Some Characterizations of Intra-regular Abel-grassmann Groupoids

In this study we introduce a new class of a non-associative algebraic structure namely intra-regular Abel Grassmann's groupoid (AG-groupoid in short). We apply generalized fuzzy ideal theory to this class and discuss its related properties. We introduce) , (k q ∨ ∈ ∈-fuzzy semiprime ideals in AG-groupoids and characterize it. Specifically we have characterized intra-regular AG-groupoids in terms of left, bi and two sided ideals and) , (k q ∨ ∈ ∈-fuzzy left, bi and two sided ideals. For support of our arguments we give examples of AG-groupoids. At the end we characterize intra-regular AG-groupoids using the properties of) , (k q ∨ ∈ ∈-fuzzy semiprime ideals. k q ∨ ∈ ∈-fuzzy ideals and) , (k q ∨ ∈ ∈-fuzzy bi-ideals, left invertive law, medial law, paramedial law

Abstract: In this study we introduce a new class of a non-associative algebraic structure namely intra-regular Abel Grassmann's groupoid (AG-groupoid in short).We apply generalized fuzzy ideal theory to this class and discuss its related properties.We introduce ) , ( -fuzzy semiprime ideals in AG-groupoids and characterize it.Specifically we have characterized intra-regular AG-groupoids in terms of left, bi and two sided ideals and ) , ( -fuzzy left, bi and two sided ideals.For support of our arguments we give examples of AG-groupoids.At the end we characterize intra-regular AG-groupoids using the properties of ) , ( -fuzzy semiprime ideals.

INTRODUCTION
The real world has a lot of different aspects which are not usually been specified.In different fields of knowledge like engineering, medical science, mathematics, physics, computer science and artificial intelligence, many problems are simplified by constructing their "models".These models are very complicated and it is impossible to find the exact solutions in many occasions.Therefore, the classical set theory, which is precise and exact, may not be suitable for such problems of uncertainty.
In today's world, many theories have been developed to deal with such uncertainties like fuzzy set theory, theory of vague sets, theory of soft ideals, theory of intuitionist fuzzy sets and theory of rough sets.The theory of soft sets has many applications in different fields such as the smoothness of functions, game theory, operations research, Riemann integration etc.The basic concept of fuzzy set theory was first given by Zadeh (1965).Zadeh (1965) discussed the relationships between fuzzy set theory and probability theory.Rosenfeld (1971) initiated the fuzzy groups in fuzzy set theory.Mordeson et al. (2003) have discussed the applications of fuzzy set theory in fuzzy coding, fuzzy automata and finite state machines.
The idea of belongingness of a fuzzy point to a fuzzy subset under the natural equivalence on a fuzzy subset has been defined by Murali (2004).Bhakat and Das (1992) gave the concept of ( ) , β α -fuzzy subgroups where -fuzzy subgroups is a generalization of fuzzy subgroupoid defined by Rosenfeld (1971).An ) , ( -fuzzy bi-ideals and ) , ( -fuzzy ideals of a semi group are defined in Shabir et al. (2010a).
In this study we have discussed the ) , ( -fuzzy bi-ideals in a new non-associative algebraic structure, that is, in AGgroupoids and developed some new results.We have characterized intra-regular AG-groupoids by the properties of their ) , ( A groupoid is S called an AG-groupoid if it satisfies the left invertive law, that is: Every AG-groupoid satisfies the medial law: It is basically a non-associative algebraic structure in between a groupoid and a commutative semi group.It is important to mention here that if an AG-groupoid contains identity or even right identity, then it becomes a commutative monoid.An AG-groupoid is not necessarily contains a left identity and if it contains a left identity then it is unique (Mushtaq and Yusuf, 1978).An AG-groupoid S with left identity satisfies the paramedial law, that is: It is important to note that an AG-groupoid can also be considered as a fuzzy subset of itself and we can write for all x in S.
Let f and g be any two fuzzy subsets of an AGgroupoid S, then the product g f  is defined by: will means the following fuzzy subsets of S: The following definitions for AG-groupoids are same as for semigroups in Shabir et al. (2010b): Definition 3: A fuzzy subset f of an AG-groupoid S is called an ) , ( the following conditions hold: Definition 5: A fuzzy subset f of an AG-groupoid S is said to be ( ) -fuzzy semiprime if it satisfies Definition 6: Let A be any subset of an AG-groupoid S, is defined as: The proofs of the following four theorems are same as in Shabir et al. (2010b).
Theorem 1: Let f be a fuzzy subset of S. Then f is an ) , ( Theorem 2: A fuzzy subset f of an AG-groupoid S is called an ) , ( -fuzzy left (right) ideal of S if: -fuzzy bi-ideal of S if and only if: Proof: Let f be a fuzzy subset of an AG-groupoid S which is ( ) In addition, we have -fuzzy semiprime.On the other hand, we have 1 ) ( Conversely, assume that f is a fuzzy subset of an AG-groupoid S such that . Now, we consider the following two cases: fuzzy semiprime as required.
Example 1: , then from the following multiplication table one can easily verify that S is an AG-groupoid: Let us define fuzzy subset f of S as: , Definition 7: An AG groupoid S is called intra-regular AG-groupoid if for each a in S there exists , the following table shows that S is an intra-regular AG-groupoid: It is easy to see that ) , ( * S is an AG-groupoid and is non-commutative and non-associative structure because ( ) ( ) Lemma 1: Let A be a non-empty subset of an AGgroupoid S, then: -fuzzy bi-ideal.
Proof: It is same as in Shabir et al. (2010a).
Lemma 2: Let A and B be non-empty subsets of an AGgroupoid S, then the following properties hold: Shabir et al. (2010a).
Theorem 5: Let S be an AG-groupoid with left identity then the following conditions are equivalent: (i) S is intra-regular (ii) For every left ideal L and for every ideal I, Assume that S is an intra-regular AG-groupoid and f and g are ) , ( -fuzzy ideal of S. Since S is intra-regular therefore for any a in S there exist y x, in S such that .) ( 2 y xa a = By using ( 4) and (1): So for any a in S there exist u and v in S such that , uv a = then: ) Let L be the left and I be an ideal of S. Then and ), 4 ( we get: Sa is both left and right ideal therefore it is an ideal containing .
Proof: Let S be an AG-groupoid and f be an ) , ( fuzzy left ideal of S. Then for any x in S there exist a and b in S such that: .
-fuzzy bi-ideal of S Theorem 6: Let S be an AG-groupoid with left identity then the following conditions are equivalent: -fuzzy generalized bi-ideal f and every ) , ( Assume that S is an intra-regular AGgroupoid with left identity and f and g are ) , ( generalized fuzzy bi-ideal of S respectively.Thus, for any a in S there exist u and v in S such that , uv a = then: . Since S is intra-regular so for any a in S there exist S y x ∈ , such that .) ( 2 y xa a = by using (4) and ) 3 ( ), 1 ( and (2) we get: Thus, we have: -fuzzy generalized bi-ideal and g be an ) , ( -fuzzy left ideal.Then by Lemma left ideal is a bi-ideal, g is also an ) , ( Let B be a bi-ideal and L be a left ideal of S. Then Since Sa is both bi-ideal and left ideal containing a. Therefore by ) (ii and using ) 2 ( ), 3 ( and ), 4 ( we obtain: Hence S is an intra-regular AG-groupoid.
Theorem 7: Let S be an AG-groupoid with left identity then the following conditions are equivalent: -fuzzy left ideals f and g, Let S be an intra-regular AG-groupoid and f and g are both ) , ( -fuzzy bi-ideals.For any , S a ∈ there exist u and v in S such that , uv a = then we get: -fuzzy left ideals.Then by Lemma left ideal is a bi-ideal, f and g are ) , ( Since Sa is both bi-ideal and left ideal containing a. Using (ii), we get: Hence S is intra-regular.
Theorem 8: Let S is an AG-groupoid with left identity then the following conditions are equivalent: Assume that S is an intra-regular AGgroupoid and f and g are ) , ( Since S is an intra-regular AG-groupoid therefore for all S a ∈ there exist y x, in S such that .) ( 2 y xa a = By using ( 4) and (1), we get: Assume that f and g are ) , ( -fuzzy left ideals and h is an ) , ( -fuzzy bi-ideal of S then f and g are ) , ( Assume that g f , are left ideals and S is a right ideal.Then by using ), (ii we get: . Hence S is intra- Let L be a left ideal and B be a bi-ideal of S. Then ), (iii Since Sa is both left and bi-ideal.Let .S a ∈ So By using (ii): .
Therefore, S is an intra-regular AG-groupoid.
The proofs of following two lemmas are easy and therefore omitted.Lemma 5: Let S be an intra-regular AG-groupoid then for any ) , ( Lemma 6: Let S be an intra-regular AG-groupoid then for any ) , ( Proof: Let S be an intra-regular AG-groupoid and f and g are any ) , ( -fuzzy-subsets then we get: .
A of an AG-groupoid S the characteristics function, C A is defined by:

Lemma 4 :
For any fuzzy subset f of an AG-groupoid S Let S be an intra-regular AG-groupoid and f and g ideals of S. Then for any a in S there exist x and y in S such , Let S be an AG-groupoid with left identity then the following conditions are equivalent:(i) S is intra-regular.(ii) For every ideals A and B, we will show that f and g are semiprime ideals.So, S is an intra-regular AG-groupoid.