DIRECT PRODUCT AND t-NORMED PRODUCT OF FUZZY ...

SOOCHOW JOURNAL OF MATHEMATICSVolume 27, No. 1, pp. 83-88, January 2001DIRECT PRODUCT AND t-NORMEDPRODUCT OF FUZZY SUBALGEBRAS INBCK-ALGEBRAS WITH RESPECT TO A t-NORMBYYOUNG BAE JUNAbstract. Using a t-norm T , the direct product and T -product of T -fuzzy subalgebrasare discussed, and their properties are investigated.1. IntroductionA BCK-algebra is an important class of logical algebras introduced by Isekiand was extensively investigated by several researchers. Zadeh [11] introducedthe notion of fuzzy sets. At present this concept has been applied to manymathematical branches, such as group, functional analysis, probability theory,topology, and so on. In 1991, Xi [9] applied this concept to BCK-algebras, andhe introduced the notion of fuzzy subalgebras(ideals) of the BCK-algebras withrespect to minimum, and since then Jun et al. studied fuzzy subalgebras andfuzzy ideals (see [2, 5, 6]). In [7], Jun et al. redened the fuzzy subalgebra of theBCK-algebras with respect to a t-norm T and hence generalized the notion in [9],and obtained some related results. In this paper, we consider the direct productand t-normed product of fuzzy subalgebras of BCK-algebras with respect to at-norm.2. PreliminariesAn algebra (X 0) of type (2 0) is said to be a BCK-algebra if it satises:for all x y z 2 X,Received September 17, 1999 revised July 26, 2000.AMS Subject Classication. 06F35, 03G35, 94D05.Key words. t-norm, T -fuzzy subalgebra, direct product, T -product.83

84 YOUNG BAE JUN(I) ((x y) (x z)) (z y) =0(II) (x (x y)) y =0,(III) x x =0(IV) 0 x =0(V) x y =0andy x =0implyx = y.Dene a binary relation on X by letting x y if and only if xy =0. Then(X ) is a partially ordered set with the least element 0. A subset S of a BCKalgebraX is called a subalgebra of X if x y 2 S whenever x y 2 S. A mapping : X ! X 0 of BCK-algebras is called a homomorphism if (x y) =(x) (y)for all x y 2 X.3. T -fuzzy SubalgebrasIn what follows, let X denote a BCK-algebra unless otherwise specied. Afuzzy subset of X is a function : X ! [0 1]. Let be a fuzzy subset of X. For 2 [0 1], the set U( ) =fx 2 X j (x) g is called an upper level set of .Denition 3.1.([1]) By a t-norm T ,we mean a function T :[0 1] [0 1] ![0 1] satisfying the following conditions:(T1) T (x 1) = x(T2) T (x y) T (x z) ify z,(T3) T (x y) =T (y x),(T4) T (x T (y z)) = T (T (x y)z)for all x y z 2 [0 1].Denition 3.2.([7]) A function : X ! [0 1] is called a fuzzy subalgebra ofX with respect to a t-norm T (briey, a T -fuzzy subalgebra of X) if (x y) T ((x)(y)) for all x y 2 X.Lemma 3.3.([1]) Let T be at-norm. Thenfor all 2 [0 1].T (T ( )T()) = T (T ( )T())Theorem 3.4. Let T be at-norm and let X = X 1 X 2 be the direct productBCK-algebra of BCK-algebras X 1 and X 2 . If 1 (resp. 2 ) is a T -fuzzy subalgebra

DIRECT PRODUCT AND t-NORMED PRODUCT OF FUZZY SUBALGEBRAS 85of X 1 (resp. X 2 ), then = 1 2 is a T -fuzzy subalgebra of X dened by(x 1 x 2 )=( 1 2 )(x 1 x 2 )=T ( 1 (x 1 ) 2 (x 2 ))for all (x 1 x 2 ) 2 X 1 X 2 .Proof. Let x =(x 1 x 2 )andy =(y 1 y 2 ) be any elements of X = X 1 X 2 .Then(x y)=((x 1 x 2 ) (y 1 y 2 )) = (x 1 y 1 x 2 y 2 )= T ( 1 (x 1 y 1 ) 2 (x 2 y 2 )) T (T ( 1 (x 1 ) 1 (y 1 ))T( 2 (x 2 ) 2 (y 2 )))= T (T ( 1 (x 1 ) 2 (x 2 ))T( 1 (y 1 ) 2 (y 2 )))= T ((x 1 x 2 )(y 1 y 2 ))= T ((x)(y)):Hence is a T -fuzzy subalgebra of X.We will generalize the idea to the product of nT-fuzzy subalgebras. We rstQneed to generalize the domain of t-norm T to n [0 1] as follows:Denition 3.5.([1]) The function T n : [0 1] ! [0 1] is dened byi=1T n ( 1 2 ::: n )=T ( i T n;1 ( 1 ::: i;1 i+1::: n ))for all 1 i n, where n 2, T 2 = T and T 1 =id(identity).nQi=1Lemma 3.6.([1]) For a t-norm T and every i iand n 2, we have2 [0 1] where 1 i nT n (T ( 1 1 )T( 2 2 ):::T( n n ))= T (T n ( 1 2 ::: n )T n ( 1 2 ::: n )):Theorem 3.7. Let T be a t-norm and let fX i g n i=1 be a nite collection ofQBCK-algebras and X = n X i the direct product BCK-algebra of fX i g. Let ii=1

86 YOUNG BAE JUNbe a T -fuzzy subalgebra of X i ,where 1 i n. Then = n QnYi=1 idened by(x 1 x 2 :::x n )=( i )(x 1 x 2 :::x n )i=1= T n ( 1 (x 1 ) 2 (x 2 )::: n (x n ))for all (x 1 x 2 :::x nQ) 2 n X i is a T -fuzzy subalgebra of the BCK-algebra X.i=1Proof. Let x =(x 1 x 2 :::x n ) and y =(y 1 y 2 :::y n ) be any elements ofQX = n X i . Theni=1(x y)=(x 1 y 1 x 2 y 2 :::x n y n )= T n ( 1 (x 1 y 1 ) 2 (x 2 y 2 )::: n (x n y n )) T n (T ( 1 (x 1 ) 1 (y 1 ))T( 2 (x 2 ) 2 (y 2 )):::T( n (x n ) n (y n )))= T (T n ( 1 (x 1 ) 2 (x 2 )::: n (x n ))T n ( 1 (y 1 ) 2 (y 2 )::: n (y n )))= T ((x 1 x 2 :::x n )(y 1 y 2 :::y n ))= T ((x)(y)):Hence is a T -fuzzy subalgebra of X.Denition 3.8. Let T be a t-norm and let and be fuzzy subsets of aBCK-algebra X. Then the T -product of and , written [ ] T , is dened by[ ] T (x) =T ((x)(x)) for all x 2 X.Theorem 3.9. Let T be at-norm and let and be T -fuzzy subalgebras ofa BCK-algebra X. IfT is a t-norm which dominates T ,i.e.,T (T ( )T()) T (T ( )T ())for all 2 [0 1] then the T -product of and , [ ] T, is a T -fuzzysubalgebra of X.Proof. For any x y 2 X we have[ ] T(x y)=T ((x y)(x y)) T (T ((x)(y))T((x)(y))) T (T ((x)(x))T ((y)(y)))= T ([ ] T(x) [ ] T(y)):

DIRECT PRODUCT AND t-NORMED PRODUCT OF FUZZY SUBALGEBRAS 87Hence [ ] T is a T -fuzzy subalgebra of X:Let f : X ! X 0 be an onto homomorphism of BCK-algebras. Let T and T be t-norms such that T dominates T . If and are T -fuzzy subalgebras of X 0 ,then the T -product of and ,[] T,isaT -fuzzy subalgebra of X 0 . Since everyonto homomorphic preimage of a T -fuzzy subalgebra is a T -fuzzy subalgebra (see[7, Theorem 3.6]), the preimages f ;1 (), f ;1 () and f ;1 ([ ] T) are T -fuzzysubalgebras of X. The next theorem provides the relation between f ;1 ([ ] T)and the T -product [f ;1 () f ;1 ()] T of f ;1 () andf ;1 ().Theorem 3.10. Let f : X ! X 0 be an onto homomorphism of BCKalgebras.Let T and T be t-norms such that T dominates T . Let and beT -fuzzy subalgebras of X 0 : If [ ] T f ;1 ()] T is the T -product of f ;1 () and f ;1 (), thenis the T -product of and and [f ;1 () f ;1 ([ ] T )=[f ;1 () f ;1 ()] T :Proof. For any x 2 X we get[f ;1 ([ ] T)](x)=[ ] T(f(x))= T ((f(x))(f(x)))= T ([f ;1 ()](x) [f ;1 ()](x))=[f ;1 () f ;1 ()] T(x)ending the proof.4. ConclusionsWe have considered the direct product and t-normed product of fuzzy subalgebrasof BCK-algebras under a t-norm. This ideas could be enable us to discussthe direct product and t-normed product of fuzzy ideals of BCK-algebras. Theyalso suggest possible problems to generalize Theorem 3.7 for an arbitrary collectionof BCK-algebras, to slacken the condition in Theorem 3.9 that one of thetriangular norms dominates the other, to study how are the converses of Theorems3.4 and 3.7, and to discuss similar results for images like f(), instead ofpreimages like f ;1 () in Theorem 3.10.

88 YOUNG BAE JUNAcknowledgmentI would like to thank the referees for carefully reading the manuscript andmaking several helpful comments to increase the quality of the paper.References[1] M. T. Abu Osman, On some product of fuzzy subgroups, Fuzzy Sets and Systems, 24(1987),79-86.[2] S. M. Hong and Y. B. Jun, Fuzzy and level subalgebras of BCK(BCI)-algebras, Pusan KyongnamMath. J. (presently, East Asian Math. J.), 7:2(1991), 185-190.[3] K. Iseki, BCK-algebras with condition (S), Math. Japon., 24:1(1979), 107-119.[4] K. Iseki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon.,23:1(1978), 1-26.[5] Y. B. Jun, S. M. Hong, S. J. Kim and S. Z. Song, Fuzzy ideals and fuzzy subalgebras ofBCK-algebras, J.Fuzzy Math., 7:2(1999), 411-418.[6] Y. B. Jun, S. M. Hong and E. H. Roh, Fuzzy characteristic subalgebras/ideals of a BCKalgebra,Pusan Kyongnam Math. J. (presently, East Asian Math. J.), 9:1(1993), 127-132.[7] Y. B. Jun and Q. Zhang, Fuzzy subalgebras of BCK-algebras with respect to a t-norm, FarEast J. Math. Sci. (FJMS), 2:3(2000), 489-495.[8] J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa Co., Korea, 1994.[9] O. G. Xi, Fuzzy BCK-algebras, Math. Japon., 36:5(1991), 935-942.[10] Y. Yu, J. N. Mordeson and S. C. Cheng, Elements of L-algebra, Lecture Notes in FuzzyMath. and Computer Science, Creighton Univ., Omaha, Nebraska 68178, USA, 1994.[11] L. A. Zadeh, Fuzzy sets, Inform. Control., 8(1965), 338-353.Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Korea.E-mail: ybjun@nongae.gsnu.ac.kr