GENERALIZATION OF ANTI P-FUZZY GROUP WITH ANTI P-FUZZY ALGEBRA FROM THE ALGEBRA A

The Algebras are generalized with Anti Partially ordered fuzzy groups. Our main objective of this work is to define Anti partially ordered algebra and its anti groups on the algebra A. Some Properties of Anti partially ordered algebra are explored. We also discuss about the condition for which Anti partially ordered algebra to be Anti Partially ordered fuzzy groups and similarly condition for Anti Partially ordered fuzzy groups to be Anti partially ordered algebra.


INTRODUCTION
In 1965, Zadeh mathematically initiated the concept of fuzzy set [4], it opened a new path of thinking to many mathematicians, engineers, physicists, chemists and many others due to its diverse applications in various fields. The Fuzzy Algebraic structures play a important role on Fuzzy Mathematics with wide applications. Rosenfeld [7] in 1971 introduced the concept of fuzzy subgroups, which was the first fuzzification of any algebraic structures. Biswas [6] in 1994 2113 GENERALIZATION OF ANTI P-FUZZY GROUP defined interval -valued fuzzy subgroups of the same nature of Rosenfeld's fuzzy subgroups and discussed some important results. In 1998, Lazlo filep [5] introduced the concept of P -Fuzzy Algebra. Biswas introduced the concept of Anti fuzzy set which is an extension of fuzzy set i.e., the complement of fuzzy set. K. H. Kim and Y. H. Yon [3] defined Anti fuzzy ideals and Anti fuzzy Rsubgroups [2] using near rings. D. Y. Li, C. Y. Zhang and S. Q. Ma [1] generalized the Intuitionistic Anti-fuzzy Subgroup in Group G. Later Fuzzy set was developed into some special types of AFS which are widely used in artificial intelligence, computer science, medical science, control engineering, decision theory, expert systems, operations research, pattern recognition, robotics and other fields.

PRELIMINARIES
Definition 2.1. A fuzzy subset µ of the set X is a function µ: X → [0,1], where X be a non-empty set.
Definition 2.2. Let A be a nonempty set and P = (P, *, 1, ) where * is minimum operation a (2, 0) type ordered algebra, satisfies the property of monoid, poset and isotone.

Definition 2.3. A mapping :
A →P is a P-fuzzy subset of A(P A ) where (P, ) is a partially ordered set and X is a nonvoid set.

Definition 2.4.
A Pfuzzy set P A along with n-ary and nullary operation is called a Pfuzzy algebra or fuzzy subalgebra on the algebra .

Converse part:
Given that (x) = (e).  Hence HK is a APFG.

Converse part: Suppose HK is a APFG. To prove that H  K or K  H.
Suppose that H is not contained in K and K is not contained in H. Then there exists element 1, 2 such that

S. PRIYADARSHINI
Our assumption is wrong H is not contained in K and K is not contained in H.  Therefore (xy -1 ) = (a1b1)( a2b2) -1 = (a1)(b1)(b2 -1 )(a2 -1 ) Therefore AB is an APFSG of group G.   the order of APFG divides order of APFA on the Algebra A. Theorem 3.11 Every P -level subsets of an APFG of a group G from APFA must be a fuzzy subgroup from APFA.
Proof: Let  be two APFSG of a group G from the APFA on the algebra A. It is clear that the above lower level set is not a Group.

CONCLUSION
The research work on P -anti Fuzzy subgroup is extended to Partially ordered fuzzy Right Algebra on the Algebra A using P -Fuzzy set. In future work it can be extended to anti P -Fuzzy bigroups, anti P -Fuzzy rings etc.