Interval-Valued 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
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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 congruenceR on a groupG , the interval-valued congruence classRe is an interval-valued fuzzy normal subgroup ofG . Second, for any interval-valued fuzzy congruenceR on a groupoidS , we show that a binary operation * anS/R is well-defined and also we obtain some results related to additional conditions forS . Also we improve that for any two interval-valued fuzzy congruencesR andQ on a semigroupS such thatR ⊂Q , there exists a unique semigroup homomorphismg : S/R → S/G .
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KEYWORD
Interval-valued fuzzy set , Interval-valued fuzzy (normal) subgroup , Interval-valued fuzzy congruence
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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 groupG , the interval-valued congruence classRe is an interval-valued fuzzy normal subgroup ofG (Proposition 3.11).(ii) For any interval-valued fuzzy congruence
R on a groupoidS , we show that a binary operation * anS/R is well-defined (Proposition 3.20) and also we obtain some results related to additional conditions forS (Theorem 3.21, Corollaries 3.21-1, 3.21-2, and 3.21-3). 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).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 ofD (I ) are generally denoted by capital lettersM ,N , …, and note thatM = [ML, MU ], whereML andMU 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 ⇔ML =NL ,MU =NU ),(ii) (∀
M, N ∈D (I )) (M =N ≤ML ≤NL ,MU ≤NU ),For every
M ∈D (I ), thecomplement ofM , denoted byMC , is defined byMC = 1 ?M = [1 ?MU , 1 ?ML ]([7, 14]).Definition 2.1 [4, 10, 14]. A mappingA :X →D (I ) is called aninterval-valued fuzzy set (IVFS ) inX , denoted byA = [AL, AU ], ifAL ,AL ∈IX such thatAL ≤AU , i.e.,AL (x ) ≤AU (x ) for eachx ∈X , whereAL (x )[respAU (x )] is called thelower [respupper ]end point of x to A. For any [a, b ] ∈D (I ), the interval-valued fuzzyA inX defined byA (x ) = [AL (x ),AU (x )] = [a, b ] for eachx ∈X is denoted byand if
a =b , then the IVFSis denoted by simply
a . In particular,denote the
interval-valued fuzzy empty set and theinterval-valued 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 ∈IX .Definition 2.2 [14]. LetA, B ∈D (I )X and let {Aα }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 {Aα }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 = [RL ,RU ] :X ∏X →D (I ) is called aninterval-valued 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 aninterval-valued fuzzy equivalence relation (IV FER ) onX if it satisfies the following conditions :(1) it is
interval-valued fuzzy reflexiv , i.e.,R (x, x ) = [1, 1], for eachx ∈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) IfR is an fuzzy equivalence relation on a setX , then [R, R ] ∈ IVE(X ).(b) If
R ∈ IVE(X ), thenRL andRU 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 [ΧR, ΧR ] ∈ 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 theinterval-valued fuzzy equivalence class ofR containing a ∈X . The set {Ra :a ∈X } is called theinterval-valued 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 {Rα }α∈Г be the family of all IVFERs onX containingR . Then ∩α∈ГRα is called the IVFERgenerated by R and denoted byRe .It is easily seen that
Re is the smallest IVFER containingR .Definition 2.9 [11]. LetX be a set and letR ∈ IVR(X ). Then theinterval-valued fuzzy transitive closure ofR , denotedR ∞, 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 anyx, y ∈X ,and
▽(x, y) = [1, 1].
It is clear that △, ▽ ∈ IVE(
X ) andR is an interval-valued 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 interval-valued 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 iinterval-valued 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 iinterval-valued 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 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]. 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 theinterval-valued fuzzy right [resp.left ]coset ofG determined byx andA .It is obvious that if
A ∈ IVNG(G ), then the interval-valued fuzzy left coset coincides with the interval-valued fuzzy right coset ofA onG . In this case, we will callinterval-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 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 well-known [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 interval-valued fuzzy compatible relation on a groupoid.
Definition 3.2 An IVFRR on a groupoidS is said to be :(1)
interval-valued fuzzy left compatible if for anyx, y, z ∈G ,RL(x, y) ≤ RL(zx, zy) and RU(x, y) ≤ RU(zx, zy),
(2)
interval-valued fuzzy right compatible if for anyx, y, z ∈G ,RL(x, y) ≤ RL(xz, yz) and RU(x, y) ≤ RU(xz, yz),
(3)
interval-valued 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 interval-valued 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 interval-valued 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 interval-valued fuzzy compatible relation onS .Lemma 3.4 LetR be a relation on a groupoidS . ThenR is left compatible if and only if [ΧR, ΧR ] is interval-valued fuzzy left compatible.Proof . (⇒) : SupposeR is left compatible. Leta, b, x ∈S .Case(1) Suppose (a, b ) ∈R . ThenΧR (a, b ) = 1. SinceR is left compatible, (xa, xb ) ∈R , for eachx ∈S . ThusΧR (xa, xb ) = 1 =ΧR (a, b ).Case(2) Suppose ¬(a, b ) ∈R . Then, for eachx ∈S , it holds thatΧR (a, b ) = 0 ≤ΧR (xa, xb ). Thus, in either cases, [ΧR, ΧR ].(?) : Suppose [
ΧR, ΧR ] is interval-valued fuzzy compatible. Leta, b, x ∈S and (a, b ) ∈R . Then, by hypothesis,ΧR (xa, xb ) ≥ΧR (a, b ) = 1. ThusΧR (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 [ΧR, ΧR ] is interval-valued fuzzy right compatible.Definition 3.6 An IVFERR on a groupoidS 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 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 interval-valued 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 [ΧR, ΧR ] ∈ IVC(S ).For any interval-valued 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 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 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 interval-valued fuzzy left(right) compatible equivalence relation.Lemma 3.13 LetP andQ be interval-valued fuzzy compatible relations on a groupoidS . ThenQ ○P is also an interval-valued 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 interval-valued fuzzy right compatible. Similarly, we can see thatQ ○P is interval-valued fuzzy left compatible. HenceQ ○P is interval-valued 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 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 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 interval-valued fuzzy compatible, by Lemma 3.13,Q ○P is interval-valued 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 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 groupoidS and leta ∈S . ThenRa ∈D (I )S is called aninterval-valued fuzzy congruence class of R containing a ∈S and we will denote the set of all interval-valued 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 mappingAR :G →D (I ) as follows : For eacha ∈G ,AR(a) = R(a, e) = Re(a).
Then
AR =Re ∈ IVNG(G ).Proof. From the definition ofAR , it is obvious thatAR ∈D (I )G . Leta, b ∈G . ThenSimilarly, we have that
On the other hand,
Moreover,
So
AR ∈ IVG(G ) such thatAR (e ) = [1, 1].Finally,
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 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 interval-valued fuzzy congruence classRx ofx ∈G byR is an interval-valued fuzzy coset ofRe . Conversely, each interval-valued fuzzy coset ofRe is an interval-valued fuzzy congruence class byR .Proof. SupposeR ∈ IVC(G ) and letx.g ∈G . ThenRx (g ) =R (x, g ). SinceR is interval-valued fuzzy left compatible, by Lemma 3.10,R (x, g ) =R (e, x ?1g ). ThusRx(g) = R(e, x?1 g) = Re(?1g) = (xRe)(g).
So
Rx =xRe . HenceRx is an interval-valued fuzzy coset ofRe .Conversely, let
A be any interval-valued 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 interval-valued fuzzy left compatible,R(e, x?1 g) = R(x, g) = Rx(g).
So
A =Rx . HenceA is an interval-valued 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 well-defined.
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 well-defined.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 ?1a =a anda ?1 =a ?1aa ?1.Corollary 3.21-1 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 ?1a =a anda ?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 andRa * (Ra )?1 *Ra =Ra *Ra ?1 *Ra =Raa ?1a =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 3.21-2 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 3.21-3 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 3.21-1, (
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 3.21-1,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
RU (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 ,RL (ax, bx ) ≥RL (a, b ) = 1 andRU (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]). 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 interval-valued 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 interval-valued 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
Rn is interval-valued fuzzy left and right compatible for eachn ≥ 1. HenceR ∞ 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 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.
Let
f :S →T be a semigroup homomorphism. Then it is well-known 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 theinterval-valued fuzzy kernel off and denoted by IVK(f ). In fact, for anya, b ∈S ,Theorem 4.2 (a ) LetR be an interval-valued 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 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 . 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.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 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 interval-valued fuzzy congruences on a semiring.
No potential conflict of interest relevant to this article was reported.