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 R_e 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 R ⊂ Q, there exists a unique semigroup homomorphism g : S/R → S/G (Theorem 4.3).

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 = [M^L, M^U], where M^L and M^U 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, 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]([7, 14]).

Definition 2.1 [4, 10, 14]. 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 a. In particular,

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 [14]. Let A, B ∈ D(I)^X and let ｛A_α｝_α∈Г ⊂ D(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].

(v) A ∩ B = [AL ？ BL, AU ？ BU].

Result 2.A [14, Theorem 1]. Let A, B, C ∈ D(I)^X and let {A_α}_α∈Г ⊂ D(I)^X. Then

(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.

(h) (Ac)c = A.

Definition 2.3 [8]. Let X be a set. Then a mapping R = [R^L , R^U] : X∏X → D(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, y ∈ X.

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

and

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).

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 x ∈ X,

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

(3) it is interval-valued fuzzy transitive, i.e., R ○ R ⊂ R.

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 R^L and R^U 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 [Χ_R, Χ_R] ∈ IVE(X).

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

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

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

Then clearly Ra ∈ D(I)^X. In this case, Ra is called the interval-valued fuzzy equivalence class of R containing a ∈ X. The set {Ra : a ∈ X} 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, b ∈ X.

(b) R(a, b) = [0, 0] if and only if Ra ∩ Rb =

for any a, b ∈ X.

(d) There exits the surjection π : X → X/R defined by π(x) = Rx for each x ∈ X.

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

It is easily seen that R^e 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 :

,where Rⁿ = R ○ R ○ … ○ R(n factors).

Definition 2.10 [11]. We define two mappings △, ▽ : X → D(I) as follows : For any x, y ∈ X,

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, B ∈ D(I)^X. Then the interval-valued fuzzy product of A and B, A ○ B is defined as follows : For each a ∈ X,

and

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

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

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 x ∈ G,

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, y ∈ G.

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 A ∈ D(I)^G. If B ∈ IVNG(G), then A ○ B = B ○ A.

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

Ax : G → D(I)

and

xA : G → D(I)

as follows, respectively : For each g ∈ G,

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.

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, b ∈ S,

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

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

(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, z ∈ G,

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

(2) interval-valued fuzzy right compatible if for any x, y, z ∈ G,

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

(3) interval-valued fuzzy compatible if for any x, y, z, t ∈ G,

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 :

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

where [λ_ij , μ_ij] ∈ D(I) such that [λ_1i , μ_1i](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 × S → D(I) be the mapping defined as the matrix :

where [λ_ij , μ_ij] ∈ D(I) such that [λ_ij , μ_ij](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 × S → D(I) be the mapping defined as the matrix :

where [λ_ij , μ_ij] ∈ 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 [Χ_R, Χ_R] is interval-valued fuzzy left compatible.

Proof. (⇒) : Suppose R is left compatible. Let a, b, x ∈ S.

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

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

(？) : Suppose [Χ_R, Χ_R] is interval-valued fuzzy compatible. Let a, b, x ∈ S and (a, b) ∈ R. Then, by hypothesis, Χ_R(xa, xb) ≥ Χ_R(a, b) = 1. Thus Χ_R(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 [Χ_R, Χ_R] 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 × S → D(I) be the mapping defined as the matrix :

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, z ∈ S. 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, t ∈ S. Then

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 [Χ_R, Χ_R] ∈ 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, t ∈ G. 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, y ∈ G.

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

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

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

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 Q○P is also an interval-valued fuzzy compatible relation on S.

Proof. Let a, b, x ∈ S. Then

By the similar arguments, we have that

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

Thus

and

So Q○P is interval-valued fuzzy right compatible. Similarly, we can see that Q○P is interval-valued fuzzy left compatible. Hence Q○P 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, b ∈ S. Then

Similarly, we have that

(Q○P)U(a, b) = (P○Q)U(a, b).

Hence Q○P = P○Q

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

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

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

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 P○Q is interval-valued fuzzy left and right compatible.

Hence P○Q ∈ IVC(S).

Let R be an IVC on a groupoid S and let a ∈ S. Then Ra ∈ D(I)^S is called an interval-valued fuzzy congruence class of R containing a ∈ S 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 Ra ○ Rb ⊂ Rab, for any a, b ∈ S.

Proof. Let x ∈ S. If x is not expressible as x = yz, then clearly (Ra ○ Rb)(x) = [0, 0]. Thus Ra ○ Rb ⊂ Rab. Suppose x is expressible as x = yz. Then

Similarly, we have that

(Ra ○ Rb)^U(x) ≤ (Rab)^U(x).

Thus Ra ○ Rb ⊂ Rab. 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 A_R : G → D(I) as follows : For each a ∈ G,

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

Then A_R = R_e ∈ IVNG(G).

Proof. From the definition of A_R, it is obvious that A_R ∈ D(I)^G. Let a, b ∈ G. Then

Similarly, we have that

On the other hand,

Moreover,

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

Finally,

Hence A_R ∈ 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 Pe ○ Qe = Qe ○ Pe.

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 x ∈ G 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.g ∈ G. 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 x ∈ G such that A = xRe. Let g ∈ G.

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, b ∈ S,

Ra * Rb = Rab.

Then * is well-defined.

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

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

Thus

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 a ∈ S has a unique inverse, i.e., there exists a unique a^？1 ∈ S 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 a ∈ S. Since S is an inverse semigroup, there exists a unique a^？1 ∈ S 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 a ∈ S.

An element a of a semigroup S is said to be regular if a ∈ aSa, i.e., there exists an x ∈ S such that a = axa. The semigroup S is said to be regular if for each a ∈ S, a is a regular element. Corresponding to a regular element a, there exists at least one a ∈ S 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 a ∈ S. Since S is a regular semigroup, there exists an x ∈ S such that a = axa. It is obvious that Rx ∈ S/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 x ∈ G. 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/R → D(I) as follows : For each x ∈ G,

π(Rx) = [(Rx)L(e), Rx)U(e)].

Then π ∈ IVG(G/R).

Proof. From the definition of π, it is clear that π = [π^L, π^U] ∈ D(I)^G/R. Let x, y ∈ G. Then

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 x ∈ G. 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, y ∈ S. Proof. By Corollary 3.21-1, S/R is an inverse semigroup with (Rx)^？1 = Rx^？1 for each x ∈ S. Let x, y ∈ S. Then

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, y ∈ G.

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

Similarly, we have that R^U(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 x ∈ S. Since R is an IVC on S,

R^L(ax, bx) ≥ R^L(a, b) = 1 and R^U(ax, bx) ≥ R^U(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 :

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

and

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, d ∈ S. Then

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* ⊂ (P ∪ Q)* and Q* ⊂ (P ∪ Q)*. Thus P* ∪ Q* ⊂ (P ∪ Q)*. Let c, d ∈ S. Then

Similarly, we have that

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

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

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

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, d ∈ S. Then

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, c ∈ S and let n ≥ 1. Then

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 {Rα}_α∈Г be the family of all IVCs on a semigroup S containing R. Then the IVFR

defined by

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

is called the IVC on S generated by R.

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

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 ∪ △ = (R ∪ R^？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 R ⊂ Q. Then, by (c) and (d) of Proposition 3.26, R* ⊂ Q* = Q. Thus (R*)^e ⊂ Q. So

This completes the proof.

Let f : S → T 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 : S → T be a semigroup homomorphism. Then R = [Χ_Ker(f), Χ_Ker(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, b ∈ S,

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

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

commutes, where [IVK(f)]^# denotes the natural mapping. Proof. (a) Let a, b ∈ S. 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 a ∈ S. Suppose [IVK(f)]a = IVK(f)]b for any a, b ∈ S. Since IVK(f)(a, b) = [1, 1], i.e. Χ_IVK(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, b ∈ S, Then

So g is a homomorphism. Let a ∈ S. 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 R ⊂ Q. Then there exists a unique semigroup S homomorphism g : S/R → S/Q such that the diagram

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

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, b ∈S. 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.

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

for the IVC

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 R ⊂ Q, there exists a unique semigroup homomorphism g : S/K → S/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.

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