Interval-Valued Fuzzy KUS-Ideals in KUS-Algebras

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1 IOSR Jornal of Mathematics (IOSR-JM) e-issn: Volme 5, Isse 4 (Jan. - Feb. 03), PP 6-66 Samy M. Mostafa, Mokhtar.bdel Naby, Fayza bdel Halim 3, reej T. Hameed 4 and Department of Mathematics, Faclty of Edcation, in Shams niversity, Roxy, Cairo, Egypt. 3 and 4 Department of Pre Mathematics, Faclty of Sciences, in Shams niversity, Cairo, Egypt. 4 Department of Mathematics, College of Edcation for Girls, niversity of Kfa, Najaf, Iraq. bstract: In this paper the notion of interval-valed fzzy KS-ideals (briefly i-v fzzy KS-ideal) in KSalgebras is introdced. Several theorems are stated and proved. The image and inverse image of i-v fzzy KSideals are defined and how the homomorphic images and inverse images of i-v fzzy KS-ideals become i-v fzzy KS-ideals in KS-algebras is stdied as well. Keywords: KS-algebras, fzzy KS-ideals, interval-valed fzzy KS-sb-algebras, interval-valed fzzy KS-ideals in KS-algebras. 000 Mathematics Sbject Classification: 06F35, 03G5, 03B5, 94D05. I. Introdction W.. Ddek and X. Zhang ([],[3]) stdied ideals and congrences of BCC-algebras. C. Prabpayak and. eerawat ([5],[6]) introdced a new algebraic strctre which is called K-algebras and investigated some related properties. The concept of a fzzy set, was introdced by.. Zadeh [8]. O.G. Xi [7] applied the concept of fzzy set to BCK-algebras and gave some of its properties. In [9],.. Zadeh made an extension of the concept of fzzy set by an interval-valed fzzy set (i.e., a fzzy set with an interval-valed membership fnction). This interval-valed fzzy set is referred to as an i-v fzzy set. He constrcted a method of approximate inference sing his i-v fzzy sets. In [], R. Biswas defined interval-valed fzzy sbgrops and investigated some elementary properties. Recently S.M. Mostafa, and et al ([4]) introdced a new algebraic strctre, called KS-algebra, They have stdied a few properties of these algebras, the notion of KS-ideals on KS-algebras was formlated and some of its properties are investigated. In this paper, sing the notion of interval-valed fzzy set by.. Zadeh, we introdce the concept of an interval-valed fzzy KS-ideals (briefly, i-v fzzy KS-ideals) of a KS-algebra, and stdy some of their properties. sing an i-v level set of an i-v fzzy set, we state a characterization of an i-v fzzy KS-ideals. We prove that every KS-ideals of a KSalgebra X can be realized as an i-v level KS-ideals of an i-v fzzy KS ideals of X. In connection with the notion of homomorphism, we stdy how the images and inverse images of i-v fzzy KS-ideals become i-v fzzy KS-ideals. II. The Strctre of KS-algebras: In this section we inclde some elementary aspects that are necessary for this paper Definition.([4]). et (X;,0) be an algebra with a single binary operation ( ). X is called a KS-algebra if it satisfies the following identities: (ks ) : (z y) (z x) = y x, (ks ) : 0 x = x, (ks 3 ) : x x = 0, (ks 4 ) : x (y z) = y (x z),for any x, y, z X, In what follows, let (X;,0) be denote a KS-algebra nless specified. For brevity we also call X a KS-algebra. In X we can define a binary relation ( ) by: x y if and only if y x = 0. emma. ([4]). In any KS-algebra (X;,0), the following properties hold: for all x, y, z X; a) x y = 0 and y x = 0 imply x = y, b) y [(y z) z] = 0, c) (0 x) (y x) = y 0, d) x y implies that y z x z and z x z y, e) x y and y z imply x z, f) x y z implies that z y x. Definition.3 ([4]). nonempty sbset I of a KS-algebra X is called a KS-ideal of X if it satisfies: for all x, y, z X, (Iks ) (0 I), 6 Page

2 (Iks ) (z y) I and (y x) I imply (z x) I. Definition.4([8]). et X be a nonempty set, a fzzy sbset μ in X is a fnction μ : X [0,]. Definition.5([4]). et X be a KS-algebra and, a fzzy sbset μ in X is called a fzzy KS-sb-algebra of X if μ(x y) min {μ(x), μ(y)}, for all x, y X. Definition.6([4]). et X be a KS-algebra, a fzzy sbset μ in X is called a fzzy KS-ideal of X if it satisfies the following conditions: for all x, y, z X, (Fks ) μ (0) μ (x), (Fks ) μ (z x) min {μ (z y), μ (y x)}. Proposition.7([4]). The intersection of any finite sets of fzzy KS-ideals of KS-algebra X is also a fzzy KS-ideal. Definition.8([9]). et X be a set and μ be a fzzy sbset of X, for t [0,], the set t ={ x X μ(x) t} is called a level sbset of μ. Theorem.9([4]). fzzy sbset μ of KS-algebra X is a fzzy KS-ideal of X if and only if, for every t [0,], t is either empty or a KS-ideal of X. Definition.0([6]).et (X ;,0) and (Y; `,0`) be nonempty sets. The mapping f : (X;,0) (Y; `,0`) is called a homomorphism if it satisfies f (x y) = f (x) ` f (y) for all x, y X. The set {xx f (x) = 0'} is called the Kernel of f and is denoted by Ker f. Definition. ([6]). et f : (X;,0) (Y; ',0') be a mapping from the set X to a set Y. If μ is a fzzy sbset of X, then the fzzy sbset β of Y defined by: sp{ ( x) : x f ( y)} if f ( y) { x X, f ( x) y} f ( )( y) 0 is said to be the image of μ nder f. Similarly if β is a fzzy sbset of Y, then the fzzy sbset μ = (β о f ) in X ( i.e the fzzy sbset defined by μ (x) = β ( f (x)) for all x X ) is called the pre-image of β nder f. Theorem.([4]). n into homomorphic pre-image of a fzzy KS-ideal is a fzzy KS-ideal. Theorem.3([4]). n into homomorphic image of a fzzy KS-ideal is a fzzy KS-ideal. III. Interval-valed fzzy KS-ideal of KS-algebra Remark 3.([9]). n interval-valed fzzy sbset (briefly i-v fzzy sbset ) defined in the set X is given by = {(x, [ (x), (x)])}, for all x X. (briefly, it is denoted by = [, ] where and are any two fzzy sbsets in X sch that (x) (x) for all x X. et ~ (x)= [ (x), (x)], for all x X and let D[0,] be denotes the family of all closed sb-interval of [0,]. It is clear that if (x) = (x) = c, where 0 c, then ~ (x) = [c, c] in D[0,], then ~ (x)[0,], for all x X. Therefore the i-v fzzy sbset is given by : = {(x, ~ (x))}, for all x X where ~ : X D[0,]. Now we define the refined minimm (briefly r min) and order on elements D = [a, b ] and D = [a, b ] of D[0, ] as follows: r min( D, D ) = [min {a,a }, min {b,b }], D D a a and b b. Similarly we can define ( ) and (=). In what follows, let X denote a KS-algebra nless specified, we begin with the following definition. Definition 3.. n i-v fzzy sbset in X is called an i-v fzzy KS-sb-algebra of X if min{ ~ (x), ~ (y)}, for all x, y X. Example 3.3. et X = {0,,, 3} in which the operation (as in example ( ) be define by the following table: ~ (x y) r 6 Page

3 Then (X;,0) is a KS-algebra. Define a fzzy sbset μ: X [0,] by 0.7 if x {0,} μ(x) =. I ={0,} is a KS-ideal of X. Rotine calclation given that μ is a fzzy 0.3 KS-ideal of X. Define ~ (x) as follows: [0.3,0.9] ~ (x) = [0.,0.6] KS-sb-algebra. if x {0,}. It is easy to check that is an i-v fzzy ~ (0) ~ (x), for all x X. Proposition 3.4. If is an i-v fzzy KS-sb-algebra of X, then Proof. For all x X, we have ~ (0) = ~ (x * x) r min{ ~ (x), ~ (x)} = r min {[ (x), (x)], [ (x), (x)]}= r min {[ (x), (x)]}= ~ (x). Proposition 3.5. et be an i-v fzzy KS-sb-algebra of X, if there exist a seqence { X n } in X sch that lim ~ (xn) = [,], then ~ (0) = [, ]. n n ~ (0) ~ (x), for all x X. Then Proof. By proposition (3.4), we have ~ (0) ~ (x n ), for every positive integer n, Consider the ineqality [,] ~ (0) lim ~ (xn) = [,]. Hence ~ (0) = [,]. Definition 3.6. n i-v fzzy sbset = {(x, ~ (x))}, x X in KS-algebra X is called an interval-valed fzzy KS-ideal (i-v fzzy KS-ideal, in short) if it satisfies the following conditions: ( ) ~ (0) ~ (x), ( ) ~ (z x) r min{ ~ (z y), ~ (y x)}, for all x, y, z X. Example 3.7. et X = {0,,, 3} as in example (3.3). Define ~ (x) as follows: ~ (x) = [0.3,0.9] [0.,0.6] if x {0,} Theorem 3.8. n i-v fzzy sbset = [ and are fzzy KS-ideals of X. Proof. If [ (z x), and (z x)]. It is easy to check that is an i-v fzzy KS-ideal of X., ] in X is an i-v fzzy KS-ideal of X if and only if are fzzy KS-ideals of X. For any x, y, z X. Observe [ min { (z y), (y x)}, min { (z y), (y x)}] = r min {[ (z y), (z y)], [ (y x), (y x)]} = r min { ~ (z y), ~ (y x)]. From what was mentioned above we can conclde that is an i-v fzzy KS-ideal of X. (z x)]= ~ (z x) = Conversely, sppose that is an i-v fzzy KS-ideal of X. For all x, y, z X we have [ ~ (z x) r min{ ~ (z y), ~ (y x)} = r min{[ (z y), (z y)], [ (y x), (y x)]} (z x), 63 Page

4 = [ min { min{ (z y), (z y), (y x)} and (z x) min{ (z y), (y x) }, min { (y x)}. (z y), (y x) }]. Therefore, (z x) Hence, we get that and are fzzy KS-ideals of X. Theorem 3.9. et and be i-v fzzy KS-ideals of a KS-algebra X. Then is an i-v fzzy KS-ideal of X. ~ Proof. (0) = [ (0), (0)] [ (x), (x)]= ~ (x). Sppose x, y, z X sch that (z*y) and (y*x). Since and are i-v fzzy KS-ideals of X, then by the theorem (3.8), we get ~ (z x) = [ (z x), (z x) ] = [min{ (z y), = [min{ ( z y), ~ = r min{ (z y), (y x)}, min{ (z y), (z y)}, min{ (y x), ~ (y x)}]. (y x)}] (y x)}] Corollary 3.0. et { iλ} be a family of i-v fzzy KS-ideal of X. Then i is also an i-v fzzy KS-ideal of X. i i Theorem 3.. et X be a KS-algebra and be an i-v fzzy sbset in X. Then is an i-v fzzy KSideal of X if and only if the nonempty set ~ (;[, ]):={ x X ~ (x) [, ]} is a KS-ideal of X, for every [, ] D[0, ]. We call ~ ( ; [, ] ) the i-v level KS-ideal of. Proof. ssme that is an i-v fzzy KS-ideal of X and let [, ] D[0, ] be sch that (z y), (y x) ~ ( ; [, ] ), then ~ (y x)} r min{[, ], [, ] } = [, ] and so (z x) ~ ( ; ~ (z x) r min{ ~ (z y), [, ] ). Then ~ ( ; [, ] ) the i-v level KS-ideal of. Conversely, assme that ~ ( ; [, ] ) is a KS-ideal of X, for every [, ] D[0, ].In the contrary, sppose that there exist x 0, y 0, z 0 X, sch that et ~ (z 0 x 0 ) < r min{ ~ (z 0 y 0 ), ~ (y 0 x 0 )}. ~ (z 0 y 0 ) = [, ], ~ ( y 0 x 0 ) = [ 3, 4 ] and ~ (z 0 x 0 ) = [, ]. If [, ] < r min{ [, ], [ 3, 4 ]} = min { min {, }, min { 3, 4 }}. So < min {, } and < min{ 3, 4 }. Consider [, ] = We find that { ~ (z 0 x 0 ) + r min{ ~ (z 0 y 0 )), ~ ( y 0 x 0 )} } [, ] = {[, ]+ r min{[, ], [ 3, 4 ]}} = {( + min{, 3 }), ( + min{, 4 })]. Therefore min {, 3 } > = ( + min{, 3 }) >, 64 Page

5 min {, 4 } > = ( + min{, 4 }) >. Hence [min {, 3 }, min {, 4 }] > [, ] > [, ] = ~ (z 0 * x 0 ), so that, (z 0 x 0 ) ~ ( ; [, ] ). which is a contradiction, since ~ (z 0 y 0 ) = [, ] [min{, 3 }, min {, 4} ] > [, ]. ~ ( y 0 x 0 ) =[ 3, 4 ] [min{, 3 }, min {, 4} ] > [, ], imply that (z 0 y 0 ),(y 0 x 0 ) ~ ( ; [, ] ). Then ~ (z x) r min{ ~ (z y), ~ (y x)}, for all x, y, z X. Theorem 3.. Every KS-ideal of a KS-algebra X can be realized as an i-v level KS-ideal of an i-v fzzy KS-ideal of X. Proof. et Y be a KS-ideal of X and let be an i-v fzzy sbset on X defined by [, ] if x y ~ (x) =. [0,0] Where, [0, ] with <. It is clear that ~ ( ; [, ]) = Y. We show that is an i-v fzzy KS-ideal of X. et x, y, z X. If (z y), (y x) Y, then (z x) Y, and therefore ~ (z x) = [, ] = r min{[, ],[, ]} =r min{ ~ (z y)), ~ (y x)}. ~ (z y) = [0,0] = ~ (y x) and so If (z y), (y x) Y, then ~ (z x) [0,0] = r min{[0,0],[0,0]} = r min{ ~ (z y), ~ (y x)}, ~ (z y) =[, ] and ~ (y x)= [0,0], then ~ (z x) [0,0] If (z y)y and (y x) Y, then =r min{[, ],[0,0]} = r min{ ~ (z y), ~ (y x)}. Similarly for the case (z y) Y and (y x) Y we get ~ (z x) r min{ ~ (z y)), ~ ( y x)}. Therefore is an i-v fzzy KS-ideal of X, the proof is complete. Proposition 3.3. et X be a KS-algebra, B be a fzzy sbset on X and let be an i-v fzzy sbset on X [, ] if x y defined by ~ (x) =. Where [0,0] <. If is an i-v fzzy KS-sb-algebra of X, then B is a fzzy KS- sb-, (0, ] with algebra of X. Proof. Clear. Theorem 3.4. If is an i-v fzzy KS-ideal of X, then the set X M ~ := {x X ~ (x)= ~ (0)} is a KS-ideal of X. Proof. et (z y), (y x) X M ~. Then ~ (z y) = ~ (0) = ~ (y x), and so ~ (z x) r min{ ~ (z y), ~ (y x)} = r min{ ~ (0), ~ (0)} = ~ (0). Combining this with condition () of definition (3.6), we get Hence X M ~ is a KS-ideal of X. ~ (z x) = ~ (0), that is (z x) X M ~. IV. Homomorphism of KS-algebra Definition 4. ([]). et f : (X;,0) (Y; ',0') be a mapping from set X into a set Y. let B be an i-v fzzy sbset in Y. Then the inverse image of B, denoted by membership fnction given by f (B), is an i-v fzzy sbset in X with the 65 Page

6 f (x) = ~ (B) B ( f (x )), for all x X. Proposition 4. ([]). et f be a mapping from set X into a set Y, let m = [m,m ], and n = [n,n ] be i-v fzzy sbsets in X and Y respectively. Then () f (n) = [ f (n ), f (n )], () f (m) = [ f (m ), f (m )]. Theorem 4.3. et f be homomorphism from a KS-algebra X into a KS-algebra Y. If B is an i-v fzzy KS-ideal of Y, then the inverse image Proof. Since B= [ ) and ( f (B) of B is an i-v fzzy KS-ideal of X. B, B ) are fzzy KS-ideals of Y. sing theorem (.), we know B ] is an i-v fzzy KS-ideal of Y, it follows that from theorem (3.8), that ( fzzy KS-ideals of X. Hence by proposition (4.), we conclde that f ( B ) and f (B) = [ f ( f ( B ), ) are B f ( B B )] is an i-v fzzy KS-ideal of X. Definition 4.4 ([9]).et f be a mapping from a set X into a set Y. let be a an i-v fzzy set in X. then the image of, denoted by f (), is the i-v fzzy sbset in Y with membership fnction denoted by : ~ sp (z) if f (y), y Y ~ (x)= z f (y) f () where f [0,0] ( y):={x X f (x) = y}. Theorem 4.5. et f be a homomorphism from a KS-algebra X into a KS-algebra Y. If is an i-v fzzy KS-ideal of X, then f () of is an i-v fzzy KS-ideal of Y. Proof. ssme that = [ ( ) and (,, ] is an i-v fzzy KS-ideal of X. it follows that from theorem (3.8), that ) are fzzy KS-ideals of X. sing theorem (.3), that the images f ( ) and f ) are fzzy KS-ideal of Y. Hence by proposition (4.), we conclde that f ()= [ f ( ), f ( ( is an i-v fzzy KS-ideal of Y. )] References [] Biswas R., Rosenfeld s fzzy sbgrops with interval valed membership, fnction, Fzzy Sets and ystems, vol.63, no. (994), [] Ddek W.. and Zhang X., On ideal and congrences in BCC-algebras, Czechoslovak Math. Jornal, vol.48, no. 3 (998), -9. [3] Ddek W.., On proper BCC-algebras, Bll. Ins. Math. cademic Science, vol. 0 (99), [4] Mostafa S. M., bdel Naby M.., bdel-halim F. and Hameed. T., Fzzy KS-ideals in KS-algebras. To appear. [5] Prabpayak C. and eerawat., On ideals and congrences in K-algebras, scientia magna jornal, vol.5, no. (009), [6] Prabpayak C. and eerawat., On isomorphisms of K-algebras, scientia magna jornal, vol.5, no.3 (009), 5-3. [7] Xi O. G., Fzzy BCK-algebra, Math. Japon., vol.36 (99) [8] Zadeh.., Fzzy sets, Inform. nd Control, vol. 8 (965) [9] Zadeh.., The concept of a lingistic variable and its application to approximate I, Information Sci. nd Control, vol.8 (975), Page