IntervalValued Fuzzy Congruences on a Semigroup
 Author: Lee Jeong Gon, Hur Kul, Lim Pyung Ki
 Organization: Lee Jeong Gon; Hur Kul; Lim Pyung Ki
 Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 13, Issue3, p231~244, 25 Sep 2013

ABSTRACT
We introduce the concept of intervalvalued fuzzy congruences on a semigroup
S and we obtain some important results: First, for any intervalvalued fuzzy congruenceR on a groupG , the intervalvalued congruence classR_{e} is an intervalvalued fuzzy normal subgroup ofG . Second, for any intervalvalued fuzzy congruenceR on a groupoidS , we show that a binary operation * anS/R is welldefined and also we obtain some results related to additional conditions forS . Also we improve that for any two intervalvalued fuzzy congruencesR andQ on a semigroupS such thatR ⊂Q , there exists a unique semigroup homomorphismg : S/R → S/G .

KEYWORD
Intervalvalued fuzzy set , Intervalvalued fuzzy (normal) subgroup , Intervalvalued fuzzy congruence

1. Introduction
As a generalization of fuzzy sets introduced by Zadeh [1], Zadeh [2] also suggested the concept of intervalvalued 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 intervalvalued fuzzy sets. Moreover, Mondal and Samanta [5] introduced the concept of intervalvalued fuzzy topology and investigated some of it’s properties. In particular, Roy and Biswas [6] introduced the notion of intervalvalued fuzzy relations and studied some of it’s properties. Recently, Jun et al. [7] investigated strong semiopenness and strong semicontinuity in intervalvalued fuzzy topology. Moreover, Min [8] studied characterizations for intervalvalued fuzzy msemicontinuous mappings, Min and Kim [9,10] investigated intervalvalued fuzzy m*continuity and m*open mappings. Hur et al. [11] studied intervalvalued fuzzy relations in the sense of a lattice theory. Also, Choi et al. [12] introduced the concept of intervalvalued smooth topological spaces and investigated some of it’s properties.
On the other hand, Cheong and Hur [13], and Lee et al. [14] studied intervalvalued fuzzy ideals/(generalized)biideals in a semigroup. In particular, Kim and Hur [15] investigated intervalvalued fuzzy quasiideals in a semigroup. Kang [16], Kang and Hur [17] applied the notion of intervalvalued fuzzy sets to algebra. Jang et al. [18] investigated intervalvalued fuzzy normal subgroups.
In this paper, we introduce the concept of intervalvalued fuzzy congruences on a semigroup
S and we obtain some important results:(i) For any intervalvalued fuzzy congruence
R on a groupG , the intervalvalued congruence classR_{e} is an intervalvalued fuzzy normal subgroup ofG (Proposition 3.11).(ii) For any intervalvalued fuzzy congruence
R on a groupoidS , we show that a binary operation * anS/R is welldefined (Proposition 3.20) and also we obtain some results related to additional conditions forS (Theorem 3.21, Corollaries 3.211, 3.212, and 3.213). Also we improve that for any two intervalvalued fuzzy congruencesR andQ on a semigroupS such thatR ⊂Q , there exists a unique semigroup homomorphismg : S/R → S/G (Theorem 4.3).2. Preliminaries
In this section, we list some concepts and wellknown 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 ofD (I ) are generally denoted by capital lettersM ,N , …, and note thatM = [M^{L}, M^{U} ], whereM^{L} andM^{U} are the lower and the upper end points respectively. Especially, we denoted ,0 = [0, 0],1 = [1, 1], anda = [a, a ] for everya ∈ (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 ), thecomplement ofM , denoted byM^{C} , is defined byM^{C} = 1 ？M = [1 ？M^{U} , 1 ？M^{L} ]([7, 14]).Definition 2.1 [4, 10, 14]. A mappingA :X →D (I ) is called anintervalvalued fuzzy set (IVFS ) inX , denoted byA = [A^{L}, A^{U} ], ifA^{L} ,A^{L} ∈I^{X} such thatA^{L} ≤A^{U} , i.e.,A^{L} (x ) ≤A^{U} (x ) for eachx ∈X , whereA^{L} (x )[respA^{U} (x )] is called thelower [respupper ]end point of x to A. For any [a, b ] ∈D (I ), the intervalvalued fuzzyA inX defined byA (x ) = [A^{L} (x ),A^{U} (x )] = [a, b ] for eachx ∈X is denoted byand if
a =b , then the IVFSis denoted by simply
a . In particular,denote the
intervalvalued fuzzy empty set and theintervalvalued fuzzy whole set inX , respectively.We will denote the set of all IVFSs in
X asD (I )^{X}. It is clear that setA = [A, A ] ∈D (I )^{X} for eachA ∈I^{X} .Definition 2.2 [14]. LetA, 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]. LetA, 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]. LetX be a set. Then a mappingR = [R^{L} ,R^{U} ] :X ∏X →D (I ) is called anintervalvalued fuzzy relation (IVFR ) onX .We will denote the set of all IVFRs on X as IVR(X).
Definition 2.4 [8]. LetR ∈ IVR(X). Then theinverse ofR ,R ^{？1} is defined byR ^{？1}(x,y ) =R (y,x ), for eachx, y ∈X .Definition 2.5 [11]. LetX be a set and letR ,Q ∈ IVR(X). Then the composition ofR andQ ,Q ○R , is defined as follows : For anyx, y ∈X ,and
Result 2.B [11, Proposition 3.4]. LetX be a set and letR ,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 IVFRR on a setX is called anintervalvalued fuzzy equivalence relation (IV FER ) onX if it satisfies the following conditions :(1) it is
intervalvalued fuzzy reflexiv , i.e.,R (x, x ) = [1, 1], for eachx ∈X ,(2) it is
intervalvalued fuzzy symmetric , i.e.,R ^{？1} =R ,(3) it is
intervalvalued 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) IfR is an fuzzy equivalence relation on a setX , then [R, R ] ∈ IVE(X ).(b) If
R ∈ IVE(X ), thenR^{L} andR^{U} are fuzzy equivalence relation onX .(c) Let
R be an ordinary relation on a setX . ThenR is an equivalence relation onX if and only if [Χ_{R}, Χ_{R} ] ∈ IVE(X ).Result 2.C [11, Proposition 3.9]. LetX be a set and letQ ,R ∈ IVE(X ). IfQ ○R =R ○Q , thenR ○Q ∈ IVE(X ).Let
R be an IVFER on a setX and leta ∈X . We define a mappingRa :X →D (I ) as follows : For eacha ∈X ,Ra(x) = R(a, x).
Then clearly
Ra ∈D (I )^{X} . In this case,Ra is called theintervalvalued fuzzy equivalence class ofR containing a ∈X . The set {Ra :a ∈X } is called theintervalvalued fuzzy quotient set of X by R and denoted byX/R .Result 2.D [11, Proposition 3.10]. LetR be an IVFER on a setX . Then the following hold :(
a )Ra =Rb if and only ifR (a, b ) = [1, 1], for anya, b ∈X .(
b )R (a, b ) = [0, 0] if and only ifRa ∩Rb =for any
a, b ∈X .(
d ) There exits the surjectionπ :X →X/R defined byπ (x ) =Rx for eachx ∈X .Definition 2.8 [11]. LetX be a set, letR ∈ IVR(X ) and let {R_{α} }_{α∈Г} be the family of all IVFERs onX containingR . Then ∩_{α∈Г}R_{α} is called the IVFERgenerated by R and denoted byR^{e} .It is easily seen that
R^{e} is the smallest IVFER containingR .Definition 2.9 [11]. LetX be a set and letR ∈ IVR(X ). Then theintervalvalued fuzzy transitive closure ofR , denotedR ^{∞}, 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 anyx, y ∈X ,and
▽(x, y) = [1, 1].
It is clear that △, ▽ ∈ IVE(
X ) andR is an intervalvalued fuzzy reflexive relation onX if and only if △ ⊂R .Result 2.E [11, Proposition 4.7]. IfR is an IVFR on a setX , thenRe = [R ∪ R？1 ∪ △]∞.
Definition 2.11 [17]. Let (X , ·) be a groupoid and letA, B ∈D (I )^{X} . Then the intervalvalued fuzzy product ofA andB ,A ○B is defined as follows : For eacha ∈X ,and
Definition 2.12 [17]. Let (X , ·) be a groupoid and letA ∈D (I )^{X}. ThenA is called an iintervalvalued fuzzy subgroupoid (IVGP ) ofX if for anyx, 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 thatDefinition 2.13 [17]. LetG be a group and letA ∈ IVGP(G ). ThenA is an iintervalvalued fuzzy subgroup (IVG ) ofG if for eachx ∈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]. LetG be a group and letA ∈ IVG(G ). ThenA is said to benormal ifA (xy ) =A (yx ), for anyx, y ∈G .We will denote the set of all intervalvalued 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]. LetG be a group and letA ∈D (I )^{G}. IfB ∈ IVNG(G ), thenA ○B =B ○A .Definition 2.15 [18]. LetG be a group, letA ∈ IVG(G ) and letx ∈G . We define two mappingsAx : 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 theintervalvalued fuzzy right [resp.left ]coset ofG determined byx andA .It is obvious that if
A ∈ IVNG(G ), then the intervalvalued fuzzy left coset coincides with the intervalvalued fuzzy right coset ofA onG . In this case, we will callintervalvalued fuzzy coset instead of intervalvalued fuzzy left coset or intervalvalued fuzzy right coset.3. IntervalValued Fuzzy Congruences
Definition 3.1 [19]. A relationR on a groupoidS is said to be:(1)
left compatible if (a, b ) ∈R implies (xa, xb ) ∈R , for anya, b ∈S ,(2)
right compatible if (a, b ) ∈R implies (ax, bx ) ∈R , for anya, b ∈S ,(3)
compatible if (a, b ) ∈R and (s, d ) ∈R imply (ab, cd ) ∈R , for anya, b, c, d ∈S ,(4) a
left [resp.right ]congruence onS if it is a left[resp. right] compatible equivalence relation.(5) a
congruence onS if it is both a left and a right congruence onS .It is wellknown [19, Proposition I.5.1] that a relation
R on a groupoidS is congruence if and only if it is both a left and a right congruence onS . We will denote the set of all ordinary congruences onS asC (S ).Now we will introduce the concept of intervalvalued fuzzy compatible relation on a groupoid.
Definition 3.2 An IVFRR on a groupoidS is said to be :(1)
intervalvalued fuzzy left compatible if for anyx, y, z ∈G ,RL(x, y) ≤ RL(zx, zy) and RU(x, y) ≤ RU(zx, zy),
(2)
intervalvalued fuzzy right compatible if for anyx, y, z ∈G ,RL(x, y) ≤ RL(xz, yz) and RU(x, y) ≤ RU(xz, yz),
(3)
intervalvalued fuzzy compatible if for anyx, 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 LetS =e, a, b be the groupoid with multiplication table :(
a ) LetR _{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 intervalvalued fuzzy left compatible relation onS .(
b ) LetR _{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 intervalvalued fuzzy right compatible relation onS .(
c ) LetR _{3} :S ×S →D (I ) be the mapping defined as the matrix :where [
λ _{ij} ,μ _{ij}] ∈D (I ) such thatλ11 ∧ 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 intervalvalued fuzzy compatible relation onS .Lemma 3.4 LetR be a relation on a groupoidS . ThenR is left compatible if and only if [Χ_{R}, Χ_{R} ] is intervalvalued fuzzy left compatible.Proof . (⇒) : SupposeR is left compatible. Leta, b, x ∈S .Case(1) Suppose (a, b ) ∈R . ThenΧ_{R} (a, b ) = 1. SinceR is left compatible, (xa, xb ) ∈R , for eachx ∈S . ThusΧ_{R} (xa, xb ) = 1 =Χ_{R} (a, b ).Case(2) Suppose ￢(a, b ) ∈R . Then, for eachx ∈S , it holds thatΧ_{R} (a, b ) = 0 ≤Χ_{R} (xa, xb ). Thus, in either cases, [Χ_{R}, Χ_{R} ].(？) : Suppose [
Χ_{R}, Χ_{R} ] is intervalvalued fuzzy compatible. Leta, b, x ∈S and (a, b ) ∈R . Then, by hypothesis,Χ_{R} (xa, xb ) ≥Χ_{R} (a, b ) = 1. ThusΧ_{R} (xa, xb ) = 1. So (xa, xb ) ∈R . HenceR is left compatible.Lemma 3.5 [The dual of Lemma 3.4]. LetR be a relation on a groupoidS . ThenR is right compatible if and only if [Χ_{R}, Χ_{R} ] is intervalvalued fuzzy right compatible.Definition 3.6 An IVFERR on a groupoidS is called an :(1)
intervalvalued fuzzy left congruence (IVLC ) if it is intervalvalued fuzzy left compatible,(2)
intervalvalued fuzzy right congruence (IVRC ) if it is intervalvalued fuzzy right compatible,(3)
intervalvalued 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 LetS =e, a, b be the groupoid defined in Example 3.3. LetR _{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 thatR ∈ IVC(S ).Proposition 3.8 LetS be a groupoid and letR ∈ IVE(S ). ThenR ∈ IVC(S ) if and only if it is both an IVLC and an IVRC.Proof. (⇒) : Suppose
R ∈ IVC(S ) and letx, y, z ∈S . ThenRL(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 letx, y, z, t ∈S . ThenBy the similar arguments, we have that
RU(x, y) ∧ RU(z, t) ≤ RU(xz, yt)
So
R is intervalvalued fuzzy compatible. HenceR ∈ 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 LetR be a relation on a groupoidS . ThenR ∈ C(S ) if and only if [Χ_{R}, Χ_{R} ] ∈ IVC(S ).For any intervalvalued fuzzy left[resp. right] compatible relation
R , it is obvious that ifG is a group, thenR (x, y ) =R (tx, ty )[resp:R (x, y ) =R (xt, yt )], for anyx, y, t ∈G . Thus we have following result.Lemma 3.10 LetR be an IVC on a groupG . ThenR(xay, xby) = R(xa, xb) = R(ay, by) = R(a, b),
for any
a, b, x, y ∈G .Example 3.11 LetV be the Klein 4group 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 anys, 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 LetR be an IVFR on a groupG . ThenR ∈ IVC(G ) if and only if it is intervalvalued fuzzy left(right) compatible equivalence relation.Lemma 3.13 LetP andQ be intervalvalued fuzzy compatible relations on a groupoidS . ThenQ ○P is also an intervalvalued fuzzy compatible relation onS .Proof. Leta, b, x ∈S . ThenBy 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 intervalvalued fuzzy right compatible. Similarly, we can see thatQ ○P is intervalvalued fuzzy left compatible. HenceQ ○P is intervalvalued fuzzy compatible.Theorem 3.14 LetP andQ be IVC on a groupoidS . Then the following are equivalent :(a) Q○P ∈ IVC(S).
(b) Q○P ∈ IVE(S).
(c) Q○P is intervalvalued fuzzy symmetric.
(d) Q○P = P○Q
Proof. It is obvious that (a ) ⇒ (b ) ⇒ (c ).(
c ) ⇒ (d ) : Suppose the condition (c) holds and leta, b ∈S . ThenSimilarly, 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 ). SinceP andQ are intervalvalued fuzzy compatible, by Lemma 3.13,Q ○P is intervalvalued fuzzy compatible. SoQ ○P ∈ IVC(S ). This completes the proof.Proposition 3.15 LetS be a groupoid and letQ, P ∈ IVC(S ). IfQ ○P =P ○Q , thenP ○Q ∈ IVC(S ).Proof. By Result 2.C, it is clear thatP ○Q ∈ IVE(S ). Letx, y, t ∈S . Then, sinceP andQ are intervalvalued 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 intervalvalued fuzzy left and right compatible.Hence
P ○Q ∈ IVC(S ).Let
R be an IVC on a groupoidS and leta ∈S . ThenRa ∈D (I )^{S} is called anintervalvalued fuzzy congruence class of R containing a ∈S and we will denote the set of all intervalvalued fuzzy congruence classes ofR asS/R .Proposition 3.16 IfR is an IVC on a groupoidS , thenRa ○Rb ⊂Rab , for anya, b ∈S .Proof. Letx ∈S . Ifx is not expressible asx =yz , then clearly (Ra ○Rb )(x ) = [0, 0]. ThusRa ○Rb ⊂Rab . Supposex is expressible asx =yz . ThenSimilarly, we have that
(
Ra ○Rb )^{U} (x ) ≤ (Rab )^{U}(x ).Thus
Ra ○Rb ⊂Rab . This completes the proof.Proposition 3.17 LetG be a group with the identitye and letR ∈ IVC(G ). We define the mappingA_{R} :G →D (I ) as follows : For eacha ∈G ,AR(a) = R(a, e) = Re(a).
Then
A_{R} =R_{e} ∈ IVNG(G ).Proof. From the definition ofA_{R} , it is obvious thatA_{R} ∈D (I )^{G}. Leta, b ∈G . ThenSimilarly, we have that
On the other hand,
Moreover,
So
A_{R} ∈ IVG(G ) such thatA_{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 LetG be a group with the identitye . IfP,Q ∈ IVNG(G ), thenPe ○Qe =Qe ○Pe .Proposition 3.19 LetG be a group with the identitye . IfR ∈ IVC(G ), then any intervalvalued fuzzy congruence classRx ofx ∈G byR is an intervalvalued fuzzy coset ofRe . Conversely, each intervalvalued fuzzy coset ofRe is an intervalvalued fuzzy congruence class byR .Proof. SupposeR ∈ IVC(G ) and letx.g ∈G . ThenRx (g ) =R (x, g ). SinceR is intervalvalued fuzzy left compatible, by Lemma 3.10,R (x, g ) =R (e, x ^{？1}g ). ThusRx(g) = R(e, x？1 g) = Re(？1g) = (xRe)(g).
So
Rx =xRe . HenceRx is an intervalvalued fuzzy coset ofRe .Conversely, let
A be any intervalvalued fuzzy coset ofRe . Then there exists anx ∈G such thatA =xRe . Letg ∈G .Then
A(g) = (xRe)(g) = Re(x？1g) = R(e, x？1 g).
Since
R is intervalvalued fuzzy left compatible,R(e, x？1 g) = R(x, g) = Rx(g).
So
A =Rx . HenceA is an intervalvalued fuzzy congruence class ofx byR .Proposition 3.20 LetR be an IVC on a groupoidS . We define the binary operation * onS/R as follows : For anya, b ∈S ,Ra * Rb = Rab.
Then * is welldefined.
Proof. SupposeRa =Rx andRb =Ry , wherea, 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 . SoRa *Rb =Rx *Ry . Hence * is welldefined.From Proposition 3.20 and the definition of semigroup, we obtain the following result.
Theorem 3.21 LetR be an IVC on a semigroupS . Then (S/R , *) is a semigroup.A semigroup
S is called aninverse semigroup [7] if eacha ∈S has a unique inverse, i.e., there exists a uniquea ^{？1} ∈S such thataa ^{？1}a =a anda ^{？1} =a ^{？1}aa ^{？1}.Corollary 3.211 LetR be an IVC on an inverse semigroupS . Then (S/R , *) is an inverse semigroup.Proof. By Theorem 3.21, (S/R , *) is a semigroup. Leta ∈S . SinceS is an inverse semigroup, there exists a uniquea ^{？1} ∈S such thataa ^{？1}a =a anda ^{？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} andRa * (Ra )^{？1} *Ra =Ra *Ra ^{？1} *Ra =Raa ^{？1}a =Ra .So
Ra ^{？1} is an inverse of Ra for eacha ∈S .An element
a of a semigroupS is said to beregular ifa ∈aSa , i.e., there exists anx ∈S such thata =axa . The semigroupS is said to beregular if for eacha ∈S ,a is a regular element. Corresponding to a regular elementa , there exists at least onea ∈S such thata =aaa and a =aaa . Such ana is called aninverse ofa .Corollary 3.212 LetR be an IVC on a regular semigroupS .Then (
S/R , *) is a regular semigroup.Proof. By Theorem 3.21, (S/R , *) is a semigroup. Leta ∈S . SinceS is a regular semigroup, there exists anx ∈S such thata =axa . It is obvious thatRx ∈S/R . Moreover,Ra *Rx *Ra =Raxa =Ra . SoRa is an regular element ofS/R . HenceS/R is a regular semigroup.Corollary 3.213 LetR be an IVC on a groupG . Then (G/R , *) is a group.Proof. By Theorem 3.21, (G/R , *) is a semigroup. Letx ∈G . ThenRx * Re = Rxe = Rx = Rex = Re * Rx.
Thus
Re is the identity inG/R with respect to *. Moreover,Rx * Rx？1 = Rxx？1 = Re = Rx？1x = Rx？1 * Rx.
So
Rx ^{？1} is the inverse ofRx with respect to *. HenceG/R is a group.Proposition 3.22 LetG be a group and letR ∈ IVC(G ). We define the mappingπ :G/R →D (I ) as follows : For eachx ∈G ,π(Rx) = [(Rx)L(e), Rx)U(e)].
Then
π ∈ IVG(G/R ).Proof. From the definition ofπ , it is clear thatπ = [π ^{L},π ^{U}] ∈D (I )^{G/R}. Letx, y ∈G . ThenSimilarly, we have that
πU(Rx * Ry) ≥ πU(Rx) ∧ πU(Ry).
By the process of the proof of Corollary 3.211, (
Rx )^{？1} =Rx ^{？1}. Thusπ((Rx)？1) = π(Rx？1) = R(x？1, e) = R(e, x) = π(Rx).
So
π ((Rx )^{？1}) =π (Rx ) for eachx ∈G . Henceπ ∈ IVG(G/R ).Proposition 3.23 IfR is an IVC on an inverse semigroupS . ThenR (x ^{？1},y ^{？1}) =R (x, y ) for anyx, y ∈S .Proof. By Corollary 3.211,S/R is an inverse semigroup with (Rx )^{？1} =Rx ^{？1} for eachx ∈S . Letx, y ∈S . ThenHence
R (x ^{？1},y ^{？1}) =R (x, y ).The following is the immediate result of Proposition 3.22
Corollary 3.23 LetR be an IVC on a groupG . ThenR(x？1, y？1) = R(x, y)
for any
x, y ∈G .Proposition 3.24 LetR be an IVC on a semigroupS . ThenR？1([1, 1]) = {(a, b) ∈ S × S : R(a, b) = [1; 1]}
is a congruence on
S .Proof. It is clear thatR ^{？1}([1, 1]) is reflexive and symmetric. Let (a, b ), (b, c ) ∈R ^{？1}([1, 1]). ThenR (a, b ) =R (b, c ) = [1, 1]. ThusSimilarly, we have that
R^{U} (a, c ) ≥ 1. SoR (a, c ) = [1, 1], i.e., (a, c ) ∈R ^{？1}([1, 1]). HenceR ^{？1}([1, 1]) is an equivalence relation onS .Now let (
a, b ) ∈R ^{？1}([1, 1]) and letx ∈S . SinceR is an IVC onS ,R^{L} (ax, bx ) ≥R^{L} (a, b ) = 1 andR^{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]). SoR ^{？1}([1, 1]) is compatible. HenceR ^{？1}([1, 1]) is a congruence onS .Let
S be a semigroup. ThenS ^{1} denotes the monoid defined as follows :Definition 3.25 LetS be a semigroup and letR ∈ IVR(S ). Then we define a mappingR * :S ×S →D (I ) as follows : For anyc, d ∈S ,and
It is obvious that
R * ∈ IVR(S ).Proposition 3.26 LetS be a semigroup and letR, 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 thatR * ⊂ (R *)*. Letc, d ∈S . ThenBy 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 )* andQ * ⊂ (P ∪Q )*. ThusP * ∪Q * ⊂ (P ∪Q )*. Letc, d ∈S . ThenSimilarly, 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 ) (⇒) : SupposeR =R * and letc, d, e ∈S . ThenSimilarly, 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 intervalvalued fuzzy left and right compatible. Letc, d ∈S . ThenSimilarly, we have that
(R*)U(c, d) ≤ RU(c, d)
Thus
R * ⊂R . SoR * =R . This completes the proof.Proposition 3.27 IfR is an IVFR on a semigroupS such that is intervalvalued fuzzy left and right compatible, then so isR ^{∞}.Proof. Leta, b, c ∈S and letn ≥ 1. ThenSimilarly, 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 intervalvalued fuzzy left and right compatible for eachn ≥ 1. HenceR ^{∞} is intervalvalued fuzzy left and right compatible.Let
R ∈ IVR(S )and let {Rα }_{α∈Г} be the family of all IVCs on a semigroupS containingR . Then the IVFRdefined by
is clearly the least IVC on
S . In this case,is called the IVC on
S generated by R .Theorem 3.28 IfR is an IVFR on a semigroupS , thenProof. By Definition 2.8, (R *)^{e} ∈ IVE(S ) such thatR * ⊂ (R *)^{e} . Then, by Proposition 3.26(a ),R ⊂ (R *)^{e} . Also, by (a ) and (b ) of Proposition 3.26R * ∪ (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 supposeQ ∈ IVC(S ) such thatR ⊂Q . Then, by (c ) and (d ) of Proposition 3.26,R * ⊂Q * =Q . Thus (R *)^{e} ⊂Q . SoThis completes the proof.
4. Homomorphisms
Let
f :S →T be a semigroup homomorphism. Then it is wellknown that the relationKer(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 Letf :S →T be a semigroup homomorphism. ThenR = [Χ _{Ker(f)},Χ _{Ker(f)}] ∈ IVC(S ).In this case,
R is called theintervalvalued fuzzy kernel off and denoted by IVK(f ). In fact, for anya, b ∈S ,Theorem 4.2 (a ) LetR be an intervalvalued fuzzy congruence on a semigroupS . Then the mappingπ :S →S/R defined same as in Result 2.D(d ) is an epimorphism.(
b ) Iff :S →T is a semigroup homomorphism, then there is a monomorphismg :S /IVK(f ) →T such that the diagramcommutes, where [IVK(
f )]^{#} denotes the natural mapping.Proof. (a ) Leta, b ∈S . Then, by the definition ofR ^{#} 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. Henceπ is an epimorphism.(
b ) We defineg :S /IVK(f ) →T byg ([IFK(f )]a ) =f (a ) for eacha ∈S . Suppose [IVK(f )]a = IVK(f )]b for anya, 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 ). Sog ([IVK(f )]a ) =f (a ) =f (b ) =g ([IVK(f )]b ). Henceg is welldefined.Suppose
f (a ) =f (b ). Then IVK(f )(a, b ) = [1, 1]. Thus, by Result 2.D(a ), [IVK(f )]a = IVK(f )]b . Sog is injective. Now leta, b ∈S , ThenSo
g is a homomorphism. Leta ∈S . Theng ([IVK(f )]^{#}(a )) =g ([IVK(f )]a ) =f (a ). Sog ○ [IVK(f )]^{#} =f . This completes the proof.Theorem 4.3 LetR andQ be IVCs on a semigroup such thatR ⊂Q . Then there exists a unique semigroupS homomorphismg :S/R →S/Q such that the diagramcommutes and (
S/R )/IVK(g ) is isomorphic toS/R , whereR ^{#} andQ ^{#} denote the natural mappings, respectively.Proof. Defineg :S/R →S/Q byg (Ra ) =Qa for eacha ∈S . SupposeRa =Rb . Then, by Result 2.D(a ),R (a, b ) = [1, 1]. SinceR ⊂Q ,1 = RL(a, b) ≤ QL(a, b) and 1 = RU(a, b) ≤ QU(a, b).
Then
Q (a, b ) = [1, 1]. ThusQa =Qb , i.e.,g (Ra ) =g (Rb ). Sog is well defined.Let
a, b ∈S . Theng(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 intervalvalued fuzzy relations in the sense of a lattice. Cheong and Hur [13], Hur et al. [14], and Kim et al. [15] investigated intervalvalued fuzzy ideals/(generalized) biideas and quasiideals in a semigroup, respectively.
In this paper, we mainly study intervalvalued fuzzy congruences on a semigroup. In particular, we obtain the result that
for the IVC
on
S generated byR for each IVFRR on a semigroupS (See Theorem 3.28). Finally, for any IVCsR andQ on a semigroupS such thatR ⊂Q , there exists a unique semigroup homomorphismg :S/K →S/Q such that (S ?R )/IVK(g ) is isomorphic toS/Q (See Theorem 4.3).In the future, we will investigate intervalvalued fuzzy congruences on a semiring.
> Conflict of Interest
No potential conflict of interest relevant to this article was reported.