Anti-fuzzy BRK-Ideal of BRK-Algebra

DOI: 10.9734/BJMCS/2015/19309 Editor(s): (1) Andrej V. Plotnikov, Department of Applied and Calculus Mathematics and CAD, Odessa State Academy of Civil Engineering and Architecture, Ukraine. (2) Paul Bracken, Department of Mathematics, The University of Texas-Pan American Edinburg, TX 78539, USA. Reviewers: (1) Francisco Bulnes, Department of Research in Mathematics and Engineering, Tecnológico de Estudios Superiores de Chalco, Mexico. (2) Xingting Wang, Department of Mathematics, University of California, San Diego, USA. (3) Piyush Shroff, Department of Mathematics, Texas State University, USA. Complete Peer review History: http://sciencedomain.org/review-history/10426


Introduction
Y. Imai and K. Iséki introduced two classes of abstract algebras: BCK-algebras and BCI-algebras [5]. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. In [4], Q. P. Hu and X. Li introduced a wide class of abstract: BCH-algebras. They have shown that the class of BCI-algebras is a proper subclass of the class of BCH-algebras. In [10], J. Neggers, S. S. Ahn and H. S. Kim introduced Qalgebras which is a generalization of BCK / BCI-algebras and obtained several results. In 2002, Neggers and Kim [9] introduced a new notion, called a B-algebra, and obtained several results. In 2007, Walendziak [11] introduced a new notion, called a BF-algebra, which is a generalization of B-algebra. In [7], C. B. Kim and H.S. Kim introduced BG-algebra as a generalization of B-algebra. In 2012, R. K. Bandaru [1] introduces a new notion, called BRK-algebra which is a generalization of BCK / BCI / BCH / Q / QS-algebras and BH/BM-algebras [6,8]. In [3] I consider the fuzzification of BRK-ideal of BRK-algebra. I introduce the

Original Research Article
notion of fuzzy BRK-ideal of BRK-algebra. In this paper I introduce a new notion which is the anti fuzzy BRK-ideal of BRK-algebras. I study the epimorphic image and the into homomorphic inverse image of an anti fuzzy BRK-ideal. I investigate some related properties. Also I introduce the Cartesian product of Anti fuzzy BRK-ideals and some related properties.

Preliminaries Definition 2.1 [1]:
A BRK-algebra is a non-empty set X with a constant 0 and a binary operation "  " satisfying the following conditions: In a BRK-algebra X, a partially ordered relation  can be defined by y (  X is a BRK-algebra, the following conditions hold:

Definition 2.3 [1]:
A non empty subset S of a BRK-algebra X is said to be BRK-subalgebra of X, if

Definition 2.4 (BRK-ideal of BRK-algebra):
A non empty subset I of a BRK-algebra X is said to be a BRK-ideal of X if it satisfies:

Definition 2.5:
Let X be a set. A fuzzy set  in X is a function

Definition 2.6:
Let ) 0 , , (  X be a BRK-algebra. A fuzzy set  in X is called a fuzzy BRK-ideal of X if it satisfies:

Anti-Fuzzy BRK-Ideal of BRK-Algebra
Definition 3.1: be a BRK-algebra. A fuzzy set  in X is called an anti fuzzy BRK-ideal of X if it satisfies:

Example 3.2:
. Define  on X as the following table: , routine calculation gives that  is an anti fuzzy BRK-ideal of BRK-algebra.

Proposition 3.3:
Let  be an anti fuzzy BRK-ideal of BRK-algebra X and if Proof. Let  be an anti fuzzy BRK-ideal of a BRK-algebra X . For any

Theorem 3.4:
A fuzzy subset  of a BRK-algebra X is a fuzzy BRK-ideal of X if and only if its complement c  is an anti fuzzy BRK-ideal of X.
Proof. Let  be a fuzzy BRK-ideal of a BRK-algebra X, and let So,  is a fuzzy BRK-ideal of a BRK-algebra X.

Theorem 3.5:
Let  be an anti fuzzy BRK-ideal of BRK-algebra X. Then the set

Definition 3.6 [5]:
Let  be a fuzzy subset of a set X ,

Definition 3.7:
Let  be a fuzzy BRK-ideal of BRK-algebra X . The BRK-ideal t  , [0,1] t  , is called a lower t-level BRK-ideal of  .

Theorem 3.8:
Let  be a fuzzy subset of a BRK-algebra X. If  is an anti fuzzy BRK-ideal of X then for each Proof. Let  be an anti fuzzy BRK-ideal of X and let

Definition 3.10:
Let f be a mapping from the set X to the set Y. If  is a fuzzy subset of X, then the fuzzy subset B of Y defined by Similarly, if B is a fuzzy subset of Y, then the fuzzy subset defined by is said to be the inverse image of B under f.

Theorem 3.11:
The epimorphic image of an anti fuzzy BRK-ideal is also an anti fuzzy BRK-ideal.
Hence μ is an anti fuzzy BRK-ideal of Y.

Theorem 3.12:
The into homomorphic inverse image of an anti fuzzy BRK-ideal is also an anti fuzzy BRK-ideal.

Proof. Let
So that (FI 1 ) holds, since The proof is complete.

Cartesian Product of Anti Fuzzy BRK-Ideal
All the definitions in this section are from [2].

Definition 5.1:
A fuzzy relation on any set S is a fuzzy subset :

Definition 5.2:
Let  and  be the fuzzy subsets of a set S . The anti Cartesian product of , for all , x y S  .

Definition 5.3:
Let  and  be fuzzy subsets of a set S . The Cartesian product of  and  is defined by

Definition 5.6:
If  is a fuzzy subset of a set S , the strongest Anti fuzzy relation on S that is an anti fuzzy relation on  is   given by for all , x y S  .

Proposition 5.7:
For a given anti fuzzy subset  of a BRK-algebra X , let   be the strongest anti fuzzy relation on X .
If   is an anti fuzzy BRK-ideal of Proof. Since   is an anti fuzzy BRK-ideal of X X  , it follows from (FT 1 ) that ) ,

Theorem 5.8:
Let  be an anti fuzzy subset of BRK-algebras X, and   be the strongest anti fuzzy relation on X. If  is an anti fuzzy BRK-ideal of X then   is an anti fuzzy BRK-ideal of Proof. Suppose that,  is an anti fuzzy subset of a BRK-ideal X, and   is the strongest anti fuzzy relation on X. Then  is an anti fuzzy BRK-ideal of X X  .

Conclusion
In this paper, we have introduced the concept of anti fuzzy BRK-ideal of BRK-algebra and studied their properties.
In our future work, we introduce the concept of Cubic fuzzy BRK-ideal of BRK-algebra, interval-valued Fuzzy BRK-ideal of BRK-algebra, intuitionistic fuzzy structure of BRK-ideal of BRK-algebra, intuitionistic fuzzy BRK-ideals of BRK-algebra, and intuitionistic L-fuzzy BRK-ideals of BRK-algebra. I hope this work would serve as a foundation for further studies on the structure of BRK-algebras.