ON INTUITIONISTIC FUZZY SUB- IMPLICATIVE IDEALS OF BCI-ALGEBRAS

International Journal of Engineering Science Invention Research & Development; Vol. IV, Issue II, AUGUST 2017 www.ijesird.com, E-ISSN: 2349-6185 ON INTUITIONISTIC FUZZY SUBIMPLICATIVE IDEALS OF BCI-ALGEBRAS Dr. R. Jayasudha Depatment of Mathematics, K. S. Rangasamy. College of Technology,Tiruchengode-637215, Tamilnadu, India. rjayasudha98@gmail.com Abstract- The aim of this paper is to introduce the notion of intuitionistic fuzzy sub-implicative ideals in BCI- algebras and to investigate some of their related properties.. Keywords: intuitionistic fuzzy sub-implicative ideal, intuitionistic fuzzy positive implicative idea , intuitionistic fuzzy p-ideal , intuitionistic fuzzy characteristic sub-implicative ideal . Mathematics Subject Classification : 06F35,03B52 1. INTRODUCTION The notion of BCK/BCI-algebras was introduced by Imai and Iseki in 1966.In the same year Iseki introduced the notion of a BCI-algebras which is a generalization of a BCK-algebras. After the introduction of the concept of fuzzy sets by L.A. Zadeh [7],several researches were conducted on the generalization of the fuzzy sets. The idea of intuitionistic fuzzy set was first introduced by K.T.Atanassov [1,2], as a generalization of the notion of fuzzy set. In this paper using Atanassov’s idea ,we establish the intuitionistic fuzzification of the concept of sub-implicative ideals in BCI-algebras and investigate some of their properties . 2. PRELIMINARIES In this section we include some elementary definitions that are necessary for this paper. By a BCI- algebra we mean an algebra (X,*,0 ) of type (2,0) satisfying the following conditions: (1) ((x * y) * (x * z)) * (z * y) = 0, (2) (x * (x * y)) * y = 0, (3) x * x = 0, (4) x * y = 0 and y * x = 0 imply x = y, for all x ,y ,zX. In a BCI-algebra X, we can define a partial ordering “  ” by putting x  y if and only if x* y = 0.A BCI-algebra X is said to be implicative if (x * (x * y) * (y * x) = y * (y * x) for all x ,y  X. A mapping f: X  Y of BCI-algebras is called a homomorphism if f(x*y) = f(x) * f(y) for all x ,y  X. In any BCI- algebra X ,the following hold : (5) ((x * z) * (y * z)) * (x * y) = 0, (6) x * (x * (x * y)) = x * y, (7) 0 * (x * y) = (0 * x) * (0 * y), (8) x * 0 = x, (9) (x * y) * z = (x * z) * y, (10) x  y implies x * z  y * z and z * y  z * x, for all x ,y, z X. Dr. R. Jayasudha ijesird, Vol. IV, Issue II, August 2017/65 International Journal of Engineering Science Invention Research & Development; Vol. IV, Issue II, AUGUST 2017 www.ijesird.com, E-ISSN: 2349-6185 Example 2.1 The set X = { 0, 1, 2, 3 } with the following Cayley table is a BCI - algebra * 0 1 2 3 0 1 2 3 0 1 2 3 0 0 2 3 0 0 0 3 3 3 3 0 Throughout this paper X always means a BCI-algebra without any specification. Definition 2.2 An non-empty subset A of X is a positive implicative ideal of X if for all x X , (1) 0 * x  A implies x  A (2) (( x * z) * z) * (y * z)  A and y A imply x * z  A. Definition 2.3. An non empty set A in X is called a P-ideal if it satisfies for all x ,y ,z  X, (1) 0  A, (2) (x * z) * (y * z)  A and yA imply xA. Definition 2.4 [ 7 ]. Let X be a non-empty set. A fuzzy set  in X is a function  : X  [ 0, 1 ]. Definition 2.5 [7 ]. Let  be a fuzzy set in X. For t  [0,1],the set t ={ x X (x) t} is called a level subset of . Definition 2.6 . A fuzzy set  in X is called a fuzzy ideal of X if (1)  ( 0)  ( x), (2)  ( x ) ≥ min{  (x * y ), ( y )} , for all x, y  X. For any elements x ,y of a BCI-algebra , xn * y denotes x * (…* (x * (x * y))…..) in which x occurs n times. Definition 2.7 [ 3 ]. A fuzzy set  in X is called a fuzzy sub-implicative ideal of X if (1)  ( 0 )   ( x ) , (2)  ( y2 * x ) ≥ min{  ( (( x2 * y) * (y * x)) * z) ,  ( z )} , for all x, y, z  X. Definition 2.8 [ 2 ]. An intuitionistic fuzzy set ( IFS ) A in a non empty set X is an object having the form A ={ < x,  A (x), A( x) > / x  X }, where the functions  A : X  [ 0,1] and A : X  [0,1] denote the degree of membership and the degree of non membership of each element x  X to the set A, respectively, and 0 ≤  A (x) + A(x) ≤ 1 for all x  X. Notation: For the sake of simplicity, we shall use the symbol A = <  A, A > for the Dr. R. Jayasudha ijesird, Vol. IV, Issue II, August 2017/66 International Journal of Engineering Science Invention Research & Development; Vol. IV, Issue II, AUGUST 2017 www.ijesird.com, E-ISSN: 2349-6185 IFS A = { < x,  A (x), A(x) > / x  X }. Definition 2.9 [ 2 ]. Let A be an intuitionistic fuzzy set of a set X. For each pair < t ,s >  [0, 1], the set A <t, s> = { x  X :  A (x) ≥ t and A(x) ≤ s } is called the level subset of A . Definition 2.10 [ 2 ]. Let A be an IFS in X and let t  [ 0, 1] .Then the sets U( A ; t) = { x  X :  A (x) ≥ t }and L(A , t ) = { x  X : A(x) ≤ t } are called a  -level t -cut and - level t-cut of A , respectively. 3. INTUITIONISTIC FUZZY SUB-IMPLICATIVE IDEALS Definition 3.1. An intuitionistic fuzzy set A in X is called an intuitionistic fuzzy sub-implicative ideal of X if it satisfies: (1)  A ( 0 )   A ( x ) , (2) A ( 0 )  A ( x ) , (3)  A ( y2 * x ) ≥ min{  A ( (( x2 * y) * (y * x) ) * z) ,  A ( z )}, (4) A ( y2 * x ) ≤ max {A ( (( x2 * y) * (y * x) ) * z) , A ( z )} for all x ,y, zX. Example 3.2.Let X = { 0,1,2 } with the following Cayley table be a BCI algebra. * 0 1 2 0 0 1 2 1 0 0 2 2 2 2 0 Let A= <  A, A > be an IFS in X defined by  A(0) =  A (1) = 0.6,  A (2) = 0.2 and A(0) = A (1) = 0.2 and A (2) = 0.6. Then A is an intuitionistic fuzzy sub-implicative ideal of X. Theorem 3.3. Let A be an intuitionistic fuzzy set in X satisfying  A (0)   A (x) and A (0)  A (x). If A is an intuitionistic fuzzy sub-implicative ideal of X , then A satisfies the following inequality  A ( y2 * x ) ≥  A (( x2 * y) * (y * x)) and A ( y2 * x ) ≤ A ( ( x2 * y) * (y * x) ) for all x ,y X. Proof. Let A be an intuitionistic fuzzy ideal of X. Then  A ( y2 * x ) ≥ min{  A ( (( x2 * y) * (y * x)) * z) ,  A ( z )}, and A ( y2 * x ) ≤ max {A ( (( x2 * y) * (y * x) ) * z) , A ( z )} Taking z = 0 we get  A ( y2 * x ) ≥ min{  A ( (( x2 * y) * (y * x) ) * 0) ,  A ( 0 )} Dr. R. Jayasudha ijesird, Vol. IV, Issue II, August 2017/67 International Journal of Engineering Science Invention Research & Development; Vol. IV, Issue II, AUGUST 2017 www.ijesird.com, E-ISSN: 2349-6185 =  A ( ( x2 * y) * (y * x) ) ,  A ( 0 )} =  A ( ( x2 * y) * (y * x) ) and A ( y2 * x ) ≤ max{ A ( (( x2 * y) * (y * x) ) * 0) , A ( 0 )} = A ( ( x2 * y) * (y * x) ) , A ( 0 )} = A ( ( x2 * y) * (y * x) ) Theorem3.4. Every intuitionistic fuzzy sub- implicative ideal of X is an intuitionistic fuzzy ideal. Proof. Let A be an intuitionistic fuzzy sub-implicative ideal of X. Then (i)  A ( 0 )   A ( x ) , (ii) A ( 0 )  A ( x ) , (iii)  A ( y2 * x ) ≥ min{  A ( (( x2 * y) * (y * x) ) * z) ,  A ( z )}, (iv ) A ( y2 * x ) ≤ max {A ( (( x2 * y) * (y * x) ) * z) , A ( z )} for all x ,y, z  X. Putting y = x in (iii)and (iv),we get  A (x ) =  A ( x2 * x ) ≥ min{ {  A ( (( x2 * x) * (x * x) ) * z) ,  A ( z )} = min{ {  A ( x * z) , A ( z )} A (x ) = A ( x2 * x )  max{ { A ( (( x2 * x) * (x * x) ) * z) , A ( z )} = max{ { A ( x * z) , A ( z )} for all x ,z  X. Hence A is an intuitionistic fuzzy ideal. Theorem 3.5. An intuitionistic fuzzy ideal of X may not be an intuitionistic fuzzy sub- implicative ideal. Proof. Let X = { 0,a,b,c } with the following Cayley table be a BCI- algebra. * 0 a b c 0 a b c 0 a b c 0 0 b c 0 0 0 c c c c 0 Let A= <  A, A > be an IFS in X defined by  A(0) = 0.7 and  A (x) = 0.2 for all x ≠ 0 and A(0) = 0.2 and A (x) = 0.7 for all x ≠ 0.Then A is an intuitionistic fuzzy ideal of X. ,but it is not an intuitionistic fuzzy sub-implicative ideal of X because  A ( a2* b) < min{  A( ((b2 * a) * ( a * b )) * 0),  A (0)} A ( a2* b) > max{ A ( ((b2 * a) * ( a * b )) * 0), A (0)} Theorem 3.6. Every intuitionistic fuzzy ideal satisfying the condition Dr. R. Jayasudha ijesird, Vol. IV, Issue II, August 2017/68 International Journal of Engineering Science Invention Research & Development; Vol. IV, Issue II, AUGUST 2017 www.ijesird.com, E-ISSN: 2349-6185  A ( y2 * x ) ≥  A (( x2 * y) * (y * x) ) and A ( y2 * x ) ≤ A (( x2 * y) * (y * x) ) is an intuitionistic fuzzy subimplicative ideal of X. Proof. Let A be an intuitionistic fuzzy ideal of X satisfying  A ( y2 * x ) ≥  A (( x2 * y) * (y * x) ) and A ( y2 * x ) ≤ A (( x2 * y) * (y * x) )  A ( y2 * x ) ≥  A (( x2 * y) * (y * x) ) ≥ min{  A ( (( x2 * x) * (x * x) ) * z) ,  A ( z )} A ( y2 * x ) ≤ A (( x2 * y) * (y * x) ) ≤ max{ A ( (( x2 * x) * (x * x) ) * z) , A ( z )} This completes the proof. Definition 3.7 An intuitionistic fuzzy set A in X is called an intuitionistic fuzzy positive implicative ideal of X if (i)  A (0)   A (x), (ii) A (0)  A (x), (iii)  A ( x * z ) ≥ min{  A ( (( x * z) * z) * (y * z) ) ,  A ( y )}, (iv ) A (x * z ) ≤ max {A ( ((x * z) * z) * (y * z) ) , A ( y )} for all x ,y, z X. Example 3.8. Let X = { 0,a,b,c } with the following Cayley table be a BCI- algebra. * 0 a b c 0 a b c 0 a b c 0 0 b c 0 0 0 c c c c 0 Let A = <  A, A > be an IFS in X defined by  A(0) = 0.7 and  A (x) = 0.2 for all x ≠ 0 and A(0) = 0.2 and A (x) = 0.7 for all x ≠ 0.Then A is an intuitionistic fuzzy positive implicative ideal of X. Theorem 3.9.Every intuitionistic fuzzy sub-implicative ideal is an intuitionistic fuzzy positive implicative ideal. Proof. Let A be an intuitionistic fuzzy sub-implicative ideal of X. Then A is an intuitionistic fuzzy ideal of X. From Theorem 3.3  A ( b2 * a ) ≥  A (( a2 * b) * (b * a)) and A ( b2 * a ) ≤ A ( ( b2 * a) * (b * a) ) for all a ,bX. Substituting x*y for a and x for b we have  A ( x * y ) =  A ( x * (x * (x * y))) =  A(b2* a) ≥  A (( a2 * b) * (b * a)) =  A(((x * y) * (( x *y) * x))) * (x * ( x * y))) =  A(((x *y) * ( x * (x * y))) * ((x * y) * x)) Dr. R. Jayasudha ijesird, Vol. IV, Issue II, August 2017/69 International Journal of Engineering Science Invention Research & Development; Vol. IV, Issue II, AUGUST 2017 www.ijesird.com, E-ISSN: 2349-6185 =  A(((x * (x * (x * y))) * y) * ((x * x) * y)) =  A(((x * y) * y) * (0 * y)) A ( x *y ) = A ( x * (x * (x * y))) = A (b2* a) ≤ A (( a2 * b) * (b * a)) = A (((x * y) * (( x * y) * x))) * (x * ( x * y))) = A (((x * y) * ( x * (x * y))) * ((x * y) * x)) = A (((x * (x * (x * y))) * y) * (( x * x) * y)) = A (((x * y) * y) * (0 * y)) Hence A is an intuitionistic fuzzy positive implicative ideal of X. Definition3.10 An intuitionistic fuzzy set A in X is called an intuitionistic fuzzy p-ideal of X if (0)  A (x), (ii) A (0)  A (x), (iii)  A (x) ≥ min{  A (( x * z) * (y * z) ) ,  A ( y )}, (iv ) A (x ) ≤ max {A ((x * z) * (y * z) ) , A ( y )} for all x ,y, z X. (i)  A Theorem 3.11 . An intuitionistic fuzzy p- ideal of X is an intuitionistic fuzzy sub-implicative ideal of X, but the converse does not hold. Proof. Suppose that A is an intuitionistic fuzzy p- ideal of X . Then it is an intuitionistic fuzzy ideal of X. Note that (02 * (y2 * x)) * (x2* y) * (y * x)) = (0 * ((x2 * y) * (y * x))) * (0 * (y2 * x)) = ((0 * ((x2 * y)) * (0 * (y * x))) * ((0 * y) * (0 * (y * x))) = (((0 * x) * (0 * ((x * y))) * (0 * (y * x))) * ((0 * y) * (0 * (y * x))) ≤ (((0 * x) * (0 * ((x * y))) * (0 * y) = ((0 * x) * (0 * y)) * (0 * (x * y)) =0 Since in an intuitionistic fuzzy ideal A = <  A, A>,  A is order reversing and and A is order preserving  A(y2* x) ≥  A(02* (y2 * x)) ≥ min{ A((02* (y2* x)) * ((x2* y) * (y * x))) ,  A((x2* y) * (y * x))} ≥ min{ A(0) ,  A((x2* y) * (y * x))} =  A((x2* y) * (y * x)) A (y2* x) ≤ A (02* (y2 * x)) ≤ max{A ((02* (y2* x) ) * ((x2* y) * (y * x))) , A ((x2* y) * (y * x))} ≤ max{A (0) , A ((x2* y) * (y * x))} = A ((x2 * y) * (y * x)) Hence A is an intuitionistic fuzzy sub-implicative ideal. Example 3.12.Let X = { 0,a,1,2,3 } with the following Cayley table be a BCI- algebra. Dr. R. Jayasudha ijesird, Vol. IV, Issue II, August 2017/70 International Journal of Engineering Science Invention Research & Development; Vol. IV, Issue II, AUGUST 2017 www.ijesird.com, E-ISSN: 2349-6185 * 0 a 1 2 3 0 a 1 2 3 0 0 3 a 0 3 1 1 0 2 2 1 3 3 2 2 2 3 0 1 1 1 2 3 0 Let A= <  A, A > be an IFS in X defined by  A(0) = 0.7,  A(a) = 0.5  A (1) =  A (2) =  A (3) = 0.2 and A(0) = 0.2, A (a) = 0.5,A (1) = A (2) = A (3) = 0.7. Then A is an intuitionistic fuzzy ideal of X. in which the inequalities  A(y2* x) ≥  A(y2* x) * (y* x)) and A (y2* x) ≤ A (y2*x)*(y*x)) hold for all x ,y X . A is an intuitionistic fuzzy sub-implicative ideal of X by Theorem 3.7. But it is not an intuitionistic fuzzy p- ideal of X, since  A(a) < min{  A((a *1) * (0 * 1)) ,  A (0)} A (a) > max{A ((a *1) * (0 * 1)) , A (0)} The proof is complete. Theorem 3.13. For any intuitionistic fuzzy sub-implicative ideal of X, the set XA={ xX   A(x) =  A(0) and A (x) = A (0) } is a sub-implicative ideal. Proof. Clearly 0XA. Let x ,y, z  X be such that ((x2 * y) * (y * x)) * z XA and zXA. Then  A ( y2 * x ) ≥ min{  A ( (( x2 * y) * (y * x) ) * z) ,  A ( z )}=  A(0)and A ( y2 * x ) ≤ max {A ( (( x2 * y) * (y * x)) * z) , A (z)}= A (0) for all x ,y, zX. which implies A ( y2 * x) = A (0) and A ( y2 * x ) = A (0).That is , y2 * x XA Therefore XA is a sub-implicative ideal of X. REFERENCES [2] [3] [4] [5] [6] [7] [1] K.T. Atanassov ,Intuitionistic fuzzy sets, Fuzzy sets and Systems 20(1) (1986), 87-96. K. T. Atanassov , New operations defined over intuitionistic fuzzy sets, Fuzzy sets and Systems 61(1994), 137-142. Y. B. Jun , Fuzzy Sub-implicative ideals of BCI-algebras, Bull. Korean Math.Soc.39(2002) No.2 pp 185-198. Y. B. Jun and J. Meng ,Fuzzy p-ideals in BCI-algebras, Math.Japonica 40(1994),No.2, 271-282 Y.L. Liu and J. Meng ,Sub-implicative ideals and sub-commutative ideals of BCI-algebras ,Sochow J. Math (to appear) Y.L. Liu and J. Meng ,Fuzzy ideals in BCI-algebras ,Fuzzy sets and systems 123(2001)227-237 L.A. Zadeh, Fuzzy sets, Information and Control, 8 (1965) 338-353. Dr. R. Jayasudha ijesird, Vol. IV, Issue II, August 2017/71