Interval-valued intuitionistic fuzzy ideals of K-algebras

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1 WSES TRNSCTIONS on MTHEMTICS Muhammad kram, Karamat H. Dar, Interval-valued intuitionistic fuzzy ideals of K-algebras MUHMMD KRM University of the Punjab Punjab University College of Information Technology, Old Campus, P. O. Box 54000, Lahore PKISTN KRMT HUSSIN DR Department of Mathematics, G. C. University Lahore Lahore-54000, PKISTN. prof BIO LONG MENG Department of Basic Courses, Xian University of Science Technology, Xian , P.R.CHIN. mengbl KR-PING SHUM Department of Mathematics, The University of Hong Kong Pokfulam Road, HONG KONG. bstract: The notion of interval-valued intuitionistic fuzzy sets was first introduced by tanassov Gargov in 1989 as a generalization of both interval-valued fuzzy sets intuitionistic fuzzy sets. In this paper we first apply the concept of interval-valued intuitionistic fuzzy sets to K-algebras. Then we introduce the notion of interval-valued intuitionistic fuzzy ideals (IIFIs, in short) of K-algebras investigate some interesting properties. We characterize rtinian Noetherian K-algebras by considering IIF Is of a K-algebra K. Characterization theorems of fully invariant characteristic IIFIs are also discussed. Key Words: K-algebras; Interval-valued intuitionistic fuzzy sets; Equivalence relations; rtinian Noetherian K-algebras; Fully invariant; Characteristic. 1 Introduction It is known that mathematical logic is a discipline used in sciences humanities with different point of view. Non-classical logic takes the advantage of the classical logic (two-valued logic) to hle information with various facts of uncertainty. The non-classical logic has become a formal useful tool for computer science to deal with fuzzy information uncertain information. The notion of logical algebras: BCK-algebras [21] was initiated by Imai Iséki in 1966 as a generalization of both classical non-classical positional calculus. In the same year, Iséki introduced BCI-algebras [22] as a super class of the class of BCK-algebras. In 1983, Hu Li introduced BCH-algebras [20]. They demonstrated that the class of BCI-algebras is a proper subclass of the class of BCH-algebras. Dar kram [11] introduced a new kind of logical algebra: K-algebra (G,,, e). K-algebra is an algebra built on a group (G,, e) with identity e by adjoining an induced binary operation on (G,, e) which is attached to an abstract algebraic system (G,,, e) [11]. It is known that this system ISSN: Issue 9, Volume 7, September 2008

2 WSES TRNSCTIONS on MTHEMTICS Muhammad kram, Karamat H. Dar, is non-commutative non-associative with a right identity e. It was proved in [11] that a K-algebra on an abelian group is equivalent to a p-semisimple BCI-algebra. For the sake of convenience, a K-algebra built on a group G was re-named as a K(G)-algebra [12]. The K(G)-algebras have been characterized by their left right mappings [12]. Dar kram further proved in [14] that a class of K-algebras is a super class of the class of nonclassical BCH/BCI/BCK-algebras [20, 21, 22] the class of B-algebras [29] when the group is abelian non-abelian, respectively. Interval-valued fuzzy sets were first introduced by Zadeh [31] as a generalization of fuzzy sets. n interval-valued fuzzy set is a fuzzy set whose membership function is many-valued forms an interval in the membership scale. This idea gives the simplest method to capture the imprecision of the membership grades for a fuzzy set. Thus, intervalvalued fuzzy sets provide a more adequate description of uncertainty than the traditional fuzzy sets. It is therefore important to use interval-valued fuzzy sets in applications. One of the main applications is in fuzzy control the most computationally intensive part of fuzzy control is defuzzification. Since the transition of interval-valued fuzzy sets usually increases the amount of computations, it is vitally important to design some faster algorithms for the necessarily defuzzification. On the other h, tanassov [5] introduced the notion of intuitionistic fuzzy sets as an extension of fuzzy set in which not only a membership degree is given, but also a non-membership degree is involved. Considering the increasing interest in intuitionistic fuzzy sets, it is useful to determine the position of intuitionistic fuzzy sets in a frame of different theories of imprecision. With the above background, tanassov Gargov [8] introduced the notion of interval-valued intuitionistic fuzzy sets which is a common generalization of intuitionistic fuzzy sets interval-valued fuzzy sets. The fuzzy structures of K-algebras was introduced in [1]. Since then, the concepts results of K-algebras have been broadened to the fuzzy setting frames (see, for example, [2-4, 10, 23]). In this paper we first apply the concept of interval-valued intuitionistic fuzzy sets to K-algebras. Then we introduce the notion of interval-valued intuitionistic fuzzy ideals (IIFIs, in short) of K-algebras investigate some interesting properties. We characterize rtinian Noetherian K-algebras by considering IIF Is of a K-algebra K. Characterization theorems of fully invariant characteristic IIFIs are also discussed. 2 Preliminaries In this section we cite some definitions properties that are necessary for this paper. Definition 1 [11] K-algebra K = (G,,, e) is an algebra of type (2, 2, 0) defined on a group (G,, e) in which each non-identity element is not of order 2 observes the following -axioms: (K1) (x y) (x z)=(x ((e z) (e y))) x, (K2) x (x y) = (x (e y)) x, (K3) x x = e, (K4) x e = x, (K5) e x = x 1. for all x, y, z G. If the group (G,, e) is abelian, then the above axioms (K1) (K2) can be replaced by: (K1) (x y) (x z) = z y. (K2) x (x y) = y. nonempty subset H of a K-algebra K is called a subalgebra [11] of the K-algebra K if a b H for all a, b H. Note that every subalgebra of a K-algebra K contains the identity e of the group (G,, e). mapping f : K 1 = (G 1,,, e 1 ) K 2 = (G 2,,, e 2 ) of K-algebras is called a homomorphism [14] if f(x y) = f(x) f(y) for all x, y K 1. We note that if f is a homomorphism, then f(e) = e. Definition 2 [1] nonempty subset I of a K-algebra K is called an ideal of K if it satisfies: (i) e I, (ii) x y I, y (y x) I x I for all x, y G. Let µ be a fuzzy set on G, i.e., a map µ : G [0, 1]. Definition 3 [28] fuzzy set µ in a group G is called a fuzzy subgroup of G if it satisfies: ( x, y G) (µ(xy) min{µ(x), µ(y)}). ( x G) (µ(x 1 ) µ(x)). Lemma 4 [28] Let µ be a fuzzy subgroup of a group G. Then µ(x 1 ) = µ(x) µ(x) µ(e) for all x G, where e is the identity element of G. ISSN: Issue 9, Volume 7, September 2008

3 WSES TRNSCTIONS on MTHEMTICS Muhammad kram, Karamat H. Dar, Lemma 5 [28] fuzzy set µ in a group G is a fuzzy subgroup of G if only if it satisfies: ( x, y G)(µ(xy 1 ) min{µ(x), µ(y)}). Definition 6 [1] fuzzy set µ in a K-algebra K is called a fuzzy subalgebra of K if it satisfies: ( x, y G) (µ(x y) min{µ(x), µ(y)}). Note that every fuzzy subalgebra µ of a K-algebra K satisfies the following inequality: ( x G) (µ(e) µ(x)). Proposition 7 [1] Let K = (G,,, e) be a K- algebra in which the operation is induced by the group operation. Then every fuzzy subgroup of (G,, e) is a fuzzy subalgebra of K vice versa. Definition 8 [1] fuzzy ideal of a K-algebra K is a mapping µ : G [0, 1] such that (i) ( x G) (µ(e) µ(x), (ii) ( x, y G) (µ(x) min{µ(x y), µ(y (y x))}). Lemma 9 [1] µ is a fuzzy ideal of a K-algebra K if only if µ is a fuzzy normal subgroup of G. By an interval number D on [0, 1], we mean an interval [a, a + ], where 0 a a + 1. The set of all closed subintervals of [0, 1] is denoted by D[0, 1]. For interval numbers D1 = [a 1, b+ 1 ] D2 = [a 2, b+ 2 ] D[0, 1], we define min(d1, D2) = min([a 1, b+ 1 ], [a 2, b+ 2 ]) = [min{a 1, a 2 }, min{b+ 1, b+ 2 }], max(d1, D2) = max([a 1, b+ 1 ], [a 2, b+ 2 ]) = [max{a 1, a 2 }, max{b+ 1, b+ 2 }], D1 + D2 = [min{a 1, a 2 }, max{b+ 1, b+ 2 }]. D1 D2 a 1 a 2 b+ 1 b+ 2, D1 = D2 a 1 = a 2 b+ 1 = b+ 2, D1 < D2 D1 D2 D1 D2, md = m[a 1, b+ 1 ] = [ma 1, mb+ 1 ], where 0 m 1. Obviously, (D[0, 1],,, ) forms a complete lattice with [0, 0] as its least element [1, 1] as its greatest element. If G( ) be a given set, then an intervalvalued fuzzy set B on G is defined by B = {(x, [µ B (x), µ+ B (x)]) : x G}, where µ B (x) µ+ B (x) are fuzzy sets of G such that µ B (x) µ+ B (x), for all x G. If µ B(x) = [µ B (x), µ+ B (x)], then B = {(x, µ B (x)) : x G}, where µ B : G D[0, 1]. We now assume that µ (x) = [µ (x), µ+ (x)] satisfies the following condition: [µ (x), µ+ (x)] < [0.5, 0.5] or [0.5, 0.5] [µ (x), µ+ (x)] for all x. For a nonempty set G, we call a mapping = ( µ, λ ) : G D[0, 1] D[0, 1] an interval-valued intuitionistic fuzzy set in G if µ + (x) + λ+ (x) 1 µ (x) + λ (x) 1 for all x G, where the mappings µ (x) = [µ (x), µ+ (x)] : G D[0, 1] λ (x) = [λ (x), λ+ (x)] : G D[0, 1] are the degree of membership functions the degree of non-membership functions, respectively. We adopt the following symbols terminology. (i) The symbol 0 is used to denote the intervalvalued fuzzy empty set 1 to denote the interval-valued fuzzy universal set in a set G, we define 0(x) = [0, 0] 1(x) = [1, 1] for all x G. (ii) The symbol 0 is used to denote the intervalvalued intuitionistic fuzzy empty set 1 is used to denote the interval-valued intuitionistic fuzzy universal set in a given set G, we define 0(x) = ( 0, 1) = ([0, 0], [1, 1]) 1(x) = ( 1, 0) = ([1, 1], [0, 0]) for all x G. (iii) We write t = [t 1, t 2 ], s = [s 1, s 2 ], s 1 = [s 3, s 4 ] t 1 = [t 3, t 4 ] D[0, 1]. 3 Interval-valued intuitionistic fuzzy ideals Definition 10 n interval-valued intuitionistic fuzzy set = ( µ, λ ) in a K-algebra K is called an interval-valued intuitionistic fuzzy ideal ( IIF I, in short) if it satisfies the following conditions: (1) µ (e) µ (x), ISSN: Issue 9, Volume 7, September 2008

4 WSES TRNSCTIONS on MTHEMTICS Muhammad kram, Karamat H. Dar, (2) λ (e) λ (x), (3) µ (x) min{ µ (x y), µ (y (y x))}, (4) λ (x) max{ λ (x y), λ (y (y x))} for all x, y G. We now give an example of an IIF I of a K-algebra K. Example 11 Consider the K-algebra K=(G,,, e) on the Dihedral group G = {< a, b >: a 4 = e = b 2 = (ab) 2 }, where u = a 2, v = a 3, x = ab, y = a 2 b, z = a 3 b, is given by the following Cayley table: e a u v b x y z e e v u a b x y z a a e v u x y z b u u a e v y z b x v v u a e z b x y b b x y z e v u a x x y z b a e v u y y z b x u a e v z z b x y v u a e We define an interval-valued intuitionistic fuzzy set = ( µ, λ ) : G D[0, 1] D[0, 1] by { [0.4, 0.5] if x = e, µ (x) := [0.2, 0.3] if x e, λ (x) := { [0.06, 0.4] if x = e, [0.07, 0.2] if x e. By routine computations, = ( µ, λ ) can be verified to be an IIF I of K. The following Propositions are obvious. Proposition 12 If = ( µ, λ ) is an IIF I of K, then the level subsets U( µ ; s) L( λ ; s) are ideals of K for every s Im( µ ) Im( λ ) D[0, 1], where Im( µ ) Im( λ ) are sets of values of µ λ, respectively. Proposition 13 If all nonempty level subsets U( µ ; s) L( λ ; s) of an interval-valued intuitionistic fuzzy set = ( µ, λ ) are ideals of K, then is an IIF I of K. Definition 14 Let = ( µ, λ ) be an intervalvalued intuitionistic fuzzy set on G. Pick s, t D[0, 1] such that s + t [1, 1]. Then the set G ( s, t) := {x G s µ (x), λ (x) t} is called an ( s, t)-level subset of. The set of all ( s, t) Im( µ ) Im( λ ) such that s + t [1, 1] is called the image of = ( µ, λ ). Obviously, G ( s, t) = U( µ, s) L( λ, t). Theorem 15 n interval-valued intuitionistic fuzzy set =( µ, λ ) of K is an IIF I of K if only if G ( s, t) is an ideal of K for every ( s, t) Im( µ ) Im( λ ) with s + t [1, 1]. Proof: We only need to prove the necessity because the sufficiency is trivial. ssume that G ( s, t) is an ideal of K = ( µ, λ ) an interval-valued intuitionistic fuzzy set on K. It is easy to see that µ (e) µ (x) λ (e) λ (x). Consider x, y G such that (x) = ( s, t) (y) = ( s 1, t 1 ), that is, µ (x) = s, λ (x) = t, µ (y) = s 1 λ (y) = t 1. Without loss of generality, we may assume that ( s, t) ( s 1, t 1 ), i.e., s s 1 t 1 t. Then G ( s 1, t 1 ) G ( s, t), i.e., x, y G( s, t). This implies that x y, y (y x) G ( s, t) since G( s, t) is an ideal of K. Hence, we deduce that µ (x) s = min{ µ (x y), µ (y (y x))}, λ (x) t = min{ λ (x y), λ (y (y x))}. This shows that = ( µ, λ ) is an IIF I of K. Theorem 16 Let = ( α, λ ) B = ( β B, ν B ) be IIF Is of K. Then the functions α β B : G D[0, 1] λ ν B : G D[0, 1] defined by ( x G) (( α β B )(x) = min{ α (x), β B (x)}), ( x G) (( λ ν B )(x) = max{ λ (x), ν B (x)}) are IIF Is of K. Proof: For every x G, we have ( α β B )(e) = min{ α (e), β B (e)} min{ α (x), β B (x)} = ( α β B )(x), ( λ ν B )(e) = max{ λ (e), ν B (e)} max{ λ (x), ν B (x)} = ( λ ν B )(x). ISSN: Issue 9, Volume 7, September 2008

5 WSES TRNSCTIONS on MTHEMTICS Muhammad kram, Karamat H. Dar, Hence, for x, y G, we deduce that ( α β B )(x) = min{ α (x), β B (x)} min{min{ α (x y), α (y (y x))}, min{ β B (x y), β B (y (y x))}} = min{min{ α (x y), β B (x y)}, min{ α (y (y x)), β B (y (y x))}} = min{( α β B )(x y), ( α β B )(y (y x))}, ( λ ν B )(x) = max{ λ (x), ν B (x)} max{max{ λ (x y), λ (y (y x))}, max{ ν B (x y), ν B (y (y x))}} = max{max{ λ (x y), ν B (x y)}, max{ λ (y (y x)), ν B (y (y x))}} = max{( λ ν B )(x y), ( λ ν B )(y (y x))}. This shows that α β B λ ν B are IIF Is of K. Notation 17 Denote the family of all IIF Is of K by IIF I(K). For any t D[0, 1], define on IIF I(K) two binary relations,say U t L t by : (, B) U t (, B) L t U( µ ; t) = U( µ B ; t) L( λ ; t) = L( λ B ; t), respectively, where = ( µ, λ ), B = ( µ B, λ B ). Obviously, U t L t are equivalent relations on IIF I(K). For any = ( µ, λ ) IIF I(K), denote the system of all equivalence classes of modulo U t (resp. L t ) by IIF I(K)/U t (resp. IIF I(K)/L t ) for any element in IIF I(K), write [] (resp. U t [] ). L t Then IIF I(K)/U t = { [] = ( µ U t, λ ) SIF I(K) } (resp. IIF I(K)/L t = { [] = ( µ L t, λ ) IIF I(K) }). Definition 18 Let I(K) be the family of all ideals of K t D[0, 1]. Define two mappings f t from IIF I(K) to I(K) { } by () = g t f t U( µ ; t), () = L( λ g t ; t) for all = ( µ, λ ) IIF I(K). Then are well-defined IIF I f t g t functions. Theorem 19 For any t D(0, 1), the IIF I- functions f t g t are surjective from IIF I(K) to I(K) { }. Proof: Let t 0 D(0, 1). Then, it is clear that = ([0, 0], [1, 1]) is an IIF I(K), where [0, 0] [1, 1] are interval-valued fuzzy sets in K defined by [0, 0](x) = [0, 0] [1, 1](x) = [1, 1], for all x K. Obviously f t ( 0) = U([0, 0]; t) = = L([1, 1]; t) = ( 0). Let B I(K). For B = (χ g t B, χ B ) IIF I(K), we have ( f t B) = U(χ B ; t) = B ( g t B) = L(χ B ; t) = B. Hence are surjective. f t g t Theorem 20 The quotient sets IIF I(K)/U t IIF I(K)/L t are equipotent to I(K) { } for every t D(0, 1). Proof: For t D(0, 1), let f t (resp. g ) be a t mapping from IIF I(K)/U t (resp. IIF I(K)/L t ) to I(K) { } defined by f t ([] ) = () U t f t (resp. g t ([] ) = L t g t () ) for all = ( µ, λ ) IIF I(K). If U( µ ; t) = U( µ B ; t) L( λ ; t) = L (β B ; t) for = ( µ, λ ) B = ( µ B, β B ) in IIF I(K), then (, B) U t (, B) L t. Thus [Ã] = [ B] [Ã] = [ B]. This proves U t U t L t L t that the mappings f t g t are injective. Now let = D I(K). For D = (χ D, χ D ) IIF I(K), we have f t ([ D] U t ) = f t ( D) = U(χ D ; t) = D g t ([ D] L t ) = g t ( D) = L(χ D ; t) = D. Finally, for 0, we get f t ([ 0] U t ) = f t ( 0) = U([0, 0]; t) = g t ([ 0] L t ) = g t ( 0) = L([1, 1]; t) =. This shows that f are surjective. t g t ISSN: Issue 9, Volume 7, September 2008

6 WSES TRNSCTIONS on MTHEMTICS Muhammad kram, Karamat H. Dar, Definition 21 For any t D[0, 1], we define another relation R t on IIF I(K) by: (, B) R t G ( t, t) = G( t, t), B where G ( t, t) = U( µ ; t) L( λ ; t) R ( t, t) B = U( µ B ; t) L( λ B ; t). The relation R t is clearly an equivalent relation on IIF I(K). Theorem 22 For any t D(0, 1) the mapping : IIF I(K) I(K) { } defined by ϕ t ϕ t () = G ( t, t) is surjective. Proof: Let t D(0, 1). Then ( 0) = G ϕ t ( t, t) = U([0, 0]; t) L([1, 1]; t) =. For any H IIF I(K), there exists H = (χ H, χ H ) IIF I(K) such that ( ϕ t H) = G ( t, t) = U(χ H; t) L(χ H ; t) = H. Hence is surjective. ϕ t Theorem 23 For any t D(0, 1), the quotient set IIF I(K)/R t is equipotent to I(K) { }. Proof: Let t D(0, 1) let ϕ t : IIF I(K)/R t I(K) { } be a mapping defined by ϕ t ([Ã] R t ) = ϕ t (Ã) for all [Ã] R t IIF I(K)/R t. If ϕ t ([Ã] R t ) = ϕ t ([ B] R t ) for any [Ã] R t, [ B] G t IIF I(K)/R t, then G ( t, t) = G ( t, t) B, i.e., (Ã, B) R t. It follows that [] = [ B] so is injective. Moreover, R t R t ϕ t ϕ t ([ 0] ) = ϕ R t t( 0) = G ( t, t) =. For any H I(K) 0 we have H = (χ H, χ H ) IIF I(K) ϕ t ([ H] ) = ( H) R t ϕ t = G H( t, t) = U(χ H ; t) L(χ H ; t) = H. This shows that ϕ is surjective. t 4 rtinian Noetherian K- algebras Definition 24 K-algebra K is said to be Noetherian if every ideal of K is finitely generated. We say that K satisfies the ascending chain condition on IIF Is if for every ascending sequence I 1 I 2 I 3 of ideals of K, there exists a natural number n such that I n = I k for all n k. We call K satisfies the Interval-valued intuitionistic fuzzy ascending chain condition if for every ascending sequence 1 2 of IIF Is in K, there exists a natural number n such that µ n = µ k, for all n k. The following lemma is obvious. Lemma 25 Let = ( µ, λ ) be an IIF I of K let s, t Im( µ ), s, t Im( λ ). Then U( µ ; s) = U( µ ; t) s = t, L( λ ; s) = L( λ ; t) s = t. Theorem 26 Let K be a K-algebra. Then every IIF I of K has finite number of values if only if K is rtinian. Proof: Suppose that every IIF I of K has finite number of values but K is not rtinian. Then there exists a strictly descending chain G = of ideals of K. Conditions (1) (2) of Definition 3 hold obviously. In order to prove (3), we consider an interval-valued intuitionistic fuzzy set = ( µ, λ ) given by µ (x) := λ (x) := [1, 1] µ (x). 1 [ n+1, n n+1 ] if x n \ n+1, n = 0, 1,... [1, 1] if x n, n=0 Let x, y K. Then x y, y (y x) n \ n+1 for some n = 0, 1, 2,, either x y / n+1 or y (y x) / n+1. We now let x, y U n \ n+1, for k n. Then, it follows that µ (x) = 1 [ n + 1, n n + 1 ] [ 1 k + 1, k k + 1 ] min( µ (x y), µ (y (y x))). Thus µ is an IIF I of K µ has infinite number of different values. In a similar way, λ is also an IIF I of K λ has infinite number of different values. Hence = ( µ, λ ) is an IIF I of K has infinite number of different values. This contradiction proves that the K algebra K is rtinian. Conversely, if the K-algebra K is rtinian then we can let = ( µ, λ ) be an IIF I of K. Suppose that ISSN: Issue 9, Volume 7, September 2008

7 WSES TRNSCTIONS on MTHEMTICS Muhammad kram, Karamat H. Dar, Im( µ ) is infinite. Then, we can observe that every subset of D[0, 1] contains either a strictly ascending sequence or a strictly descending sequence. Now, let [s 1, t 1 ] < [s 2, t 2 ] < [s 3, t 3 ] < be a strictly ascending sequence in Im( µ ). Then U( µ ; [s 1, t 1 ]) U( µ ; [s 2, t 2 ]) U( µ ; [s 3, t 3 ]) is strictly descending chain of ideals of K. Since K is rtinian, there exists a natural number i such that U( µ ; [s i, t i ]) = U( µ ; [s i+n, t i+n ]) for all n 1. Since [s i, t i ] Im( µ ) for all i, by applying Lemma 25, we have s i = s i+n, t i = t i+n, for all n 1. This is a contradiction since s i, t i are distinct. On the other h, if [s 1, t 1 ] > [s 2, t 2 ] > [s 3, t 3 ] > is a strictly descending sequence in Im( µ ), then U( µ ; [s 1, t 1 ]) U( µ ; [s 2, t 2 ]) U( µ ; [s 3, t 3 ]) is an ascending chain of ideals of K. Since K is rtinian, there exists a natural number j such that U( µ ; [s j, t j ]) = U( µ ; [s j+n, t j+n ]) for all n 1. Since [s j, t j ] Im( µ ) for all j, by Lemma 25 s j = s j+n,t j = t j+n for all n 1, which is again a contradiction since s j, t j are distinct. This shows that Im( µ ) is finite. For Im( λ ), the proof is similar. The proof of the following theorem is routine we hence omit the proof. Theorem 27 Let a K-algebra K be rtinian. If = ( µ, λ ) is an IIF I of K, then U µ = Im( µ ) L λ = Im( λ ), where U µ L λ are families of all level ideals of K with respect to µ λ, respectively. Theorem 28 Let a K-algebra K be rtinian. If = ( µ, λ ) B = ( ν B, η B ) are IIF Is of K, then the following statements hold: (i) U µ = U νb Im( µ )=Im( ν B ) if only if µ = ν B, (ii) L λ = L ηb Im( λ )=Im( η B ) if only if λ = η B. Proof: (i) If µ = ν B, then U µ = U νb Im( µ ) = Im( ν B ). Now we suppose that U µ = U νb Im( µ ) = Im( ν B ). By Theorems 26 27, Im( µ ) = Im( ν B ) are finite U µ = Im( µ ) U νb = Im( ν B ). Let Im( µ ) = { t 1, t 2,, t n } Im( ν B ) = { s 1, s 2,, s n }, where t 1 < t 2 < < t n s 1 < s 2 < < s n. This shows that t i = s i for all i. We now prove that U( µ ; t i ) = U( ν B ; t i ) for all i. Note that U( µ ; t 1 ) = K = U( ν B ; t 1 ). Consider U( µ ; t 2 ) U( ν B ; t 2 ). Suppose that U( µ ; t 2 ) U( ν B ; t 2 ). Then U( µ ; t 2 ) = U( ν B ; t k ) for some k > 2 U( ν B ; t 2 ) = U( µ ; t j ) for some j > 2. If there exist x G such that µ (x) = t 2, then µ (x) < t j j > 2. (1) Since U( µ ; t 2 ) = U( ν B ; t k ), x U( ν B ; t k ). This implies that ν B (x) t k > t 2, k > 2. Thus x U( ν B ; t 2 ). Since U( ν B ; t 2 ) = U( µ ; t j ), x U( µ ; t j ). Hence µ (x) t j for some j > 2. (2) Clearly, (1) (2) contradict each other. Hence U( µ ; t 2 ) = U( ν B ; t 2 ). Continuing in this way, we eventually obtain U( µ ; t i ) = U( ν B ; t i ), for all i. Now let x G. Suppose that µ (x) = t i for some i. Then x / U( µ ; t j ) for all i + 1 j n. This implies that x / U( ν B ; t j ) for all i + 1 j n. But then we have ν B (x) < t j for all i + 1 j n. Suppose that ν B (x) = t m for some i m i. If i m, then x U( ν B ; t i ). On the other h, since µ (x) = t i, x U( µ ; t i ) = U( ν B ; t i ). Thus we arrive a contradiction. Hence i = m µ (x) = t i = ν B (x). Consequently µ = ν B. (ii) The proof is similar is omitted. We now characterize the Noetherian K-algebra in the following theorem. Theorem 29 K-algebra K is Noetherian if only if the set of values of IIF Is of K are well ordered subsets of D[0, 1]. Proof: Suppose that = ( µ, λ ) is an IIF I of K whose set of values is not a well ordered subset of D[0, 1]. Then there exists a strictly decreasing sequence [s n, t n ] such that µ (x n ) = [s n, t n ]. Denote by U n the set {x G µ (x) [s n, t n ]}. Then U 1 U 2 U 3 ISSN: Issue 9, Volume 7, September 2008

8 WSES TRNSCTIONS on MTHEMTICS Muhammad kram, Karamat H. Dar, is a strictly ascending chain of ideals of K. This clearly contradicts that K is Noetherian. Hence, Im( µ ) must be a well-ordered subset of D[0, 1]. Similarly, for Im( λ ). Conversely, assume that the set of values of an IIF I of K is a well ordered subset of D[0, 1] K is not Noetherian K-algebra. Then there exists a strictly ascending chain U 1 U 2 U 3 ( ) of ideals of K. Define an interval-valued intuitionistic fuzzy set = ( µ, λ ) on K by putting 1 [ k+1, 1 k ] for x k\ k 1, µ (x) := [0, 0] for x k, λ (x) := [1, 1] µ (x), k=1 Then, we can easily prove that = ( µ, λ ) is an IIF I of K. Since the ascending chain ( ) is not terminating, has a strictly descending sequence of values, this contradicts that the value set of an IIF I is well ordered. Consequently, K is Noetherian. Finally, we state a theorem of Noetherian K algebra. Since the proof is straightforward, we omit the proof. Theorem 30 If K is a Noetherian K-algebra, then every IIF I of K is finite valued. 5 Fully invariant characteristic IIFIs Definition 31 n ideal F of a K-algebra K is said to be a fully invariant ideal if f(f ) F for all f End(K), where End(K) is the set of all endomorphisms of K. n IIF I = ( µ, λ ) of L is called a fully invariant if µ f (x) = µ (f(x)) µ (x) λ f (x) = λ (f(x)) λ (x) for all x G f End(K). Theorem 32 If { i i I} is a family of IIF I fully invariant ideals of K, then i I i = ( i I µ i, λ i I i ) is an interval-valued intuitionistic fully invariant ideal of K, where µ i (x) = inf{ µ i (x) i I, x L}, i I i I λ i (x) = sup{ λ i (x) i I, x L}. Proof: It can be easily seen that i = ( µ i, λ i ) is an IIF I of K. Let x G f End(K). Then ( i I µ i ) f (x) = ( i I µ i )(f(x)) = inf{ µ i (f(x)) i I} inf{ µ i (x) i I} = ( i I µ i )(x), ( i I λ i ) f (x) = ( i I λ i )(f(x)) = sup{ λ i (f(x)) i I} sup{ λ i (x) i I} = ( i I λ i )(x). Hence i I i = ( i I µ i, λ i I i ) is an intervalvalued intuitionistic fully invariant ideal of K. Theorem 33 Let H be a nonempty subset of a algebra K = ( µ, λ ) an IIF Is defined by µ (x) = λ (x) = { [s2, t 2 ] if x H, [s 1, t 1 ] otherwise, { [α2, β 2 ] if x H, [α 1, β 1 ] otherwise, where [0, 0] [s 1, t 1 ] < [s 2, t 2 ] [1, 1], [0, 0] [α 2, β 2 ] < [α 1, β 1 ] [1, 1], [0, 0] [s i, t i ] + [α i, β i ] [1, 1] for i = 1, 2. If H is an interval-valued intuitionistic fully invariant ideal of K, then = ( µ, λ ) is an interval-valued intuitionistic fully invariant ideal of K. Proof: We can easily see that = ( µ, λ ) is an IIF I of K. Let x G f End(K). If x H, then f(x) f(h) H. Thus, we have µ f (x) = µ (f(x)) µ (x) = [s 2, t 2 ], λ f (x) = λ (f(x)) λ (x) = [α 2, β 2 ]. For if otherwise, then we have µ f (x) = µ (f(x)) µ (x) = [s 1, t 1 ], λ f (x) = λ (f(x)) λ (x) = [α 1, β 1 ]. Thus, we have verified that = ( µ, λ ) is an interval-valued intuitionistic fully invariant ideal of K. ISSN: Issue 9, Volume 7, September 2008

9 WSES TRNSCTIONS on MTHEMTICS Muhammad kram, Karamat H. Dar, Definition 34 n IIF I = ( µ, λ ) of K has the same type as an IIF I B = ( µ B, λ B ) of K if there exists f End(K) such that = B f, i.e., µ (x) µ B (f(x)), λ (x) λ B (f(x)) for all x G. Theorem 35 IIF Is of K have same type if only if they are isomorphic. Proof: We only need to prove the necessity part because the sufficiency part is obvious. If an IIF I = ( µ, λ ) of K has the same type as B = ( µ B, λ B ), then there exists ϕ End(K) such that µ (x) µ B (ϕ(x)), λ (x) λ B (ϕ(x)) x G. Let f : (K) B(K) be a mapping defined by f((x)) = B(ϕ(x)) for all x G, that is, f( µ (x)) = µ B (ϕ(x)), f( λ (x)) = λ B (ϕ(x)) x G. Then, it is clear that f is a surjective homomorphism. lso, f is injective because f( µ (x)) = f( µ (y)) for all x, y G implies µ B (ϕ(x)) = µ B (ϕ(y)). Whence µ (x) = µ B (y). Likewise, from f( λ (x)) = f( λ (y)) we conclude λ (x) = λ B (y) for all x G. Hence = ( µ, λ ) is isomorphic to B = ( µ B, λ B ). This completes the proof. Definition 36 n ideal C of K is said to be characteristic if f(c) = C for all f ut(k), where ut(k) is the set of all automorphisms of K. n IIF I = ( µ, λ ) of K is called a characteristic if µ (f(x)) = µ (x) λ (f(x)) = λ (x) for all x G f ut(k). Lemma 37 Let = ( µ, λ ) be an IIF I of K let x G. Then µ (x) = t, λ (x) = s if only if x U( µ ; t), x / U( µ ; s) x L( λ, s), x / L( λ, t) for all s > t. Proof: Straightforward. Theorem 38 n IIF I is characteristic if only if each its level set is a characteristic ideal. Since for each x U( µ ; t) there exists y G such that f(y) = x we have µ (y) = µ (f(y)) = µ (x) t, whence we conclude y U( µ ; t). Consequently x = f(y) f(u( µ ; t)). Hence f(u( µ ; t) = U( µ ; t). Similarly, f(l( λ ; s)) = L( λ ; s). This proves that U( µ ; t) L( λ ; s) are characteristic. Conversely, if all levels of = ( µ, λ ) are characteristic ideals of K, then for x G, f ut(k) µ (x) = t < s = λ (x), by Lemma 37, we have x U( µ ; t), x / U( µ ; s) x L( λ ; s), x / L( λ ; t). Thus f(x) f(u( µ ; t)) = U( µ ; t) f(x) f(l( λ ; s)) = L( λ ; s), i.e., µ (f(x)) t λ (f(x)) s. For µ (f(x)) = t 1 > t, λ (f(x)) = s 1 < s we have f(x) U( µ ; t 1 ) = f(u( µ ; t 1 ), f(x) L( λ, s 1 ) = f(l( λ ; s 1 )), whence x U( µ ; t 1 ), x L( µ ; s 1 ). This is a contradiction. Thus µ (f(x)) = µ (x) λ (f(x)) = λ (x). So, = ( µ, λ ) is characteristic. s a consequence of the above results we obtain the following theorem. Theorem 39 If = ( µ, λ ) is a fully invariant IIF I, then it is characteristic. 6 Conclusions In the present paper, we have presented some properties of interval-valued intuitionistic fuzzy ideals of K-algebras. It is clear that the most of these results can be simply extended to interval-valued intuitionistic (T, S)-fuzzy ideals, where S T are given imaginable triangular norms. In our opinion the future study of (interval-valued intuitionistic) fuzzy ideals of K-algebras can be connected with (1) investigating (α, β)- fuzzy ideals; (2) finding interval-valued intuitionistic fuzzy sets triangular norms. The obtained results can be used in various fields such as artificial intelligence, signal processing, multiagent systems, pattern recognition, robotics, computer networks, genetic algorithm, neural networks, expert systems, decision making, automata theory medical diagnosis. Proof: Let an IIF I = ( µ, λ ) be characteristic, t Im( µ ), f ut(k), x U( µ ; t). Then µ (f(x)) = µ (x) t], which means that f(x) U( µ ; t). Thus f(u( µ ; t)) U( µ ; t). References: [1] M. kram, K. H. Dar, Y. B. Jun E. H. Roh, Fuzzy structures of K(G)-algebra, South- ISSN: Issue 9, Volume 7, September 2008

10 WSES TRNSCTIONS on MTHEMTICS Muhammad kram, Karamat H. Dar, east sian Bulletin of Mathematics Vol. 31, Issue 4, 2007, pp [2] M. kram K. H. Dar, Fuzzy ideals of K- algebras, nnals of University of Craiova, Math. Comp. Sci. Ser Vol. 34, 2007, pp [3] M. kram, Bifuzzy ideals of K-algebras, WSES Transactions on Mathematics Vol.7, Issue 5, 2008, pp [4] M. kram, (, q) fuzzy ideals of K- algebras, rs Combinatoria Vol. 89, 2008, pp [5] K. T. tanassov, Intuitionistic fuzzy sets, VII ITKRs Session, Sofia (deposed in Central Science-Technical Library of Bulgarian cademy of Science 1697/84), 1983 (in Bulgarian). [6] K. T. tanassov, Intuitionistic fuzzy sets, Fuzzy Sets Systems Vol. 20, Issue 1, 1986, pp [7] K. T. tanassov, Intuitionistic Fuzzy Sets: Theory pplications, Studies in fuzziness soft computing, vol. 35, Heidelberg, New York, Physica-Verl., [8] K. T. tanassov G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets Systems, Vol. 31, Issue 3, 1989, pp [9] R. Biswas, Rosenfeld s fuzzy subgroups with interval-valued membership functions, Fuzzy sets Systems Vol. 63, 1994, pp [10] Y. U. Cho Y. B. Jun, Fuzzy algebras on K(G)-algebras, Journal of pplied Mathematics Computing Vol. 22, 2006, pp [11] K. H. Dar M. kram, On a K-algebra built on a group, Southeast sian Bulletin of Mathematics Vol. 29, Issue 1, 2005, pp [12] K. H. Dar M. kram, Characterization of a K(G)-algebras by self maps, Southeast sian Bulletin of Mathematics Vol. 28, Issue 4, 2004, pp [13] K. H. Dar M. kram, On K- homomorphisms of K-algebras, International Mathematical Forum Vol. 2, Issue 46, 2007, pp [14] K. H. Dar M. kram, On subclasses of K(G)-algebras, nnals of University of Craiova, Math. Comp. Sci. Ser Vol. 33, 2006, pp [15] S. K. De, R. Biswas. R. Roy, n application of intuitionistic fuzzy sets in medical diagnosis, Fuzzy Sets Systems, Vol. 117, Issue 2, 2001, pp [16] G. Deschrijver, rithmetric operators in interval-valued fuzzy theory, Information Sciences Vol. 177, 2007, pp [17] G. Deschrijver E.E. Kerre, On the relationship between some extensions of fuzzy set theory, Fuzzy Sets Systems Vol. 133, 2003, pp [18] D. Dubois, S. Gottwald, P. Hajek, J. Kacprzyk H. Prade, Terminological difficulties in fuzzy set theory The case of Intuitionistic Fuzzy Sets, Fuzzy Sets Systems Vol. 156, Issue 3, 2005, pp [19] D. Dubois H. Prade, Fuzzy sets systems:theory pplications, cademic Press, 1997, US. [20] Q. P. Hu X. Li, On BCH-algebras, Math. Seminar Notes Vol. 11, 1983, pp [21] Y. Imai K. Iséki, On axiom system of propositional calculi XIV, Proc. Japonica cad Vol. 42, 1966, pp [22] K. Iséki, n algebra related with a propositional calculus, Proc. Japan cad Vol. 42, 1966, pp [23] Y. B. Jun C. H. Park, Fuzzy idelas of K(G)- algebras, Honam Mathematical Journal Vol. 28, 2006, pp [24] J. Meng Y. B. Jun, BCK-algebras, Kyung Moon Sa Co. Seoul, Korea, [25] J. Meng X. Guo, On fuzzy ideals of BCK/BCI-algebras, Fuzzy Sets Systems Vol. 149, Issue 3, 2005, pp [26] D. Minchev, n intuitionistic fuzzy sets application in multiagent systems of metamorphic robotic systems, Intelligent Systems, Proceedings First International IEEE Symposium, Vol. 3, 2002, pp [27] N. Kuroki, On fuzzy semigroups, Information Sciences Vol. 53, 1991, pp [28]. Rosenfeld, Fuzzy groups, J. Math. nal. ppl. Vol. 35, 1971, pp [29] J. Neggers H. S. Kim, On B-algebras, Matematicki Vesnik Vol. 54, 2002, pp [30] L.. Zadeh, Fuzzy sets, Information Control Vol. 8, 1965, pp [31] L.. Zadeh, The concept of a lingusistic variable its application to approximate reasoning, Information Sciences Vol. 8, 1975, pp [32] L.. Zadeh, Toward a generalized theory of uncertainty (GTU)-an outine, Information Sciences Vol. 172, 2005, pp ISSN: Issue 9, Volume 7, September 2008