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, R_{1}, R_{2}, R_{3}, Q_{1}, Q_{2} ∈ 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 R^{n} = 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 R_{1} : 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),
[λ_{21} , μ_{21}] and [λ_{31} , μ_{31}] 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 R_{1} is an interval-valued fuzzy left compatible relation on S.
(b) Let R_{2} : 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 R_{2} is an interval-valued fuzzy right compatible relation on S.
(c) Let R_{3} : 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 R_{3} 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 R_{1} : 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^{？1}a = a and a^{？1} = a^{？1}aa^{？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^{？1}a = a and a^{？1} = a^{？1}aa^{？1}. Moreover, it is clear that (Ra)^{？1} = Ra^{？1}. Then (Ra)^{？1} * Ra * (Ra)^{？1} = Ra^{？1} * Ra * Ra^{？1} = Ra^{？1}aa^{？1} = Ra^{？1} and Ra * (Ra)^{？1} * Ra = Ra * Ra^{？1} * Ra = Raa^{？1}a = 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 S^{1} 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 R^{n} 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.