Let c, dS. Then

    image

    Similarly, we have that

    (R*)U(c, d) ≤ RU(c, d)

    Thus R* ⊂ R. So R* = R. This completes the proof.

    Proposition 3.27 If R is an IVFR on a semigroup S such that is interval-valued fuzzy left and right compatible, then so is R. Proof. Let a, b, cS and let n ≥ 1. Then

    image

    Similarly, we have that

    (Rn)U(a, b) ≤ (Rn)U(ac, bc).

    By the similar arguments, we have that

    (Rn)L(a, b) ≤ (Rn)U(ca, cb)

    and

    (Rn)U(a, b) ≤ (Rn)U(ca, cb).

    So Rn is interval-valued fuzzy left and right compatible for each n ≥ 1. Hence R is interval-valued fuzzy left and right compatible.

    Let R ∈ IVR(S)and let {}α∈Г be the family of all IVCs on a semigroup S containing R. Then the IVFR

    image

    defined by

    image

    is clearly the least IVC on S. In this case,

    image

    is called the IVC on S generated by R.

    Theorem 3.28 If R is an IVFR on a semigroup S, then

    image

    Proof. By Definition 2.8, (R*)e ∈ IVE(S) such that R* ⊂ (R*)e. Then, by Proposition 3.26(a), R ⊂ (R*)e. Also, by (a) and (b) of Proposition 3.26 R* ∪ (R*)?1 ∪ △ = (RR?1 ∪ △)*. Thus, by Proposition 3.26(f) and Result 2.E, R* ∪ (R*)?1 ∪ △ is left and right compatible. So, by Proposition 3.27, (R*)e = [R* ∪ (R*)?1 ∪ △] is left and right compatible. Hence, by Proposition 3.8, (R*)e ∈ IVC(S) . Now suppose Q ∈ IVC(S) such that RQ. Then, by (c) and (d) of Proposition 3.26, R* ⊂ Q* = Q. Thus (R*)eQ. So

    image

    This completes the proof.

    4. Homomorphisms

    Let f : ST be a semigroup homomorphism. Then it is well-known that the relation

    Ker(f) = {(a, b) ∈ S × S : f(a) = f(b)}

    is a congruence on S.

    The following is the immediate result of Theorem 3.9.

    Proposition 4.1 Let f : ST be a semigroup homomorphism. Then R = [ΧKer(f), ΧKer(f)] ∈ IVC(S).

    In this case, R is called the interval-valued fuzzy kernel of f and denoted by IVK(f). In fact, for any a, bS,

    image

    Theorem 4.2 (a) Let R be an interval-valued fuzzy congruence on a semigroup S. Then the mapping π : SS/R defined same as in Result 2.D(d) is an epimorphism.

    (b) If f : ST is a semigroup homomorphism, then there is a monomorphism g : S/IVK(f) → T such that the diagram

    image

    commutes, where [IVK(f)]# denotes the natural mapping. Proof. (a) Let a, bS. Then, by the definition of R# and Theorem 3.21,

    π(ab) = Rab = Ra * Rb = π(a) * π(b).

    So π is a homomorphism. By Result 2.D(d), π is surjective. Hence π is an epimorphism.

    (b) We define g : S/IVK(f) → T by g([IFK(f)]a) = f(a) for each aS. Suppose [IVK(f)]a = IVK(f)]b for any a, bS. Since IVK(f)(a, b) = [1, 1], i.e. ΧIVK(f)(a, b) = 1. Thus (a, b) ∈ Ker(f). So (a, b) ∈ Ker(f). So g([IVK(f)]a) = f(a) = f(b) = g([IVK(f)]b). Hence g is well-defined.

    Suppose f(a) = f(b). Then IVK(f)(a, b) = [1, 1]. Thus, by Result 2.D(a), [IVK(f)]a = IVK(f)]b. So g is injective. Now let a, bS, Then

    image

    So g is a homomorphism. Let aS. Then g([IVK(f)]#(a)) = g([IVK(f)]a) = f(a). So g ○ [IVK(f)]# = f. This completes the proof.

    Theorem 4.3 Let R and Q be IVCs on a semigroup such that RQ. Then there exists a unique semigroup S homomorphism g : S/RS/Q such that the diagram

    image

    commutes and (S/R)/IVK(g) is isomorphic to S/R, where R# and Q# denote the natural mappings, respectively. Proof. Define g : S/RS/Q by g(Ra) = Qa for each aS. Suppose Ra = Rb. Then, by Result 2.D(a), R(a, b) = [1, 1]. Since RQ,

    1 = RL(a, b) ≤ QL(a, b) and 1 = RU(a, b) ≤ QU(a, b).

    Then Q(a, b) = [1, 1]. Thus Qa = Qb, i.e., g(Ra) = g(Rb). So g is well- defined.

    Let a, bS. Then

    g(Ra * Rb) = g(Rab) = Qab = Qa * Qb = g(Ra) * g(Rb).

    So g is a semigroup homomorphism. The remainders of the proofs are easy. This completes the proof.

    5. Conclusion

    Hur et al. [11] studied interval-valued fuzzy relations in the sense of a lattice. Cheong and Hur [13], Hur et al. [14], and Kim et al. [15] investigated interval-valued fuzzy ideals/(generalized) bi-ideas and quasi-ideals in a semigroup, respectively.

    In this paper, we mainly study interval-valued fuzzy congruences on a semigroup. In particular, we obtain the result that

    image

    for the IVC

    image

    on S generated by R for each IVFR R on a semigroup S (See Theorem 3.28). Finally, for any IVCs R and Q on a semigroup S such that RQ, there exists a unique semigroup homomorphism g : S/KS/Q such that (S?R)/IVK(g) is isomorphic to S/Q (See Theorem 4.3).

    In the future, we will investigate interval-valued fuzzy congruences on a semiring.

      >  Conflict of Interest

    No potential conflict of interest relevant to this article was reported.

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