On generalized fuzzy ideals of ordered AG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {AG}$$\end{document}-groupoids

We introduced (∈γ,∈γ∨qδ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\in _{\gamma }, \in _{\gamma }\vee q_{\delta })$$\end{document}-fuzzy (left, right, bi-) ideals of an ordered Abel Grassman’s groupoids (AG-groupoid) and characterized intra-regular ordered AG-groupoids in terms of these generalized fuzzy ideals.


Introduction
The fundamental concept of a fuzzy set was introduced by Zadeh in his classic paper [29], which provides a natural framework for generalizing some of the basic notions of algebra. Kuroki [10] introduced the notion of fuzzy biideals in semigroups. A new type of fuzzy subgroup, that is ða; bÞ-fuzzy subgroup, was introduced in an earlier paper of Bhakat and Das [3] by using the notions of ''belongingness and quasi-coincidence'' of fuzzy points and fuzzy sets. The concepts of an ð2; 2 _qÞ-fuzzy subgroup is a useful generalization of Rosenfeld's fuzzy subgroups [19]. It is now natural to investigate similar type of generalizations of existing fuzzy sub-systems of other algebraic structures. The concept of an ð2; 2 _qÞ-fuzzy sub-near rings of a near ring introduced by Davvaz in [6]. Kazanci and Yamak [11] studied ð2; 2 _qÞ-fuzzy bi-ideals of a semigroup. Shabir et al. [20] characterized regular semigroups by the properties of ð2; 2 _qÞ-fuzzy ideals, fuzzy bi-ideals and fuzzy quasi-ideals. Kazanci and Yamak [11] defined ð2; 2 _ qÞ-fuzzy bi-ideals in semigroups. Many other researchers used the idea of generalized fuzzy sets and gave several characterizations results in different branches of algebra. Generalizing the concept of x t qf ; Shabir and Jun [21], defined x t q k f as f ðxÞ þ t þ k [ 1, where k 2 ½0; 1Þ: In [21], semigroups are characterized by the properties of their 2; 2 _q k ð Þ -fuzzy ideals. Faisal and Khan [15] introduced the concept of an ordered AG-groupoid and provided the basic theory for an ordered AG-groupoid in terms of fuzzy subsets. The generalization of an ordered AG-groupoid was also given by Faisal et al. [27] and they introduced the notion of an ordered C-AG ÃÃ -groupoid.
The concept of a left almost semigroup (LA-semigroup) was first introduced by Kazim and Naseeruddin [12] in 1972. In [7], the same structure was called a left invertive groupoid. Protic and Stevanovic [18] called it an Abel-Grassmann's groupoid (AG-groupoid). An AG-groupoid is a groupoid S whose elements satisfy the left invertive law ðabÞc ¼ ðcbÞa; 8a; b; c 2 S: In an AG-groupoid, the medial law [12] ðabÞðcdÞ ¼ ðacÞðbdÞ; 8a; b; c; d 2 S holds. An AG-groupoid may or may not contains a left identity. The left identity of an AG-groupoid allow us to introduce the inverses of elements in an AG-groupoid. If an AG-groupoid contains a left identity, then it is unique [16]. In an AGgroupoid S with left identity, the paramedial law ðabÞðcdÞ ¼ ðdcÞðbaÞ; 8a; b; c; d 2 S holds. If an AG-F. Yousafzai School of Mathematical Sciences, University of Science and Technology of China, Hefei, China e-mail: yousafzaimath@gmail.com groupoid contains a left identity, then by using medial law, we get aðbcÞ ¼ bðacÞ; 8a; b; c 2 S: If an AG-groupoid S satisfies aðbcÞ ¼ bðacÞ; 8a; b; c 2 S without left identity, then S is called an AG ÃÃ -groupoid. Several examples and interesting properties of AG-groupoids can be found in [28], [16] and [22].
Motivated by the study of Khan et al. [14], Yin and Zhan [25] and Yin et al. [26] on generalized fuzzy ideals in ordered semigroups, we study the theory of ð2 c ; 2 c _q d Þfuzzy sets in ordered AG-groupoids. AG-groupoids are the generalization of the concept of semigroups and it was difficult to handle the results on ð2 c ; 2 c _q d Þ-fuzzy sets in ordered AG-groupoids. In this paper we introduce ð2 c ; 2 c _q d Þ-fuzzy ideals in an ordered AG-groupoid and introduce some new results which are infect the generalization of the concept of ð2 c ; 2 c _q d Þ-fuzzy ideals in an ordered semigroup. We characterize an intra-regular ordered AG-groupoid by the properties of its ð2 c ; 2 c _q d Þfuzzy ideals.

Preliminaries and examples
In this section, we will present some basic definitions needed for next section.
Definition 1 An ordered AG-groupoid (po-AG-groupoid) is a structure ðG; :; Þ in which the following conditions hold [15]: An ordered AG-groupoid is the generalization of an ordered semigroup because if an ordered AG-groupoid has a right identity then it becomes an ordered semigroup.
Let A be a non-empty subset an ordered AG-groupoid G, then A ð ¼ t 2 S j t a; for some a 2 A f g : For A ¼ fag; we usually written as a ð : Definition 2 Let G be an ordered AG-groupoid. By a left (right) ideal of G, we mean a non-empty subset A of G such that ðGA AððAG A). By two-sided ideal or simply ideal, we mean a non-empty subset A of G which is both a left and a right ideal of G.

Definition 4 A non-empty subset
A fuzzy subset f of a given set G is described as an arbitrary function f : G À! ½0; 1, where ½0; 1 is the usual closed interval of real numbers. For any two fuzzy subsets f and g of G, f g means that, f ðxÞ gðxÞ; 8x 2 G. Let f and g be any fuzzy subsets of an ordered AG-groupoid G, then the product f g is defined by Let F ðGÞ denotes the collection of all fuzzy subsets of an ordered AG-groupoid G, then ðF ðGÞ; Þ becomes an ordered AG-groupoid [15].
The characteristic function v A for a non-empty subset A of an ordered AG-groupoid G is defined by Definition 9 For a fuzzy point x r and a fuzzy subset f of an ordered AG-groupoid G, we say that: Now we introduce a new relation on F ðGÞ, denoted as '' _q ðc;dÞ '', as follows.
For any f ; g 2 FðGÞ; by f _q ðc;dÞ g; we mean that x r 2 c f ¼) x r 2 c _q d g; 8x 2 G and r 2 ðc; 1: Moreover f and g are said to be ðc; dÞ-equal, denoted by f ¼ ðc;dÞ g; if f _q ðc;dÞ g and g _q ðc;dÞ f . Example 2 Let G ¼ f1; 2; 3g be an ordered AG-groupoid with the multiplication table and order below: Define a fuzzy subset f : G ! ½0; 1 as follows: Then by routine calculation it is easy to observe the following: (i) f is an ð2 0:3 ; 2 0:3 _q 0:4 Þ-fuzzy ideal of G.
Example 3 Let S ¼ f0; 1; 2; 3g be an ordered AG-groupoid with the multiplication table and order below: Again define a fuzzy subset f : S ! ½0; 1 as follows: Then f is an ð2 0:2 ; 2 0:2 _q 0:5 Þ-fuzzy bi-ideal. Lemma CaseðbÞ: if r ! d; then Hence A B: ð2Þ : Assume that v d cA and v d cB are any fuzzy subsets of an ordered AG-groupoid G: Here arises two cases for ðiiiÞ.
CaseðaÞ: Let r d; then Case ðbÞ: Let r [ d then Hence Here arises two cases for ðvÞ. CaseðaÞ: if r d; then CaseðbÞ: if r [ d; then Hence Hence Hence Theorem 2 A fuzzy subset f of an ordered AG-groupoid G is called an ð2 c ; 2 c _q d Þ-fuzzy left (right) ideal of G if for all a; b 2 G and c; d 2 ½0; 1; the following conditions hold: (1) maxff ðaÞ; cg ! minff ðbÞ; dg with a b: (2) maxff ðabÞ; cg ! minff ðbÞ; dg: Proof ðiÞ , ð1Þ : It is same as in Theorem 1. ðiiÞ ) ð2Þ : Assume that a; b 2 G and t; s 2 ðc; 1 such that maxff ðabÞ; cg\t minff ðbÞ; dg:   ) In an ordered AG-groupoid G, the following are true.

Main results
This section contains the main results of the paper.
Definition 14 An element a of an ordered AG-groupoid G is called an intra-regular element of G if there exists x 2 G such that a ðxa 2 Þy and G is called an intra-regular, if every element of G is an intra-regular or equivalently, A ððGA 2 ÞG; 8 A G [15].
From now onward, G will denote an ordered AG-groupoid unless otherwise specified.
Theorem 4 For G with left identity, the following conditions are equivalent.  Therefore G is an intra-regular. h Theorem 5 For G with left identity, the following conditions are equivalent.
(i) G is an intra-regular.
Proof ðiÞ ) ðiiÞ: Assume that f is an ð2 c ; 2 c _q d Þ-fuzzy left ideal, h is an ð2 c ; 2 c _q d Þ-fuzzy right ideal and g is an ð2 c ; 2 c _q d Þ-fuzzy bi-ideal of an intra-regular G with left identity. Then for any a 2 G there exist x; y 2 G such that which shows that f g _ q ðc;dÞ f \ g and similarly we can show that g f _ q ðc;dÞ f \ g: Therefore ðf gÞ \ ðg f Þ _ q ðc;dÞ f \ g: vi ð Þ ¼ ) v ð Þ ¼) iv ð Þ ) iii ð Þ ¼ ) ii ð Þ are obvious cases. ii ð Þ ¼ ) i ð Þ : By using the given assumption, it is easy to show that f g _ q ðc;dÞ f \ g and therefore by using Lemma 7, G is an intra-regular. h

Conclusion
Fuzzy set theory introduced by Zadeh is a generalization of classical set theory. Fuzzy set theory has been advanced to a powerful mathematical theory. In more than 30,000 publications, it has been applied to many mathematical areas, such as algebra, analysis, clustering, control theory, graph theory, measure theory, optimization, operations research, topology, artificial intelligence, computer science, medicine, control engineering, decision theory, expert systems, logic, management science, pattern recognition, robotics and so on. In the present paper we applied the more general form of fuzzy set theory to the theory of AG-groupoids to discuss the fuzzy ideals in AGgroupoids. We introduced ð2 c ; 2 c _q d Þ-fuzzy (left, right, bi-) ideals of an ordered Abel Grassman's groupoids and characterized intra-regular ordered AG-groupoids in terms of these generalized fuzzy ideals. In our future study, may be the following topics should be considered: