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Interval-Valued Fuzzy Congruences on a Semigroup
Interval-Valued Fuzzy Congruences on a Semigroup
International Journal of Fuzzy Logic and Intelligent Systems. 2013. Sep, 13(3): 231-244
Copyright ©2013, Korean Institute of Intelligent Systems
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • Received : January 02, 2013
  • Accepted : September 10, 2013
  • Published : September 25, 2013
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Jeong Gon Lee
Kul Hur
kulhur@wonkwang.ac.kr
Pyung Ki Lim

Abstract
We introduce the concept of interval-valued fuzzy congruences on a semigroup S and we obtain some important results: First, for any interval-valued fuzzy congruence R on a group G , the interval-valued congruence class Re is an interval-valued fuzzy normal subgroup of G . Second, for any interval-valued fuzzy congruence R on a groupoid S , we show that a binary operation * an S/R is well-defined and also we obtain some results related to additional conditions for S . Also we improve that for any two interval-valued fuzzy congruences R and Q on a semigroup S such that R Q , there exists a unique semigroup homomorphism g : S/R → S/G .
Keywords
1. Introduction
As a generalization of fuzzy sets introduced by Zadeh [1] , Zadeh [2] also suggested the concept of interval-valued fuzzy sets. After that time, Biswas [3] applied it to group theory, and Gorzalczany [4] introduced a method of inference in approximate reasoning by using interval-valued fuzzy sets. Moreover, Mondal and Samanta [5] introduced the concept of interval-valued fuzzy topology and investigated some of it’s properties. In particular, Roy and Biswas [6] introduced the notion of interval-valued fuzzy relations and studied some of it’s properties. Recently, Jun et al. [7] investigated strong semi-openness and strong semicontinuity in interval-valued fuzzy topology. Moreover, Min [8] studied characterizations for interval-valued fuzzy m-semicontinuous mappings, Min and Kim [9 , 10] investigated intervalvalued fuzzy m*-continuity and m*-open mappings. Hur et al. [11] studied interval-valued fuzzy relations in the sense of a lattice theory. Also, Choi et al. [12] introduced the concept of interval-valued smooth topological spaces and investigated some of it’s properties.
On the other hand, Cheong and Hur [13] , and Lee et al. [14] studied interval-valued fuzzy ideals/(generalized)bi-ideals in a semigroup. In particular, Kim and Hur [15] investigated interval-valued fuzzy quasi-ideals in a semigroup. Kang [16] , Kang and Hur [17] applied the notion of interval-valued fuzzy sets to algebra. Jang et al. [18] investigated interval-valued fuzzy normal subgroups.
In this paper, we introduce the concept of interval-valued fuzzy congruences on a semigroup S and we obtain some important results:
(i) For any interval-valued fuzzy congruence R on a group G , the interval-valued congruence class Re is an interval-valued fuzzy normal subgroup of G (Proposition 3.11).
(ii) For any interval-valued fuzzy congruence R on a groupoid S , we show that a binary operation * an S/R is well-defined (Proposition 3.20) and also we obtain some results related to additional conditions for S (Theorem 3.21, Corollaries 3.21-1, 3.21-2, and 3.21-3). Also we improve that for any two intervalvalued fuzzy congruences R and Q on a semigroup S such that R Q , there exists a unique semigroup homomorphism g : S/R → S/G (Theorem 4.3).
2. Preliminaries
In this section, we list some concepts and well-known results which are needed in later sections.
Let D ( I ) be the set of all closed subintervals of the unit interval [0, 1]. The elements of D ( I ) are generally denoted by capital letters M , N , …, and note that M = [ ML, MU ], where ML and MU are the lower and the upper end points respectively. Especially, we denoted , 0 = [0, 0], 1 = [1, 1], and a = [ a, a ] for every a ∈ (0, 1), We also note that
(i) (∀ M, 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 ), the complement of M , denoted by MC , is defined by MC = 1 – M = [1 – MU , 1 – ML ]( [7 , 14] ).
Definition 2.1 [4 , 10 , 14] . A mapping A : X D ( I ) is called an interval-valued fuzzy set ( IVFS ) in X , denoted by A = [ AL, AU ], if AL , AL IX such that AL AU , i.e., AL ( x ) ≤ AU ( x ) for each x X , where AL ( x )[resp AU ( x )] is called the lower [resp upper ] end point of x to A. For any [ a, b ] ∈ D ( I ), the interval-valued fuzzy A in X defined by A ( x ) = [ AL ( x ), AU ( x )] = [ a, b ] for each x X is denoted by
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and if a = b , then the IVFS
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is denoted by simply ã . In particular,
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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 IX .
Definition 2.2 [14] . Let A, B D ( I ) X and let { Aα α∈Г D ( I ) X . Then
  • (i)A⊂BiffAL≤BLandAU≤BU.
  • (ii)A=BiffA⊂BandB⊂A.
  • (iii)AC= [1 –AU, 1 –AL].
  • (iv)A∪B= [AL˅BL,AU˅BU].
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  • (v)A∩B= [AL˄BL,AU˄BU].
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Result 2.A [ 14 , Theorem 1]. Let A, B, C D ( I ) X and let { Aα } α∈Г D ( I ) X . Then
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  • (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.
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  • (h) (Ac)c=A.
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Definition 2.3 [8] . Let X be a set. Then a mapping R = [ RL , RU ] : X X D ( I ) is called an interval-valued fuzzy relation ( IVFR ) on X .
  • We will denote the set of all IVFRs onXas 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 ,
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and
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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) IfR1⊂R2andQ1⊂Q2, thenR1○Q1⊂R2○Q2.
  • In particular, ifQ1⊂Q2, thenR1○Q1⊂R1○Q2.
  • (c)R1(R2∪R3) = (R1○R2) ∪ (R1○R3),
  • R1(R2∩R3) = (R1○R2) ∩ (R1○R3).
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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 RL and RU are fuzzy equivalence relation on X .
(c) Let R be an ordinary relation on a set X . Then R is an equivalence relation on X if and only if [ ΧR, ΧR ] ∈ IVE( X ).
Result 2.C [ 11 , Proposition 3.9]. Let X be a set and let Q , R ∈ IVE( X ). If 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 =
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for any a, b X .
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( 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 { Rα } α∈Г be the family of all IVFERs on X containing R . Then ∩ α∈Г Rα is called the IVFER generated by R and denoted by Re .
It is easily seen that Re is the smallest IVFER containing R .
Definition 2.9 [11] . Let X be a set and let R ∈ IVR( X ). Then the interval-valued fuzzy transitive closure of R , denoted R , is defined as followings :
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,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 ,
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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 ,
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and
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Definition 2.12 [17] . Let ( X , ·) be a groupoid and let A D ( I ) X . Then A is called an i interval-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
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Definition 2.13 [17] . Let G be a group and let A ∈ IVGP( G ). Then A is an i interval-valued fuzzy subgroup ( IVG ) of G if for each x G ,
  • A(x–1) ≥A(x),
i.e.,
  • AL(x–1) ≥AL(x) andAU(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) andxA(g) =A(x–1g).
Then Ax [resp: xA ] is called the interval-valued fuzzy right [resp. left ] coset of G determined by x and A .
It is obvious that if A ∈ IVNG( G ), then the interval-valued fuzzy left coset coincides with the interval-valued fuzzy right coset of A on G . In this case, we will call interval-valued fuzzy coset instead of interval-valued fuzzy left coset or intervalvalued fuzzy right coset.
3. Interval-Valued Fuzzy Congruences
Definition 3.1 [19] . A relation R on a groupoid S is said to be:
(1) left compatible if ( a, b ) ∈ R implies ( xa, xb ) ∈ R , for any a, 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) andRU(x, y) ≤RU(zx, zy),
(2) interval-valued fuzzy right compatible if for any x, y, z G ,
  • RL(x, y) ≤RL(xz, yz) andRU(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 :
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( a ) Let R 1 : S × S D ( I ) be the mapping defined as the matrix :
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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 :
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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 :
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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 [ ΧR, ΧR ] is interval-valued fuzzy left compatible.
Proof . (⇒) : Suppose R is left compatible. Let a, b, x S .
Case(1) Suppose ( a, b ) ∈ R . Then ΧR ( a, b ) = 1. Since R is left compatible, ( xa, xb ) ∈ R , for each x S . Thus ΧR ( xa, xb ) = 1 = ΧR ( a, b ).
Case(2) Suppose ¬( a, b ) ∈ R . Then, for each x S , it holds that ΧR ( a, b ) = 0 ≤ ΧR ( xa, xb ). Thus, in either cases, [ ΧR, ΧR ].
(⇐) : Suppose [ ΧR, ΧR ] is interval-valued fuzzy compatible. Let a, b, x S and ( a, b ) ∈ R . Then, by hypothesis, ΧR ( xa, xb ) ≥ ΧR ( a, b ) = 1. Thus ΧR ( xa, xb ) = 1. So ( xa, xb ) ∈ R . Hence R is left compatible.
Lemma 3.5 [The dual of Lemma 3.4]. Let R be a relation on a groupoid S . Then R is right compatible if and only if [ ΧR, ΧR ] is interval-valued fuzzy right compatible.
Definition 3.6 An IVFER R on a groupoid S is called an :
(1) interval-valued fuzzy left congruence ( IVLC ) if it is intervalvalued fuzzy left compatible,
(2) interval-valued fuzzy right congruence ( IVRC ) if it is interval-valued fuzzy right compatible,
(3) interval-valued fuzzy congruence ( IVC ) if it is intervalvalued fuzzy compatible.
We will denote the set of all IVCs[resp. IVLCs and IVRCs] on S as IVC( S ) [resp: IVLC( S ) and IVRC( S )].
Example 3.7 Let S = e, a, b be the groupoid defined in Example 3.3. Let R 1 : S × S D ( I ) be the mapping defined as the matrix :
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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
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By the similar arguments, we have that
  • RU(x, y) ∧RU(z, t) ≤RU(xz, yt)
So R is interval-valued fuzzy compatible. Hence R ∈ IVC( S ).
The following is the immediate result of Remark 2.7(c), Lemmas 3.4 and 3.5, and Proposition 3.5.
Theorem 3.9 Let R be a relation on a groupoid S . Then R ∈ C( S ) if and only if [ ΧR, ΧR ] ∈ IVC( S ).
For any interval-valued fuzzy left[resp. right] compatible relation R , it is obvious that if G is a group, then R ( x, y ) = R ( tx, ty )[resp: R ( x, y ) = R ( xt, yt )], for any x, y, 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 :
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Let R : V × V D ( I ) be the mapping defined as the matrix :
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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
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By the similar arguments, we have that
  • (Q○P)U(ax, bx) ≥PU(a, c) ∧QU(c, b) for eachc∈S:
Thus
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and
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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○Pis 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
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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,
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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
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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 AR : G D ( I ) as follows : For each a G ,
  • AR(a) =R(a, e) =Re(a).
Then AR = Re ∈ IVNG( G ).
Proof. From the definition of AR , it is obvious that AR D ( I ) G . Let a, b G . Then
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Similarly, we have that
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On the other hand,
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Moreover,
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So AR ∈ IVG( G ) such that AR ( e ) = [1, 1].
Finally,
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Hence AR ∈ IVNG( G ). This completes the proof.
The following is the immediate result of Proposition 3.17 and Result 2.F. Proposition 3.18 Let G be a group with the identity e . If P,Q ∈ IVNG( G ), then 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–1g) =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–1g).
Since R is interval-valued fuzzy left compatible,
  • R(e, x–1g) =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
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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 á S such that a = aáa and á = áaá . Such an á 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
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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
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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
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Similarly, we have that RU ( a, c ) ≥ 1. So R ( a, c ) = [1, 1], i.e., ( a, c ) ∈ R –1 ([1, 1]). Hence R –1 ([1, 1]) is an equivalence relation on S .
Now let ( a, b ) ∈ R –1 ([1, 1]) and let x S . Since R is an IVC on S ,
RL ( ax, bx ) ≥ RL ( a, b ) = 1 and RU ( ax, bx ) ≥ RU ( a, b ) = 1.
Then R ( ax, bx ) = [1, 1]. Thus ( ax, bx ) ∈ R –1 ([1, 1]). Similarly, ( xa, xb ) ∈ R –1 ([1, 1]). So R –1 ([1, 1]) is compatible. Hence R –1 ([1, 1]) is a congruence on S .
Let S be a semigroup. Then S 1 denotes the monoid defined as follows :
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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 ,
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and
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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) IfP⊂Q, thenP* ⊂Q*.
  • (d) (P*)* =P*.
  • (e) (P∪Q)* =P* ∪Q*.
  • (f)R=R* if and only ifRis 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
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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
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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
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Similarly, we have that
  • RU(ec, ed) ≥RU(c, d).
By the similar arguments, we have that
  • RL(ce, de) ≥RL(c, d) andRU(ce, de) ≥RU(c, d).
(⇐) : Suppose R is interval-valued fuzzy left and right compatible. Let c, d S . Then
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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
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Similarly, we have that
  • (Rn)U(a, b) ≤ (Rn)U(ac, bc).
By the similar arguments, we have that
  • (Rn)L(a, b) ≤ (Rn)U(ca, cb)
and
  • (Rn)U(a, b) ≤ (Rn)U(ca, cb).
So Rn is interval-valued fuzzy left and right compatible for each n ≥ 1. Hence R is interval-valued fuzzy left and right compatible.
Let R ∈ IVR( S )and let { } α∈Г be the family of all IVCs on a semigroup S containing R . Then the IVFR
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defined by
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is clearly the least IVC on S . In this case,
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is called the IVC on S generated by R .
Theorem 3.28 If R is an IVFR on a semigroup S , then
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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
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This completes the proof.
4. Homomorphisms
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 ,
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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
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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
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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
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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.
5. Conclusion
Hur et al. [11] studied interval-valued fuzzy relations in the sense of a lattice. Cheong and Hur [13] , Hur et al. [14] , and Kim et al. [15] investigated interval-valued fuzzy ideals/(generalized) bi-ideas and quasi-ideals in a semigroup, respectively.
In this paper, we mainly study interval-valued fuzzy congruences on a semigroup. In particular, we obtain the result that
Lager Image
for the IVC
Lager Image
on S generated by R for each IVFR R on a semigroup S (See Theorem 3.28). Finally, for any IVCs R and Q on a semigroup S such that 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.
- Conflict of Interest
No potential conflict of interest relevant to this article was reported.
Acknowledgements
This work was supported by the research grant of theWonkwang University in 2013.
References
Zadeh L. A. 1963 “Fuzzy sets” Information and Control 8 338 - 353
Zadeh L. A. 1975 “The concept of a linguistic variable and its application to approximate reasoning-I” Information Science 8 (3) 199 - 249    DOI : 10.1016/0020-0255(75)90036-5
Biswas R. 1995 “Rosenfeld’s fuzzy subgroups with intervalvalued membership functions” Fuzzy Set and Systems 63 (1) 87 - 90    DOI : 10.1016/0165-0114(94)90148-1
Gorzalczany M. B. 1987 “A method of inference in approximate reasoning based on interval-values fuzzy sets” Fuzzy Sets and Systems 21 (1) 1 - 17    DOI : 10.1016/0165-0114(87)90148-5
Mondal T. K. , Samanta S. K. 1999 “Topology of intervalvalued fuzzy sets” Indian Journal of Pure Applied Mathematics 30 (1) 20 - 38
Roy M. K. , Biswas R. 1992 “I-v fuzzy relations and Sanchez’s approach for medical diagnosis” Fuzzy Set and Systems 47 (1) 35 - 38    DOI : 10.1016/0165-0114(92)90057-B
Jun Y. B. , Bae J. H. , Cho S. H. , Kim C. S. 2006 “Intervalvalued fuzzy strong semi-openness and interval-valued fuzzy strong semi-continuity” Honam Mathematical Journal 28 (3) 417 - 431
Min W. K. 2009 “Characterizations for interval-valued fuzzy m-semicontinuous mappings on interval-valued fuzzy minimal spaces” Journal of Korean Institute of Intelligent Systems 19 (6) 848 - 851
Min W. K. , Kim M. H. 2009 “Interval-valued fuzzy mcontinuity and interval-valued fuzzy m*-open mappings” International Journal of Fuzzy Logic and Intelligent Systems 9 (1) 47 - 52
Min W. K. , Kim M. H. 2009 “On interval-valued fuzzy weakly m*-continuous mappings on interval-valued fuzzy minimal spaces” International Journal of Fuzzy Logic and Intelligent Systems 9 (2) 133 - 136
Hur K. , Lee J. G. , Choi J. Y. 2009 “Interval-valued fuzzy relations” Journal of Korean Institute of Intelligent Systems 19 (3) 425 - 432
Choi J. Y. , Kim S. R. , Hur K. 2010 “Interval-valued smooth topological spaces” Honam Mathematical Journal 32 (4) 711 - 738
Cheong M. , Hur K. 2011 “Interval-valued fuzzy ideals and bi-ideals of a semigroup” International Journal of Fuzzy Logic and Intelligent Systems 11 (4) 259 - 266    DOI : 10.5391/IJFIS.2011.11.4.259
Lee K. C. , Kang H. , Hur K. 2011 “Interval-valued fuzzy generalized bi-ideals of a semigroup” Honam Mathematical Journal 33 (4) 603 - 611    DOI : 10.5831/HMJ.2011.33.4.603
Kim S. M. , Hur K. 2012 “Interval-valued fuzzy quasiideals in semigroups” International Journal of Fuzzy Logic and Intelligent Systems 12 (3) 215 - 225    DOI : 10.5391/IJFIS.2012.12.3.215
Kang H. W. 2011 “Interval-valued fuzzy subgroups and homomorphisms” Honam Mathematical Journal 33 (4) 499 - 518    DOI : 10.5831/HMJ.2011.33.4.499
Kang H. , Hur K. 2010 “Interval-valued fuzzy subgroups and rings” Honam Mathematical Journal 32 (4) 593 - 617
Jang S. Y. , Hur K. , Lim P. K. 2012 “Interval-valued fuzzy normal subgroups” International Journal of Fuzzy Logic and Intelligent Systems 12 (3) 205 - 214    DOI : 10.5391/IJFIS.2012.12.3.205
Howie J. M. 1976 An Introduction to Semigroup Theory Academic Press New York