Glol Journl of Mthemtil Sienes: Theory nd Prtil. ISSN 974-32 Volume 9, Numer 3 (27), pp. 387-397 Interntionl Reserh Pulition House http://www.irphouse.om On Implitive nd Strong Implitive Filters of Lttie Wjserg Algers A. Irhim* nd D. Srvnn** * P.G. nd Reserh Deprtment of Mthemtis, H. H. The Rjh s College, Pudukkotti, Tmilndu, Indi. ** Deprtment of Mthemtis, Chikknn Government Arts College, Tirupur, Tmilndu, Indi. Astrt In this pper, we disuss properties of implitive filter. We introdue the notion of strong implitive filter nd investigte some properties with interesting illustrtions. Also, we introdue the dul of kernel denoted s ker nd derive some properties of it. We otin the reltion etween n implitive filter nd strong implitive filter in lttie Wjserg lger. Keywords: Wjserg lger, Lttie Wjserg lger, Implitive filter, Strong implitive filter, Implition homomorphism, Lttie implition homomorphism, dul of kernel (ker). Mthemtil Sujet Clssifition 2: 3G, 3E7.. INTRODUCTION Mordhj Wjsreg [6] introdued the onept of Wjserg lgers in 935 nd studied y Font, Rodriguez nd Torrens in [4]. They [4] defined lttie struture of Wjserg lgers. Also, they [4] introdued the notion of implitive filter of lttie Wjserg lgers nd disussed some of their properties. Bsheer Ahmed nd Irhim [, 2] introdued the definitions of fuzzy implitive filter nd n nti fuzzy implitive filter of lttie Wjserg lgers nd otined some properties with illustrtions.
388 A. Irhim nd D. Srvnn In the present pper, we disuss the properties of implitive filter. We introdue the notion of strong implitive filter of lttie Wjserg lger, nd disuss some properties with exmples. Further, we disuss some relted properties of implitive nd strong Implitive filters of lttie Wjserg lger with useful illustrtions. Finlly, we define dul of kernel (ker) of lttie implition homomorphism nd investigte the properties of implitive nd strong implitive filters relted to ker. 2. PRELIMINARIES In this setion, we review some si definitions nd properties whih re helpful to develop our min results. Definition 2.[4] Let (A,,, ) e n lger with qusi omplement nd inry opertion is lled Wjserg lger if nd only if it stisfies the following xioms for ll x, y, z A. x = x (ii) (x y) (( y z) (x z)) = (iii) (x y) y = (y x ) x (iv) (x y ) (y x) = Proposition 2.2[4] The Wjserg lger (A,,, ) stisfies the following equtions nd implitions for ll x, y, z A. x x = (ii) If x y = y x = then x = y (iii) x = (iv) x (y x) = (v) If x y = y z = then x z = (vi) (x y) ((z x) (z y)) = (vii) x ( y z) = y (x z) (viii) x = x = x (ix) (x) (x ) = x x y = y x
On Implitive nd Strong Implitive Filters of Lttie Wjserg Algers 389 Proposition 2.3[4] TheWjserg lger (A,,, ) stisfies the following equtions nd implitions for ll x, y, z A. (ii) (iii) If x y then x z y z If x y then z x z y x y z iff y x z (iv) (x y) = (x y ) (v) (x y) = (x y ) (vi) (x y) z = (x z) (y z) (vii) x (y z) = (x y) (x z) (viii) (x y) (y x) = (ix) x (y z) = (x y) (x z) (x) (x y) z = (x y) (x z) (xi) (x y) z = (x z) (y z) (xii) (x y) z =(x y) (x z) Definition 2.4[4] The Wjserg lger A is lled lttie Wjserg lger if it stisfies the following onditions for ll x, y A. (ii) (iii) A prtil ordering " "on lttie Wjserg lger A, suh tht x y if nd only if x y = (x y) = (x y) y (x y) =((x y ) y ). Thus, we hve (A,,,,, ) is lttie Wjserg lger with lower ound nd upper ound. Definition 2.5[4] Let A e lttie Wjserg lger. A suset F of A is lled n implitive filter of A if it stisfies the following xioms for ll x, y A. F (ii) x F nd x y F imply y F. Definition 2.6[3] Let A nd A2 e lttie Wjserg lgers, f : A A2 e mpping from Ato A2, if for ny x, y A, f (x y) = f (x) f (y) holds, then f is lled n implition homomorphism from Ato A2.
39 A. Irhim nd D. Srvnn Definition 2.7[3] Let A nd A2 e lttie Wjserg lgers, f : A A2 e n implition homomorphism from A to A2, is lled lttie implition homomorphism from A to A2 if it stisfies the following xioms for ll x, y A f (x y) = f (x) f (y) (ii) f (x y) = f (x) f (y) (iii) f (x ) = [f(x)] Definition 2.8[3] Let f : A A2 e n implition homomorphism, the kernel of f written s Ker( f ) is defined s Ker( f ) = {x A / f (x)= }. 3. MAIN RESULTS 3.. Properties of implitive filters In this setion, we disuss nd investigte some properties of implitive filter of lttie Wjserg lger. Proposition 3... Let F e n implitive filter of lttie Wjserg lger A then x y nd x F imply y F for ll x, y A. Proof. Let F e n implitive filter of A. By the definition 2.5 of n implitive filter x y if nd only if x y = F, x y nd x F imply y F. Proposition 3..2. Let F e non-empty suset of A. Then F is n implitive filter of A if nd only if it stisfies for ll x, y F nd z A, x y z implies z F. Proof. Let F e implitive filter of A. By the Proposition 3.., x y nd x F imply y F. Let z A, we hve x y z. Suppose x x for ll x F. By the definition 2.5 of implitive filter, we hve F. Let x y F nd x F, x (y y) =, x x y implies y F. Therefore, we get F is n implitive filter. Proposition 3..3.Let A e lttie Wjserg lger, F A. F is n implitive filter of A if nd only if it stisfies the following: F (ii) For ny x, y, z A, x y F, y z F imply x z F.
On Implitive nd Strong Implitive Filters of Lttie Wjserg Algers 39 Proof. Suppose tht F is n implitive filter of A, x y F, y z F. By (x y) ((y z) (x z)) = F, it follows tht (x z) F. Conversely, if F A stisfies nd (ii) we hve x nd x y F. It follows tht x F, x y F, nd hene, we get y = y F. Proposition 3..4.Let F e non-empty suset of lttie Wjserg lger A. F is n implitive filter of A if nd only if it stisfies the following: F (ii) For ny x, y, z A, (z y) x F, y F imply z x F. Proof. Suppose tht nd (ii) hold. If x y F nd x F, then ( x) y F, whih implies y = y F nd F is n implitive filter. Conversely, if F is n implitive filter, nd (z y) x F, y F, then y x z x nd y z y. It follows tht (z y) x (z y) (z x) y (z x) nd hene, we hve y (z x) F, whih implies z x F. Proposition 3..5. Let And A2 e ny two lttie Wjserg lgers, f is lttie implition homomorphism from Ato A2. If F2 A2 is n implitive filter of A2, then f (F2) is n implitive filter of A. Proof. Let nd 2 e the gretest elements ofa nd A2, respetively. Sine f ( ) = f ( ) = 2 F2, implies f (F2). If x nd x y f (F2), then, we hve f (x) F2 nd f (x) f (y) = f (x y) F2. It follows tht f (y) F2, nd hene y f (F2). Therefore, f (F2) is n implitive filter of A. Definition 3..6. Let And A2 e lttie Wjserg lgers, nd 2e the gretest elements of A nd A2, respetively. Let f : A A2 e n implition homomorphism, then dul of kernel of f written s ker (f)is defined y ker (f)= {x / x A, f (x) =2}. Proposition 3..7.Let And A2 e ny two lttie Wjserg lgers, nd 2e the gretest elements of And A2respetively, f is n implition homomorphism from Ato A2, ker (f) is n implitive filter of A.
392 A. Irhim nd D. Srvnn Proof. Given tht f is n implition homomorphism, then f () =2, nd hene ker (f). If x nd x y ker (f), then, we hve f (x) =2 nd f (x y) = 2 Now, f (y) = 2 f (y) = f (x) f (y) = f (x y) = 2, sine f is implition homomorphism Hene, we get y ker (f). Therefore, ker (f)is n implitive filter of A. Note. Let f e n implition epimorphism from Ato A2. f is lttie implition isomorphism if nd only if ker (f) = {}, where is the gretest element of A. 3.2. Strong Implitive Filters In this setion, we introdue strong implitive filters of lttie Wjserg lger nd investigte some properties with illustrtions. Definition 3.2.. Let A e lttie Wjserg lger. A suset F of A is lled strong implitive filter of A if it stisfies the following xioms for ll x, y, z A. F (ii) x (y z) F nd x y F imply x z F. Exmple 3.2.2.Let A = {,,, } e set with Figure () s prtil ordering. Define unry opertion nd inry opertion on A s in the Tle () nd Tle (2). x x Tle () Tle (2) Figure ()
On Implitive nd Strong Implitive Filters of Lttie Wjserg Algers 393 Define nd opertions on A s follow: (x y) = x (y y) nd (x y) = ((x y ) y ) for ll x, y A. Then, we hve A is lttie Wjserg lger. Consider the suset F = {,, } of A, then F is strong implitive filter of A. But, the suset G = {,, } of A, then G is not strong implitive filter of A. Sine ( ) = G. Exmple 3.2.3. Let A = {,,,, } with < < < <, we define " " nd " " s x y = min{x, y} nd x y = mx{x, y} for ll x, y A. Also, define unry opertion nd inry opertion on A s in the Tle (3) nd Tle (4). x x Tle (3) Tle (4) Then, we hve A is lttie Wjserg lger. Consider the suset F = {,, } of A, then F is strong implitive filter of A. But, the suset G = {,, } of A, then G is not strong implitive filter of A. Sine ( ) = G. Exmple 3.2.4. Let A = {,,,, d, } e set with Figure (2) s prtil ordering. Define unry opertion nd inry opertion on A s in the Tle (5) nd Tle (6). x x d d d d d d d Tle (5) Tle (6) Figure (2)
394 A. Irhim nd D. Srvnn Define nd opertions on A s follow: (x y) = x (y y) nd (x y) = ((x y ) y ) for ll x, y A. Then, we hve A is lttie Wjserg lger. Consider the suset F = {,, } of A, then F is strong implitive filter of A. But the suset G = {, d, } of A, then G is not strong implitive filter of A. Sine ( d) = G. Proposition 3.2.5. Every strong implitive filter F of A is n implitive filter. Proof. Let F e strong implitive filter of A nd x y F nd x F for ll x, y A. Reple z y y in the definition 3.2.. Then, we hve x (y z) = x (y y) = x = x F. Also, x y F. Then, we get x z F. Thus, every strong implitive filter is n implitive filter. Note. The onverse my not e true. In Exmple 3.2.3, {} is n implitive filter ut not strong implitive filter sine ( )= = {}, nd = {}. Proposition 3.2.6. Let F e n implitive filter of A suh tht x (y (y z)) F nd x F imply x z for ll x, y, z A. Then F is strong implitive filter of A. Proof. Let x (y (y z)) F nd x y F for ll x, y, z A. From (vii) of definition 2.2 nd (ii) of definition 2., we hve x (y z) = y (x z) (x y) (x (x z)). By the Proposition 3.. we get, (x y) (x (x z)) F. Sine x y, we hve x z F.Therefore, F is strong implitive filter. Next, we show the definition of the losed intervl [, ]. Definition 3.2.7. Let A e lttie Wjserg lger, A. The intervl [, ] defined s [, ] = {x / x A, x} of A denoted s I().
On Implitive nd Strong Implitive Filters of Lttie Wjserg Algers 395 Proposition 3.2.8. Let A e lttie Wjserg lger, A, then {} is strong implitive filter of A if nd only if I() is n implitive filter of A for ny A. Proof. Suppose tht {} is strong implitive filter of A. For ny A, A is trivil. If x I() nd x y I(), then x, x y, Tht is, x = {} nd (x y)= {}. It follows tht y {}, y, nd hene y I(). Thus, I() is n implitive filter of A. Conversely, ssume tht I() is n implitive filter of A for ny A. For ny x, y, z A, if x (y z) {} nd x y {}, then x y z, x y, it follows tht x z euse I(x)is n implitive filter, hene x z = {}. Therefore, we hve {} is strong implitive filter of A. Proposition 3.2.9. Let A e lttie Wjserg lger, F A. The following sttements re equivlent (ii) (iii) (iv) Proof. (ii). F is strong implitive filter F is n implitive filter nd for ny x, y A, x (x y) F implies x y F F is n implitive filter nd for ny x, y, z A, x (y z) F implies (x y) (x z) F F nd for ny x, y, z A, z (x (x y)) F nd z F imply x y F. For ny x, y A, if x (x y) F, sine x x = F, from (ii) of definition 3.2. we hve x y F. (ii) (iii). Assume tht (ii) holds. For ny x, y, z A, suppose x (y z) F, from (vii) of proposition 2.2,, (ii) of proposition 2.3 nd (vi) of proposition 2.2, we hve x (y z) x ((x y) (x z)). Therefore, from proposition 3.. nd (vii) of proposition 2.2, we get x (x ((x y) z)) = x ((x y) (x z)) F. From (ii) nd (vii) of proposition 2.2, we hve (iii) (iv). x ((x y) z) = (x y) (x z) F.
396 A. Irhim nd D. Srvnn Now ssume tht (iii) holds nd we prove (iv). In ft, F is trivil. For ny x, y, z A, suppose z (x (x y)) F nd z F, then x (y z) F. From (ii) of definition 2.5, we hve x y = (x y) = (x x) (x y) F. (iv). Suppose x F nd x y F, then we get x ( ( y)) = x y F, It follows tht y = y F nd hene F is n implitive filter. For ny x, y, z A, x (y z) F nd x y F, Now, (x (y z)) ((x y) (x (x z))) = (x y) ((x (y z)) (x (x z))) = (x y) ((y (x z)) (x (x z))) = (x y) (x (y (x z))) = (x y) ((x y) (x (x z))) = F. It follows tht (x y) (x (x z) F nd hene x z F y x y F nd (iv). Proposition 3.2.. Let A e lttie Wjserg lger, F nd F2 re ny two implitive filters of A, F F2. If F is strong implitive filter, so is F2. Proof. Suppose x (x y) F2, we only to prove x y F2. Now x (x ((x (x y)) y)) = (x (x y)) (x (x y)) = F. It follows tht x ((x (x y)) y) F F2. Tht is, (x (x y)) (x y) F2 nd hene x y F2. Therefore, we hve F2 is strong implitive filter. 4. CONCLUSION In this pper, we hve introdued the notion of strong implitive filter of lttie Wjserg lger, nd disussed some of their properties with exmples. Further, we hve shown tht some relted properties of implitive nd strong implitive filters of lttie Wjserg lger with useful illustrtions. Finlly, we hve defined ker of lttie implition homomorphism nd investigte the properties of implitive nd strong implitive filters relted to ker.
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398 A. Irhim nd D. Srvnn