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 R_e 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 . As a generalization of fuzzy sets introduced by Zadeh [1] , Zadeh [2] also suggested the concept of interval-valued fuzzy sets. After that time, Biswas [3] applied it to group theory, and Gorzalczany [4] introduced a method of inference in approximate reasoning by using interval-valued fuzzy sets. Moreover, Mondal and Samanta [5] introduced the concept of interval-valued fuzzy topology and investigated some of it’s properties. In particular, Roy and Biswas [6] introduced the notion of interval-valued fuzzy relations and studied some of it’s properties. Recently, Jun et al. [7] investigated strong semi-openness and strong semicontinuity in interval-valued fuzzy topology. Moreover, Min [8] studied characterizations for interval-valued fuzzy m-semicontinuous mappings, Min and Kim [9 , 10] investigated intervalvalued fuzzy m*-continuity and m*-open mappings. Hur et al. [11] studied interval-valued fuzzy relations in the sense of a lattice theory. Also, Choi et al. [12] introduced the concept of interval-valued smooth topological spaces and investigated some of it’s properties. On the other hand, Cheong and Hur [13] , and Lee et al. [14] studied interval-valued fuzzy ideals/(generalized)bi-ideals in a semigroup. In particular, Kim and Hur [15] investigated interval-valued fuzzy quasi-ideals in a semigroup. Kang [16] , Kang and Hur [17] applied the notion of interval-valued fuzzy sets to algebra. Jang et al. [18] investigated interval-valued fuzzy normal subgroups. In this paper, we introduce the concept of interval-valued fuzzy congruences on a semigroup S and we obtain some important results: (i) For any interval-valued fuzzy congruence R on a group G , the interval-valued congruence class R_e is an interval-valued fuzzy normal subgroup of G (Proposition 3.11). (ii) For any interval-valued fuzzy congruence R on a groupoid S , we show that a binary operation * an S/R is well-defined (Proposition 3.20) and also we obtain some results related to additional conditions for S (Theorem 3.21, Corollaries 3.21-1, 3.21-2, and 3.21-3). Also we improve that for any two intervalvalued fuzzy congruences R and Q on a semigroup S such that R ⊂ Q , there exists a unique semigroup homomorphism g : S/R → S/G (Theorem 4.3). In this section, we list some concepts and well-known results which are needed in later sections. Let D ( I ) be the set of all closed subintervals of the unit interval [0, 1]. The elements of D ( I ) are generally denoted by capital letters M , N , …, and note that M = [ M^L, M^U ], where M^L and M^U are the lower and the upper end points respectively. Especially, we denoted , 0 = [0, 0], 1 = [1, 1], and a = [ a, a ] for every a ∈ (0, 1), We also note that (i) (∀ M, N ∈ D ( I )) ( M = N ⇔ M^L = N^L , M^U = N^U ), (ii) (∀ M, N ∈ D ( I )) ( M = N ≤ M^L ≤ N^L , M^U ≤ N^U ), For every M ∈ D ( I ), the complement of M , denoted by M^C , is defined by M^C = 1 – M = [1 – M^U , 1 – M^L ]( [7 , 14] ). Definition 2.1 [4 , 10 , 14] . A mapping A : X → D ( I ) is called an interval-valued fuzzy set ( IVFS ) in X , denoted by A = [ A^L, A^U ], if A^L , A^L ∈ I^X such that A^L ≤ A^U , i.e., A^L ( x ) ≤ A^U ( x ) for each x ∈ X , where A^L ( x )[resp A^U ( x )] is called the lower [resp upper ] end point of x to A. For any [ a, b ] ∈ D ( I ), the interval-valued fuzzy A in X defined by A ( x ) = [ A^L ( x ), A^U ( x )] = [ a, b ] for each x ∈ X is denoted by and if a = b , then the IVFS is denoted by simply ã . In particular, denote the interval-valued fuzzy empty set and the interval-valued fuzzy whole set in X , respectively. We will denote the set of all IVFSs in X as D ( I ) ^X . It is clear that set A = [ A, A ] ∈ D ( I ) ^X for each A ∈ I^X . Definition 2.2 [14] . Let A, B ∈ D ( I ) ^X and let ｛ A_α ｝ _α∈Г ⊂ D ( I ) ^X . Then Result 2.A [ 14 , Theorem 1]. Let A, B, C ∈ D ( I ) ^X and let { A_α } _α∈Г ⊂ D ( I ) ^X . Then Definition 2.3 [8] . Let X be a set. Then a mapping R = [ R^L , R^U ] : X ∏ X → D ( I ) is called an interval-valued fuzzy relation ( IVFR ) on X . Definition 2.4 [8] . Let R ∈ IVR(X). Then the inverse of R , R ^–1 is defined by R ^–1 ( x,y ) = R ( y,x ), for each x, y ∈ X . Definition 2.5 [11] . Let X be a set and let R , Q ∈ IVR(X). Then the composition of R and Q , Q ○ R , is defined as follows : For any x, y ∈ X , and Result 2.B [ 11 , Proposition 3.4]. Let X be a set and let R , R ₁ , R ₂ , R ₃ , Q ₁ , Q ₂ ∈ IVR( X ). Then Definition 2.6 [11] . An IVFR R on a set X is called an interval-valued fuzzy equivalence relation ( IV FER ) on X if it satisfies the following conditions : (1) it is interval-valued fuzzy reflexiv , i.e., R ( x, x ) = [1, 1], for each x ∈ X , (2) it is interval-valued fuzzy symmetric , i.e., R ^–1 = R , (3) it is interval-valued fuzzy transitive , i.e., R ○ R ⊂ R . We will denote the set of all IVFERS on X as IVE( X ). From Definition 2.6, we can easily see that the following hold. Remark 2.7 (a) If R is an fuzzy equivalence relation on a set X , then [ R, R ] ∈ IVE( X ). (b) If R ∈ IVE( X ), then R^L and R^U are fuzzy equivalence relation on X . (c) Let R be an ordinary relation on a set X . Then R is an equivalence relation on X if and only if [ Χ_R, Χ_R ] ∈ IVE( X ). Result 2.C [ 11 , Proposition 3.9]. Let X be a set and let Q , R ∈ IVE( X ). If Q ○ R = R ○ Q , then R ○ Q ∈ IVE( X ). Let R be an IVFER on a set X and let a ∈ X . We define a mapping Ra : X → D ( I ) as follows : For each a ∈ X , Then clearly Ra ∈ D ( I ) ^X . In this case, Ra is called the interval-valued fuzzy equivalence class of R containing a ∈ X . The set { Ra : a ∈ X } is called the interval-valued fuzzy quotient set of X by R and denoted by X/R . Result 2.D [ 11 , Proposition 3.10]. Let R be an IVFER on a set X . Then the following hold : ( a ) Ra = Rb if and only if R ( a, b ) = [1, 1], for any a, b ∈ X . ( b ) R ( a, b ) = [0, 0] if and only if Ra ∩ Rb = for any a, b ∈ X . ( d ) There exits the surjection π : X → X/R defined by π ( x ) = Rx for each x ∈ X . Definition 2.8 [11] . Let X be a set, let R ∈ IVR( X ) and let { R_α } _α∈Г be the family of all IVFERs on X containing R . Then ∩ _α∈Г R_α is called the IVFER generated by R and denoted by R^e . It is easily seen that R^e is the smallest IVFER containing R . Definition 2.9 [11] . Let X be a set and let R ∈ IVR( X ). Then the interval-valued fuzzy transitive closure of R , denoted R ^∞ , is defined as followings : ,where Rⁿ = R ○ R ○ … ○ R (n factors). Definition 2.10 [11] . We define two mappings △, ▽ : X → D ( I ) as follows : For any x, y ∈ X , and 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 Definition 2.11 [17] . Let ( X , ·) be a groupoid and let A, B ∈ D ( I ) ^X . Then the interval-valued fuzzy product of A and B , A ○ B is defined as follows : For each a ∈ X , and Definition 2.12 [17] . Let ( X , ·) be a groupoid and let A ∈ D ( I ) ^X . Then A is called an i interval-valued fuzzy subgroupoid ( IVGP ) of X if for any x, y ∈ X , and We will denote the set of all IVGPs of X as IVGP( X ). Then it is clear that Definition 2.13 [17] . Let G be a group and let A ∈ IVGP( G ). Then A is an i interval-valued fuzzy subgroup ( IVG ) of G if for each x ∈ G , i.e., 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 and as follows, respectively : For each g ∈ G , Then Ax [resp: xA ] is called the interval-valued fuzzy right [resp. left ] coset of G determined by x and A . It is obvious that if A ∈ IVNG( G ), then the interval-valued fuzzy left coset coincides with the interval-valued fuzzy right coset of A on G . In this case, we will call interval-valued fuzzy coset instead of interval-valued fuzzy left coset or intervalvalued fuzzy right coset. Definition 3.1 [19] . A relation R on a groupoid S is said to be: (1) left compatible if ( a, b ) ∈ R implies ( xa, xb ) ∈ R , for any a, b ∈ S , (2) right compatible if ( a, b ) ∈ R implies ( ax, bx ) ∈ R , for any a, b ∈ S , (3) compatible if ( a, b ) ∈ R and ( s, d ) ∈ R imply ( ab, cd ) ∈ R , for any a, b, c, d ∈ S , (4) a left [resp. right ] congruence on S if it is a left[resp. right] compatible equivalence relation. (5) a congruence on S if it is both a left and a right congruence on S . It is well-known [ 19 , Proposition I.5.1] that a relation R on a groupoid S is congruence if and only if it is both a left and a right congruence on S . We will denote the set of all ordinary congruences on S as C ( S ). Now we will introduce the concept of interval-valued fuzzy compatible relation on a groupoid. Definition 3.2 An IVFR R on a groupoid S is said to be : (1) interval-valued fuzzy left compatible if for any x, y, z ∈ G , (2) interval-valued fuzzy right compatible if for any x, y, z ∈ G , (3) interval-valued fuzzy compatible if for any x, y, z, t ∈ G , and Example 3.3 Let S = e, a, b be the groupoid with multiplication table : ( a ) Let R ₁ : S × S → D ( I ) be the mapping defined as the matrix : where [ λ _ij , μ _ij ] ∈ D ( I ) such that [ λ _1i , μ _1i ]( i = 1, 2, 3), [ λ ₂₁ , μ ₂₁ ] and [ λ ₃₁ , μ ₃₁ ] are arbitrary, and Then we can see that R ₁ is an interval-valued fuzzy left compatible relation on S . ( b ) Let R ₂ : S × S → D ( I ) be the mapping defined as the matrix : where [ λ _ij , μ _ij ] ∈ D ( I ) such that [ λ _ij , μ _ij ]( i, j = 1, 2, 3) is arbitrary and Then we can see that R ₂ is an interval-valued fuzzy right compatible relation on S . ( c ) Let R ₃ : S × S → D ( I ) be the mapping defined as the matrix : where [ λ _ij , μ _ij ] ∈ D ( I ) such that Then we can see that R ₃ is an interval-valued fuzzy compatible relation on S . Lemma 3.4 Let R be a relation on a groupoid S . Then R is left compatible if and only if [ Χ_R, Χ_R ] is interval-valued fuzzy left compatible. Proof . (⇒) : Suppose R is left compatible. Let a, b, x ∈ S . Case(1) Suppose ( a, b ) ∈ R . Then Χ_R ( a, b ) = 1. Since R is left compatible, ( xa, xb ) ∈ R , for each x ∈ S . Thus Χ_R ( xa, xb ) = 1 = Χ_R ( a, b ). Case(2) Suppose ¬( a, b ) ∈ R . Then, for each x ∈ S , it holds that Χ_R ( a, b ) = 0 ≤ Χ_R ( xa, xb ). Thus, in either cases, [ Χ_R, Χ_R ]. (⇐) : Suppose [ Χ_R, Χ_R ] is interval-valued fuzzy compatible. Let a, b, x ∈ S and ( a, b ) ∈ R . Then, by hypothesis, Χ_R ( xa, xb ) ≥ Χ_R ( a, b ) = 1. Thus Χ_R ( xa, xb ) = 1. So ( xa, xb ) ∈ R . Hence R is left compatible. Lemma 3.5 [The dual of Lemma 3.4]. Let R be a relation on a groupoid S . Then R is right compatible if and only if [ Χ_R, Χ_R ] is interval-valued fuzzy right compatible. Definition 3.6 An IVFER R on a groupoid S is called an : (1) interval-valued fuzzy left congruence ( IVLC ) if it is intervalvalued fuzzy left compatible, (2) interval-valued fuzzy right congruence ( IVRC ) if it is interval-valued fuzzy right compatible, (3) interval-valued fuzzy congruence ( IVC ) if it is intervalvalued fuzzy compatible. We will denote the set of all IVCs[resp. IVLCs and IVRCs] on S as IVC( S ) [resp: IVLC( S ) and IVRC( S )]. Example 3.7 Let S = e, a, b be the groupoid defined in Example 3.3. Let R ₁ : S × S → D ( I ) be the mapping defined as the matrix : Then it can easily be checked that R ∈ IVE( S ). Moreover we can see that R ∈ IVC( S ). Proposition 3.8 Let S be a groupoid and let R ∈ IVE( S ). Then R ∈ IVC( S ) if and only if it is both an IVLC and an IVRC. Proof. (⇒) : Suppose R ∈ IVC( S ) and let x, y, z ∈ S . Then and Also, and Thus R is both an IVLC and an IVRC. (⇐) : Suppose R is both an IVLC and an IVRC. and let x, y, z, t ∈ S . Then By the similar arguments, we have that So R is interval-valued fuzzy compatible. Hence R ∈ IVC( S ). The following is the immediate result of Remark 2.7(c), Lemmas 3.4 and 3.5, and Proposition 3.5. Theorem 3.9 Let R be a relation on a groupoid S . Then R ∈ C( S ) if and only if [ Χ_R, Χ_R ] ∈ IVC( S ). For any interval-valued fuzzy left[resp. right] compatible relation R , it is obvious that if G is a group, then R ( x, y ) = R ( tx, ty )[resp: R ( x, y ) = R ( xt, yt )], for any x, y, t ∈ G . Thus we have following result. Lemma 3.10 Let R be an IVC on a group G . Then for any a, b, x, y ∈ G . Example 3.11 Let V be the Klein 4-group with multiplication table : Let R : V × V → D ( I ) be the mapping defined as the matrix : Then we can see that R ∈ IVC( V ). Furthermore, it is easily checked that Lemma 3.10 holds : For any s, t, x, y ∈ V , R ( xsy, xty ) = R ( xs, xt ) = R ( sy, ty ) = R ( s, t ) The following is the immediate result of Proposition 3.8 and Lemma 3.10. Theorem 3.12 Let R be an IVFR on a group G . Then R ∈ IVC( G ) if and only if it is interval-valued fuzzy left(right) compatible equivalence relation. Lemma 3.13 Let P and Q be interval-valued fuzzy compatible relations on a groupoid S . Then Q ○ P is also an interval-valued fuzzy compatible relation on S . Proof. Let a, b, x ∈ S . Then By the similar arguments, we have that Thus and So Q ○ P is interval-valued fuzzy right compatible. Similarly, we can see that Q ○ P is interval-valued fuzzy left compatible. Hence Q ○ P is interval-valued fuzzy compatible. Theorem 3.14 Let P and Q be IVC on a groupoid S . Then the following are equivalent : Proof. It is obvious that ( a ) ⇒ ( b ) ⇒ ( c ). ( c ) ⇒ ( d ) : Suppose the condition (c) holds and let a, b ∈ S . Then Similarly, we have that Hence Q ○ P = P ○ Q ( d ) ⇒ ( a ) : Suppose the condition (d) holds. Then , by Result 2.C, Q ○ P ∈ IVE( S ). Since P and Q are interval-valued fuzzy compatible, by Lemma 3.13, Q ○ P is interval-valued fuzzy compatible. So Q ○ P ∈ IVC( S ). This completes the proof. Proposition 3.15 Let S be a groupoid and let Q, P ∈ IVC( S ). If Q ○ P = P ○ Q , then P ○ Q ∈ IVC( S ). Proof. By Result 2.C, it is clear that P ○ Q ∈ IVE( S ). Let x, y, t ∈ S . Then, since P and Q are interval-valued fuzzy right compatible, Similarly, we have that By the similar arguments, we have that and So P ○ Q is interval-valued fuzzy left and right compatible. Hence P ○ Q ∈ IVC( S ). Let R be an IVC on a groupoid S and let a ∈ S . Then Ra ∈ D ( I ) ^S is called an interval-valued fuzzy congruence class of R containing a ∈ S and we will denote the set of all interval-valued fuzzy congruence classes of R as S/R . Proposition 3.16 If R is an IVC on a groupoid S , then Ra ○ Rb ⊂ Rab , for any a, b ∈ S . Proof. Let x ∈ S . If x is not expressible as x = yz , then clearly ( Ra ○ Rb )( x ) = [0, 0]. Thus Ra ○ Rb ⊂ Rab . Suppose x is expressible as x = yz . Then Similarly, we have that ( Ra ○ Rb ) ^U ( x ) ≤ ( Rab ) ^U ( x ). Thus Ra ○ Rb ⊂ Rab . This completes the proof. Proposition 3.17 Let G be a group with the identity e and let R ∈ IVC( G ). We define the mapping A_R : G → D ( I ) as follows : For each a ∈ G , Then A_R = R_e ∈ IVNG( G ). Proof. From the definition of A_R , it is obvious that A_R ∈ D ( I ) ^G . Let a, b ∈ G . Then Similarly, we have that On the other hand, Moreover, So A_R ∈ IVG( G ) such that A_R ( e ) = [1, 1]. Finally, Hence A_R ∈ IVNG( G ). This completes the proof. The following is the immediate result of Proposition 3.17 and Result 2.F. Proposition 3.18 Let G be a group with the identity e . If P,Q ∈ IVNG( G ), then Pe ○ Qe = Qe ○ Pe . Proposition 3.19 Let G be a group with the identity e . If R ∈ IVC( G ), then any interval-valued fuzzy congruence class Rx of x ∈ G by R is an interval-valued fuzzy coset of Re . Conversely, each interval-valued fuzzy coset of Re is an interval-valued fuzzy congruence class by R . Proof. Suppose R ∈ IVC( G ) and let x.g ∈ G . Then Rx ( g ) = R ( x, g ). Since R is interval-valued fuzzy left compatible, by Lemma 3.10, R ( x, g ) = R ( e, x ^–1 g ). Thus 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 Since R is interval-valued fuzzy left compatible, 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 , Then * is well-defined. Proof. Suppose Ra = Rx and Rb = Ry , where a, b, x, y ∈ S . Then, by Result 2.D( a ), Thus Similarly, we have that 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 Thus Re is the identity in G/R with respect to *. Moreover, 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 , Then π ∈ IVG( G/R ). Proof. From the definition of π , it is clear that π = [ π ^L , π ^U ] ∈ D ( I ) ^G/R . Let x, y ∈ G . Then Similarly, we have that By the process of the proof of Corollary 3.21-1, ( Rx ) ^–1 = Rx ^–1 . Thus So π (( Rx ) ^–1 ) = π ( Rx ) for each x ∈ G . Hence π ∈ IVG( G/R ). Proposition 3.23 If R is an IVC on an inverse semigroup S . Then R ( x ^–1 , y ^–1 ) = R ( x, y ) for any x, y ∈ S . Proof. By Corollary 3.21-1, S/R is an inverse semigroup with ( Rx ) ^–1 = Rx ^–1 for each x ∈ S . Let x, y ∈ S . Then Hence R ( x ^–1 , y ^–1 ) = R ( x, y ). The following is the immediate result of Proposition 3.22 Corollary 3.23 Let R be an IVC on a group G . Then for any x, y ∈ G . Proposition 3.24 Let R be an IVC on a semigroup S . Then is a congruence on S . Proof. It is clear that R ^–1 ([1, 1]) is reflexive and symmetric. Let ( a, b ), ( b, c ) ∈ R ^–1 ([1, 1]). Then R ( a, b ) = R ( b, c ) = [1, 1]. Thus Similarly, we have that R^U ( a, c ) ≥ 1. So R ( a, c ) = [1, 1], i.e., ( a, c ) ∈ R ^–1 ([1, 1]). Hence R ^–1 ([1, 1]) is an equivalence relation on S . Now let ( a, b ) ∈ R ^–1 ([1, 1]) and let x ∈ S . Since R is an IVC on S , R^L ( ax, bx ) ≥ R^L ( a, b ) = 1 and R^U ( ax, bx ) ≥ R^U ( a, b ) = 1. Then R ( ax, bx ) = [1, 1]. Thus ( ax, bx ) ∈ R ^–1 ([1, 1]). Similarly, ( xa, xb ) ∈ R ^–1 ([1, 1]). So R ^–1 ([1, 1]) is compatible. Hence R ^–1 ([1, 1]) is a congruence on S . Let S be a semigroup. Then S ¹ denotes the monoid defined as follows : Definition 3.25 Let S be a semigroup and let R ∈ IVR( S ). Then we define a mapping R * : S × S → D ( I ) as follows : For any c, d ∈ S , and It is obvious that R * ∈ IVR( S ). Proposition 3.26 Let S be a semigroup and let R, P,Q ∈ IVR( S ). Then : Proof. From Definition 3.25, the proofs of ( a ), ( b ) and ( c ) are clear. ( d ) By ( a ) and ( c ), it is clear that R * ⊂ ( R *)*. Let c, d ∈ S . Then By the similar arguments, we have that Thus ( R *)* ⊂ R *. So ( R *)* = R *. ( e ) By ( c ), R * ⊂ ( P ∪ Q )* and Q * ⊂ ( P ∪ Q )*. Thus P * ∪ Q * ⊂ ( P ∪ Q )*. Let c, d ∈ S . Then Similarly, we have that Thus ( P ∪ Q )* ⊂ P * ∪ Q *. So ( P ∪ Q )* = P * ∪ Q *. ( f ) (⇒) : Suppose R = R * and let c, d, e ∈ S . Then Similarly, we have that By the similar arguments, we have that (⇐) : Suppose R is interval-valued fuzzy left and right compatible. Let c, d ∈ S . Then Similarly, we have that Thus R * ⊂ R . So R * = R . This completes the proof. Proposition 3.27 If R is an IVFR on a semigroup S such that is interval-valued fuzzy left and right compatible, then so is R ^∞ . Proof. Let a, b, c ∈ S and let n ≥ 1. Then Similarly, we have that By the similar arguments, we have that and So Rⁿ is interval-valued fuzzy left and right compatible for each n ≥ 1. Hence R ^∞ is interval-valued fuzzy left and right compatible. Let R ∈ IVR( S )and let { Rα } _α∈Г be the family of all IVCs on a semigroup S containing R . Then the IVFR defined by is clearly the least IVC on S . In this case, is called the IVC on S generated by R . Theorem 3.28 If R is an IVFR on a semigroup S , then Proof. By Definition 2.8, ( R *) ^e ∈ IVE( S ) such that R * ⊂ ( R *) ^e . Then, by Proposition 3.26( a ), R ⊂ ( R *) ^e . Also, by ( a ) and ( b ) of Proposition 3.26 R * ∪ ( R *) ^–1 ∪ △ = ( R ∪ R ^–1 ∪ △)*. Thus, by Proposition 3.26( f ) and Result 2.E, R * ∪ ( R *) ^–1 ∪ △ is left and right compatible. So, by Proposition 3.27, ( R *) ^e = [ R * ∪ ( R *) ^–1 ∪ △] ^∞ is left and right compatible. Hence, by Proposition 3.8, ( R *) ^e ∈ IVC( S ) . Now suppose Q ∈ IVC( S ) such that R ⊂ Q . Then, by ( c ) and ( d ) of Proposition 3.26, R * ⊂ Q * = Q . Thus ( R *) ^e ⊂ Q . So This completes the proof. Let f : S → T be a semigroup homomorphism. Then it is well-known that the relation is a congruence on S . The following is the immediate result of Theorem 3.9. Proposition 4.1 Let f : S → T be a semigroup homomorphism. Then R = [ Χ _Ker(f) , Χ _Ker(f) ] ∈ IVC( S ). In this case, R is called the interval-valued fuzzy kernel of f and denoted by IVK( f ). In fact, for any a, b ∈ S , Theorem 4.2 ( a ) Let R be an interval-valued fuzzy congruence on a semigroup S . Then the mapping π : S → S/R defined same as in Result 2.D( d ) is an epimorphism. ( b ) If f : S → T is a semigroup homomorphism, then there is a monomorphism g : S /IVK( f ) → T such that the diagram commutes, where [IVK( f )] ^# denotes the natural mapping. Proof. ( a ) Let a, b ∈ S . Then, by the definition of R ^# and Theorem 3.21, So π is a homomorphism. By Result 2.D( d ), π is surjective. Hence π is an epimorphism. ( b ) We define g : S /IVK( f ) → T by g ([IFK( f )] a ) = f ( a ) for each a ∈ S . Suppose [IVK( f )] a = IVK( f )] b for any a, b ∈ S . Since IVK( f )( a, b ) = [1, 1], i.e. Χ _IVK(f) ( a, b ) = 1. Thus ( a, b ) ∈ Ker( f ). So ( a, b ) ∈ Ker ( f ). So g ([IVK( f )] a ) = f ( a ) = f ( b ) = g ([IVK( f )] b ). Hence g is well-defined. Suppose f ( a ) = f ( b ). Then IVK( f )( a, b ) = [1, 1]. Thus, by Result 2.D( a ), [IVK( f )] a = IVK( f )] b . So g is injective. Now let a, b ∈ S , Then So g is a homomorphism. Let a ∈ S . Then g ([IVK( f )] ^# ( a )) = g ([IVK( f )] a ) = f ( a ). So g ○ [IVK( f )] ^# = f . This completes the proof. Theorem 4.3 Let R and Q be IVCs on a semigroup such that R ⊂ Q . Then there exists a unique semigroup S homomorphism g : S/R → S/Q such that the diagram commutes and ( S/R )/IVK( g ) is isomorphic to S/R , where R ^# and Q ^# denote the natural mappings, respectively. Proof. Define g : S/R → S/Q by g ( Ra ) = Qa for each a ∈ S . Suppose Ra = Rb . Then, by Result 2.D( a ), R ( a, b ) = [1, 1]. Since R ⊂ Q , 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 So g is a semigroup homomorphism. The remainders of the proofs are easy. This completes the proof. Hur et al. [11] studied interval-valued fuzzy relations in the sense of a lattice. Cheong and Hur [13] , Hur et al. [14] , and Kim et al. [15] investigated interval-valued fuzzy ideals/(generalized) bi-ideas and quasi-ideals in a semigroup, respectively. In this paper, we mainly study interval-valued fuzzy congruences on a semigroup. In particular, we obtain the result that for the IVC on S generated by R for each IVFR R on a semigroup S (See Theorem 3.28). Finally, for any IVCs R and Q on a semigroup S such that R ⊂ Q , there exists a unique semigroup homomorphism g : S/K → S/Q such that ( S ? R )/IVK( g ) is isomorphic to S/Q (See Theorem 4.3). In the future, we will investigate interval-valued fuzzy congruences on a semiring. No potential conflict of interest relevant to this article was reported.