POSITIVE IMPLICATIVE AND ASSOCIATIVE FILTERS OF LATTICE IMPLICATION ALGEBRAS

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1 Bull. Koren Mth. So. 35 (998), No., pp POSITIVE IMPLICATIVE AND ASSOCIATIVE FILTERS OF LATTICE IMPLICATION ALGEBRAS YOUNG BAE JUN*, YANG XU AND KEYUN QIN ABSTRACT. We introue the onepts of positive implitive filter n n ssoitive filter in lttie implition lger. We prove tht (i) every positive implitive filter is n implitive filter, n (ii) every ssoitive filter is filter. We provie equivlent onitions for oth positive implitive filter n n ssoitive filter.. Introution In orer to reserh the logil system whose propositionl vlue is given in lttie, Y. Xu [Xu2] propose the onept of lttie implition lgers, n isusse their some properties in [Xu] n [Xu2]. Y. Xu n K. Y. Qin [XQ] introue the notions of filter n n implitive filter in lttie implition lger, n investigte their properties. Y. B. Jun [Ju] gve some equivlent onitions tht filter is n implitive filter in lttie implition lger, n estlishe n extension property for implitive filter. In this pper, We introue the onepts of positive implitive filter n n ssoitive filter in lttie implition lger. We prove tht every positive implitive filter is n implitive filter, n hene filter, n tht every ssoitive filter is filter. We give n exmple to show tht filter my not e n ssoitive filter. We provie equivlent onitions for oth positive implitive filter n n ssoitive filter. 2. Preliminries DEFINITION 2. (Xu [Xu2]). By lttie implition lger we men Reeive Mrh, Mthemtis Sujet Clssifition: 3G, 6B. Key wors n phrses: Lttie implition lger, positive implitive filter, ssoitive filter. *Supporte y the LG Yonm Fountion, 995.

2 Y. B. Jun, Y. Xu n K. Y. Qin oune lttie (L,,,, ) with orer-reversing involution n inry opertion stisfying the following xioms for ll x, y, z L: (I) x (y z) = y (x z), (I2) x x =, (I3) x y = y x, (I4) x y = y x = x = y, (I5) (x y) y = (y x) x, (L) (x y) z = (x z) (y z), (L2) (x y) z = (x z) (y z). We n efine prtil orering on lttie implition lger L y x y if n only if x y =. EXAMPLE 2.2. Let L := {,,,,}. Define the prtilly orere reltion on L s < < < <, n efine x y := min{x, y}, x y := mx{x, y} for ll x, y L n n s follows: x x Then (L,,,, ) is lttie implition lger. OBSERVATION (Xu [Xu2]). In lttie implition lger L, the following hol for ll x, y, z L: () x =, x = x n x =, (2) x y implies y z x z n z x z y, (3) (x y) ((y z) (x z)) =, (4) x ((x y) y) =. In wht follows, L woul men lttie implition lger unless otherwise speifie. In [XQ], Y. Xu n K. Y. Qin efine the notions of filter n n implitive filter in lttie implition lger. 54

3 Positive implitive n ssoitive filters DEFINITION 2.3 (Xu n Qin [XQ]). Let (L,,,, ) e lttie implition lger. A sus F of L is lle filter of L if it stisfies for ll x, y L: (F) F, (F2) x F n x y F imply y F. A suset F of L is lle n implitive filter of L, if it stisfies (F) n (F3) x (y z) F n x y F imply x z F for ll x, y, z L. LEMMA 2.4 (Jun [Ju]). Let F e non-empty suset of L. Then F is filter of L if n only if it stisfies for ll x, y F n z L: (i) x y z implies z F. LEMMA 2.5 (Jun [Ju]). Every filter F of L hs the following property: x y n x F imply y F. 3. Positive Implitive Filters MAIN DEFINITION. A suset F of L is lle positive implitive filter of L if it stisfies (F) n (F4) x ((y z) y) F n x F imply y F for ll x, y, z L. We first give n exmple of positive implitive filter of lttie implition lger. EXAMPLE 3.. Let L := {,,,,,}esetwithfiguresprtil orering. Define unry opertion n inry opertion onls follows (Tles n 2, respetively): 55

4 Y. B. Jun, Y. Xu n K. Y. Qin x x.. Figure Tle Define -n -opertions on L s follows: Tle 2 x y := (x y) y, x y := ((x y ) y ), for ll x, y L. Then L is lttie implition lger. It is esy to hek tht F := {,,}is positive implitive filter of L. THEOREM 3.. Every positive implitive filter of L is filter. Proof. Let F e positive implitive filter of L n let x y F n x F for ll x, y L. Putting z = y in (F4), we hve x ((y y) y) = x y F n x F. It follows from (F4) tht y F, whene F is filter of L. REMARK 3.2. The onverse of Theorem 3. my not e true. In ft, onsier lttie implition lger L s in Exmple 2.2. We know tht {} is filter of L, ut it is not positive implitive filter, sine (( ) ) = {}n / {}. Also, in Exmple 3., we know tht the suset G := {,}is filter, ut not positive implitive filter of L, sine (( ) ) = G for / G. Now we give n equivlent onition tht every filter is positive implitive filter. 56

5 Positive implitive n ssoitive filters THEOREM 3.3. Let F efilterofl.thenfis positive implitive filter of L if n only if for ll x, y L, (F5) (x y) x F implies x F. Proof. Assume tht F is positive implitive filter of L n let (x y) x F for ll x, y L. Thenwehve ((x y) x) = (x y) x F. Sine F, it follows from (F4) tht x F, n (F5) hols. Conversely, suppose tht F stisfies (F5). Let x ((y z) y) F n x F for ll y, z L. Then(y z) y Fy (F2), whih implies y F y (F5). Hene F is positive implitive filter of L n the proof is omplete. THEOREM 3.4. Let F e non-empty suset of L. If F is positive implitive filter of L, then it is n implitive filter of L. Proof. Let x (y z) F n x y F for ll x, y, z L. Then x (y z)=y (x z) [y (I)] (x y) (x (x z)). [y (I) n (3)] Sine F is filter of L (see Theorem 3.), it follows from Lemm 2.4 tht x (x z) F. On the other hn, using (I) n (I5) we hve ((x z) z) (x z) = x (((x z) z) z) = x ((z (x z)) (x z)) = x ((x (z z)) (x z)) = x ((x ) (x z)) = x ( (x z)) = x (x z) F. By Theorem 3.3, we get x z F. This ompletes the proof. OPEN PROBLEM. Does the onverse of Theorem 3.4 hol? 57

6 Y. B. Jun, Y. Xu n K. Y. Qin 4. Assoitive Filters MAIN DEFINITION 2. Let x e fixe element of L. A suset F of L is lle n ssoitive filter of L with respet to x if it stisfies (F) n (A) x (y z) F n x y F imply z F for ll y, z L. An ssoitive filter of L with respet to ll x is lle n ssoitive filter of L. Clerly, n ssoitive filter with respet to is the whole lger L. An ssoitive filter with respet to is oinient with filter. EXAMPLE 4.. Let L := {,,,,,}esetwithfigure2sprtil orering. Define unry opertion n inry opertion only Tles 3 n 4, respetively, n efine -n -opertions on L s follows: x y := (x y) y n x y := ((x y ) y ), respetively, for ll x, y L. x x.. Figure 2 Tle 3 Tle 4 Then L is lttie implition lger. It is routine to verify tht F := {,,} is n ssoitive filter of L with respet to n, ut not with respet to n, sine n ( )= = F, = Fut / F, ( ) = = F, = F ut / F. 58

7 Positive implitive n ssoitive filters PROPOSITION 4.. Everyssoitivefilterwithrespetto x ontins x itself. Proof. If x =, then it is trivil. Assume x. Let F e n ssoitive filter of L with respet to x. Note tht x ( x) = x x = F n x = F. It follows from (A) tht x F. THEOREM 4.. Every ssoitive filter is filter. Proof. Let F e n ssoitive filter of L n let x y F n x F for ll x, y L. Then x=x Fn (x y) = x y F. It follows from (A) tht y F. Hene F is filter of L REMARK 4.2. The onverse of Theorem 4. my not e true. In ft, we know tht, in Exmple 2.2, {} is filter of L. But it is not n ssoitive filter of L, sine ( )= = {}n = {},ut / {}. Now we give equivlent onitions tht every filter is n ssoitive filter. THEOREM 4.3. Let F e filter of L. ThenFis n ssoitive filter if n only if it stisfies (A2) x (y z) F implies (x y) z F for ll x, y, z L. Proof. If filter F of L stisfies the onition (A2), then lerly F is ssoitive. Conversely, let F e n ssoitive filter of L n let x (y z) F for ll x, y, z L. Then x ((y z) ((x y) z)) = (y z) (x ((x y) z)) [y (I)] = (y z) ((x y) (x z)) [y (I)] = F, [y (I) n (3)] whih implies from (A) tht (x y) z F. This ompletes the proof. THEOREM 4.4. Let F e filter of L. ThenFis n ssoitive filter if n only if it stisfies (A3) x (x y) F implies y F for ll x, y L. 59

8 Y. B. Jun, Y. Xu n K. Y. Qin Proof. Let F efilterofl. It is suffiient to show tht (A2) n (A3) re equivlent. Putting x = y in (A2) n using (I2) n (), we otin (A3). Assume tht (A3) hols n let x (y z) F for ll x, y, z L. Using (I), (I2), (), (2) n (3), we hve (x (y z)) (x (x ((x y) z))) = ((x (y z)) (x (x ((x y) z)))) = ((y z) (x ((x y) z))) ((x (y z)) (x (x ((x y) z)))) = (x (y z)) (((y z) (x ((x y) z))) (x (x ((x y) z)))) (x (y z)) (x (y z)) =, whih implies tht (x (y z)) (x (x ((x y) z))) = F. Sine x (y z) F n sine F is filter, it follows tht x (x ((x y) z)) F. By using (A3), we onlue tht (x y) z F. This ompletes the proof. ACKNOWLEDGEMENTS. The uthors express their thnks to the referee for his/her vlule suggestions. Referenes [Ju] Y. B. Jun, Implitive filters of lttie implition lgers, Bull. Koren Mth. So. 34 (997), [JX] Y. B. Jun n Y. Xu, Fuzzy filters of lttie implition lgers, sumitte. [Xu] Y. Xu, Homomorphisms in lttie implition lgers, Pro. of 5th Mny-Vlue Logil Congress of Chin (992), [Xu2], Lttie implition lgers, J. of Southwest Jiotong Univ. (993), [XQ] Y. Xu n K. Y. Qin, On filters of lttie implition lgers, J. Fuzzy Mth. (993),

9 Positive implitive n ssoitive filters [YW] B. Yun n W. Wu, Fuzzy iels on istriutive lttie, Fuzzy Sets n Systems 35 (99), YOUNG BAE JUN DEPARTMENT OF MATHEMATICS,GYEONGSANG NATIONAL UNIVERSITY,CHINJU 66-7, KOREA E-mil: yjun@nonge.gsnu..kr YANG XU DEPARTMENT OFAPPLIED MATHEMATICS,SOUTHWEST JIAOTONG UNIVERSITY,CHENGDU,SICHUAN 63, P. R. CHINA E-mil: yxu@enter2.swjtu.eu.n KEYUN QIN DEPARTMENT OF MATHEMATICS,HENAN NORMAL UNIVERSITY,XINXIANG,HENAN 4532, P. R. CHINA 6