BM-algebras defined by bipolar-valued sets

In this note, by using the concept of Bipolar-valued fuzzy set, the notion of bipolar-valued fuzzy BM-algebra is introduced. Moreover, the notions of (strong) negative s-cut (strong) positive t-cut are introduced and the relationship between these notions and crisp sub-algebras are studied.

Introduction Imai and Iseki introduced two classes of abstract algebras: BCK-algebras and BCI-algebras (Iseki & Tanaka, 1978;Iseki, 1980). It is known that the class of BCK-algebras is a proper subclass of the class of BCIalgebras. Li (1983 &1985) introduced a wide class of abstract algebras: BCH-algebras. They have shown that the class of BCI-algebras is a proper subclass of the class of BCH-algebras. Neggers and Kim (1999) introduced the notion of d−algebras which is another generalization of BCKalgebras, and also they introduced the notion of Balgebras (Neggers & Kim, 2002a). Moreover, Jun et al. (1998) introduced a new notion, called a BH-algebra, which is a generalization of BCH/BCI/BCK-algebras (Meng & Jun, 1994). Walendziak (2006) obtained another equivalent axiom for B-algebra. Kim et al. (2004) introduced the notion a (pre-) Coxeter algebra and showed that Coxeter algebra is equivalent to an Abelian group all of whose elements have order 2, i.e., a Boolean group. Kim & Kim (2006) introduced the notion of a BMalgebra which is a specialization of B-algebras (Neggers & Kim, 2002b). Zadeh (1965) introduced the notion of a fuzzy subset of a set; fuzzy sets are a kind of useful mathematical structure to represent a collection of objects whose boundary is vague. Since then it has become a vigorous area of research in different domains. There have been a number of generalizations of this fundamental concept such as intuitionistic fuzzy sets, interval-valued fuzzy sets, vague sets, soft sets etc. Lee (2000) introduced the notion of bipolar-valued fuzzy sets. Bipolar-valued fuzzy sets are an extension of fuzzy sets whose membership degree range is enlarged from the interval [0,1] to [ 1,1]  . In a bipolar-valued fuzzy set, the membership degree 0 means that elements are irrelevant to the corresponding property, the membership degree (0,1] indicates that elements somewhat satisfy the property, and the membership degree [ 1, 0)  indicates that elements somewhat satisfy the implicit counter-property. Bipolar-valued fuzzy sets and intuitionistic fuzzy sets look similar each other. However, they are different each other (Lee, 2000;Lee, 2004). Now, in this note we use the notion of Bipolar-valued fuzzy set to establish the notion of bipolar-valued fuzzy BM-algebras; then we obtain some-related which have been mentioned in the abstract.

Preliminary
In this section, we present now some preliminaries on the theory of bipolar-valued fuzzy set. In his pioneer work (Zadeh, 1965 and v  (x) = 0, it is the situation that x is regarded as having only positive satisfaction for It is possible for an element x to be such that µ + (x) 0 and v  (x) 0 when the membership function of the property overlaps that of its counter property over some portion of G. For the sake of simplicity, we shall use the symbol B = (µ + ,  ) for the bipolar-valued fuzzy set Definition 2. (Kim & Kim, 2006) Let X be a non-empty set with a binary operation "  " and a constant " 0 ". Then ( ,*,0) X is called a BM-algebra if it satisfies the following conditions: for all , , x y z X  .
We can define a partial ordering  by x y  if and for all , , x y z X  .
A nonempty subset S of X is called a sub-algebra of X if x*yS, for all x, yS. for all x, yX.

Bipolar-valued fuzzy subalgebras of BM-algebras
From now on ( , , 0) X  is a -algebra, unless otherwise is stated.
 is said to be a bipolar-valued fuzzy sub-algebra of a BMalgebra X if it satisfies the following conditions: Proof. Let x X  and assume that n is odd. Then 2 1 n k   for some positive integer k . We prove by induction, definition and above lemma imply that Then by assumption Which proves (i). Similarly we can prove (ii). x and Proof: By above lemma we have x y X  .

Proposition 7:
A bipolar-valued fuzzy set B of X is a bipolar-valued fuzzy sub-algebra of X if and only if   is a fuzzy sub-algebras and   is an anti fuzzy subalgebras of X .
Proof: The proof is straightforward.
The set of all ( , ) Im( ) Im( ) , x y x y x y X  . Now the proof is completed.
Theorem 10: Each sub-algebra of X is a level subalgebra of a bipolar-valued fuzzy sub-algebra of X . Proof: Let Y be subalgebra of X and B be a bipolarvalued fuzzy subset of X which is defined is defined by: x y X  . We consider the following cases: Therefore B is a bipolar-valued fuzzy sub-algebra of X .
Theorem 11: Let S be a subset of X and B be a bipolar-valued subset of X which is given in the proof of Theorem 3. 10. If B is a bipolar-valued fuzzy sub-algebra of X then S is a sub-algebra of X .
Proof: Let B be a bipolar-valued fuzzy sub-algebra of X and , x y S  . Then Now we generalize the Theorem 3. 10 Theorem 12: Let X be a -algebra. Then for any chain of sub-algebras There exists a bipolar-valued fuzzy sub-algebra B of X whose level sub-algebras are exactly the sub-algebras of this chain. Proof: Consider the following sets of numbers We prove that is a bipolar-valued fuzzy sub-algebra of X . Let , x y X  , we consider the following cases: Proof. Suppose that each bipolar fuzzy sub-algebra of Bipolar-valued fuzzy set is a generalization of fuzzy sets. In the present paper, we have introduced the concept of bipolar-valued fuzzy sub-algebras of BMalgebras and investigated some of their useful properties. In our opinion, these definitions and main results can be similarly extended to some other algebraic systems such as groups, semigroups, rings, nearrings, semirings (hemirings), lattices and Lie algebras. It is our hope that this work would other foundations for further study of the theory of -algebras. In our future study of fuzzy structure of BM-algebras may be the following topics should be considered: To establish a bipolar-valued fuzzy ideals of BMalgebras; To consider the structure of quotient BM-algebras by using these bipolar-valued fuzzy ideals; To get more results in bipolar-valued fuzzy BMalgebras and application.