International Mathematical Forum, 5, 2010, no. 34, 1685-1690 Redefined Fuzzy BH-Subalgebra of BH-Algebras Hyoung Gu Baik School of Computer and Information Ulsan college, Ulsan 682-090, Korea hgbaik@mail.uc.ac.kr Chul Hwan Park Department of Mathematics University of Ulsan, Ulsan 680-749, Korea skyrosemary@gmail.com Abstract The notion of an T H -interval-valued fuzzy BH-subalgebra in a BHalgebra is introduced, and related properties are investigated. Mathematics Subject Classification: 06F35, 03G25 Keywords: BH-algebra, T H -interval-valued fuzzy BH-subalgebra 1 Introduction The notion of BH-algebra is introduced by Jun et al. [1] Since then, several authors have studied BH-algebras. In particular, Zhang et al. [7] studied the fuzzy theory in BH-algebras. C.H.Park [3] established the interval-valued fuzzification of the concept of BH-subalgebras in BH-algebra. As a generalization of interval-valued fuzzy BH-subalgebra, we introduce the notion of T H -interval-valued fuzzy BH-subalgebra in a BH-algebra by using triangular norms. 2 Preliminaries Definition 2.1. [1] A BH-algebra is a nonempty set X with a constant 0 and a binary operation satisfying the following conditions: This work was supported by Ulsan college Research Fund of 2009.
1686 H. G. Baik and C. H. Park (I) x x =0, (II) x y =0and y x =0imply x = y (III) x 0=x for all x, y X. Definition 2.2. [1] A nonempty subset S of a BH-algebra X is called a BH-subalgebra of X if x y S for all x, y S. Definition 2.3. [5] Let X be a set. A fuzzy set A defined on X is given by A = {(x, μ A (x)])}, where μ A : X [0, 1]. Definition 2.4. [7] A fuzzy set μ in a BH-algebra X is called a fuzzy BHsubalgebra of X if it satisfies: ( x, y X)(μ(x y) min{μ(x),μ(y)}). Lemma 2.5. [7] If A = {(x, μ A (x)])} is a fuzzy BH-subalgebra of X, then (i) ( x X)(μ A (0 x) μ A (x)). (i) ( x, y X)(μ A (x (0 y)) min{μ A (x),μ A (y)}). Definition 2.6. [6] An interval-valued fuzzy set A defined on X is given by A = {(x, [μ A (x), μ A (x)])}, x X (briefly, denoted by A =[μ A, μ A ]), where μ A and μ A are two fuzzy sets in X such that μ A (x) μ A (x) for all x X. Let μ A (x) = [ μ A (x), μ A (x) ], x X and let D[0, 1] denotes the family of all closed subintervals of [0, 1]. Then the interval-valued fuzzy set A is given by A = {(x, μ A (x))}, x X, where μ A : X D[0, 1]. Let D 1 := [a 1,b 1 ], D 2 := [a 2,b 2 ] D[0, 1]. The refined minimum (briefly, rmin) is defined by rmin(d 1,D 2 ) = [min{a 1,a 2 }, min{b 1,b 2 }]. Definition 2.7. [3] An interval-valued fuzzy set A in X is called an intervalvalued fuzzy BH-subalgebra of X if it satisfies: ( x, y X)( μ A (x y) rmin{ μ A (x), μ A (y)}). Definition 2.8. [2] A binary operation T on [0, 1] is called a t-norm if it satisfies the following conditions: (T1) ( a [0, 1]) (T (a, 1) = a, T (0, 0) = 0.) (T2) ( a, b, c [0, 1]) (T (a, T (b, c)) = T (T (a, b),c)), (T3) ( a, b [0, 1]) (T (a, b) =T (b, a)),
Redefined fuzzy BH-subalgebra of BH-algebras 1687 (T4) ( a, b, c [0, 1]) (b c T (a, b) T (a, c)), A few t-norms which are frequently encountered are T l,t m, and T w defined by T l (a, b) = max{a+b 1, 0}(Lukasiewicz),T m (a, b) = min{a, b}(minimum) and { min{a, b} if a =1orb =1, T w (a, b) := 0 otherwise(weak). Lemma 2.9. [2] For any t-norm T. The following statements holds T l (a, b) T (a, b) T m (a, b) for all a, b [0, 1]. Let T be a t-norm, if for arbitrary x [0, 1], if it satisfies T (x, x) =x,the T is called a idempotent t-norm. Definition 2.10. [4] Let T be an idempotent t-norm, Defined mapping T H : D[0, 1] D[0, 1] D[0, 1] by T H (D 1,D 2 )=[T (a 1,a 2 ),T(b 1,b 2 )] for all D 1 := [a 1,b 1 ], D 2 := [a 2,b 2 ] D[0, 1], the T H is called an idemportent interval t- norm. 3 T H -Interval-valued fuzzy BH-subalgebras Throughout this section X stands for a BH-algebra unless otherwise specified. Definition 3.1. Let T H is an idemportent interval t-norm. An intervalvalued fuzzy set A in X is called an T H -interval-valued fuzzy BH-subalgebra of X if it satisfies: ( x, y X)( μ A (x y) T H { μ A (x), μ A (y)}). Lemma 3.2. Every T H -interval-valued fuzzy BH-subalgebra of X is an intervalvalued fuzzy BH-subalgebra Proof. Let A =[μ A, μ A ]beat H -interval-valued fuzzy BH-subalgebra of X. Then μ A (x y) T H ( μ A (x), μ A (y)). Since T H is idemportent interval t- norm, we have min{μ A (x),μ A (y)} = T H (min{μ A (x),μ A (y)}, min{μ A (x),μ A (y)}) T H (μ A (x),μ A (y)) min{μ A (x),μ A (y)} Hence μ A (x y) T H ( μ A (x), μ A (y)) = rmin{ μ A (x), μ A (y)}. Therefore A is an interval-valued fuzzy BH-subalgebra X.
1688 H. G. Baik and C. H. Park Theorem 3.3. An interval-valued fuzzy set A =[μ A, μ A ] in X is an T H - interval-valued fuzzy BH-subalgebra of X if and only if μ A and μ A are fuzzy BH-subalgebras of X. Proof. Assume that A is an T H -interval-valued fuzzy BH-algerba of X. Let x, y X. Then by Lemma3.2, we have μ A (x y) = T H { μ A (x), μ A (y)} = rmin { μ A (x), μ A (y)} = rmin {[ μ A (x), μ A (x) ], [ μ A (y), μ A (y) ]} = [ min { μ A (x),μ A (y) }, min {μ A (x), μ A (y)} ]. Hence μ A (x y) min { μ A (x),μ A (y) } and μ A (x y) min {μ A (x), μ A (y)}. Thereforee μ A and μ A are fuzzy BH-algebras of X. Conversely,suppose that μ A and μ A are fuzzy BH-subalgebras of X. Let x, y X. Then μ A (x y) = [ μ A (x y), μ A (x y) ] [ min{μ A (x),μ A (y)}, min{μ A (x), μ A (y)} ] [ T H {μ A (x),μ A (y)},t H {μ A (x), μ A (y)} ] {[ = T H μa (x), μ A (x) ], [ μ A (y), μ A (y) ]} = T H { μ A (x), μ A (y)}. Hence A is an T H -interval-valued fuzzy BH-algebra of X. Let X be a nonempty set and A is an interval-valued fuzzy subset of X. Then we recall L (A;[δ 1,δ 2 ]) := {x X μ A (x) [δ 1,δ 2 ]} Theorem 3.4. Let A =[μ A, μ A ] be an interval-valued fuzzy set in X and T H an idemportent interval t-norm such that L (A;[δ 1,δ 2 ]) is a BH-subalgebra of X for every [δ 1,δ 2 ] D[0, 1]. Then A =[μ A, μ A ] is an T H -interval-valued fuzzy BH-subalgebra of X Proof. Assume that L (A;[δ 1,δ 2 ]) ( ) is a BH-subalgebra of X for every [δ 1,δ 2 ] D[0, 1]. For every x, y X if we put [t, s] =T H ( μ A (x), μ A (y)] then x, y L (A;[t,s]). Since L (A;[δ 1,δ 2 ]) is a BH-subalgebra of X for every [δ 1,δ 2 ] D[0, 1], we have x y L (A;[t,s]) Therefore μ A (x y) [t, s] =T H { μ A (x), μ A (y)} for all x, y X. Theorem 3.5. If A =[μ A, μ A ] is an T H -interval-valued fuzzy BH-subalgebra of X, then the non-empty set L (A;[0,1]) is a BH-subalgebra of X. Proof. Assume that A =[μ A, μ A ]isant H -interval-valued fuzzy BH-subalgebra of X Let x, y L (A;[0,1]). Then μ A (x) [0.1] and μ A (x) [0.1] Hence μ A (z y) T H ( μ A (x), μ A (y)) T H ([0, 1], [0, 1]) = [T H (0, 0),T H (1, 1)] = [0, 1]. and so x y L (A;[0,1]). Thus L (A;[0,1]) is a BH-subalgebra of X.
Redefined fuzzy BH-subalgebra of BH-algebras 1689 Theorem 3.6. If A =[μ A, μ A ] is an T H -interval-valued fuzzy BH-subalgebra of X, then the non-empty set L (A;[δ 1,δ 2 ]) is a BH-subalgebra of X for every [δ 1,δ 2 ] D[0, 1]. Proof. Assume that A =[μ A, μ A ]isant H -interval-valued fuzzy BH-algebra of X and let [δ 1,δ 2 ] D[0, 1] be such that x, y L (A;[δ 1,δ 2 ]). We have μ A (x) [δ 1,δ 2 ] and μ A (x) [δ 1,δ 2 ] This implies that μ A (z y) T H ( μ A (x), μ A (y)) T H ([δ 1,δ 2 ], [δ 1,δ 2 ]) = [T H (δ 1,δ 1 ),T H (δ 2,δ 2 )] = [δ 1,δ 2 ]. Hence x y L (A;[δ 1,δ 2 ]). Therefore L (A;[δ 1,δ 2 ]) is a BH-subalgebra of X. Theorem 3.7. If A =[μ A, μ A ] is an T H -interval-valued fuzzy BH-subalgebra of X, then (i) ( x X)( μ A (0 x) μ A (x)). (i) ( x, y X)( μ A (x (0 y)) T H { μ A (x), μ A (y)}). Proof. Since by Theorem 3.3 and Lemma 2.5, we have μ A (0 x) = [ μ A (0 x), μ A (0 x) ] [ μ A (x), μ A (x) ] = μ A (x) for all x X, and μ A (x (0 y)) = [ μ A (x (0 y)), μ A (x (0 y)) ] [ min{μ A (x),μ A (y)}, min{μ A (x), μ A (y)} ] [ T H {μ A (x),μ A (y)},t H {μ A (x), μ A (y)} ] {[ = T H μa (x), μ A (x) ], [ μ A (y), μ A (y) ]} = T H { μ A (x), μ A (y)} for all x, y X. References [1] Y. B. Jun, E. H. Roh and H. S. Kim, On BH-algebras, Scientiae Mathematicae 1(1) (1998), 347 354. [2] B.Schweizer,A.Sklar.Associative functions and abstrct semigroup Publ.Math.Debrecen,10,(1963),69 81 [3] C.H.Park,Interval-valued fuzzy ideal in BH-algebtras Advance in fuzzy set and systems 1(3)(2006),231 240 [4] Li Xiaoping and Wang Guijun, TH-interval valued fuzzy subgroups. J. Lanzhou University 32 (1996), pp. 96?99. [5] L. A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965), 338 353.
1690 H. G. Baik and C. H. Park [6] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-i, Inform. Sci. 8 (1975), 199 249. [7] Q. Zhang, E. H. Roh and Y. B. Jun, On fuzzy BH-algebras, J. Huanggang Normal Univ. 21(3) (2001), 14 19. Received: January, 2010