1 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 bi-ideals in semigroups. A new type of fuzzy subgroup, that is
\((\alpha , \beta )\)-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
\((\in , \in \vee q)\)-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
\((\in , \in \vee q)\)-fuzzy sub-near rings of a near ring introduced by Davvaz in [
6]. Kazanci and Yamak [
11] studied
\((\in , \in \vee q)\)-fuzzy bi-ideals of a semigroup. Shabir et al. [
20] characterized regular semigroups by the properties of
\((\in , \in \vee q)\)-fuzzy ideals, fuzzy bi-ideals and fuzzy quasi-ideals. Kazanci and Yamak [
11] defined
\((\overline{\in }, \overline{\in } \vee \overline{q})\)-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
\(f(x)+t+k>1\), where
\(k\in [0,1).\) In [
21], semigroups are characterized by the properties of their
\(\left( \in , \in \vee q_{k}\right)\)-fuzzy ideals.
Faisal and Khan [
15] introduced the concept of an ordered
\(\mathcal {AG}\)-groupoid and provided the basic theory for an ordered
\(\mathcal {AG}\)-groupoid in terms of fuzzy subsets. The generalization of an ordered
\(\mathcal {AG}\)-groupoid was also given by Faisal et al. [
27] and they introduced the notion of an ordered
\(\Gamma\)-
\(\mathcal {AG}^{**}\)-groupoid.
The concept of a left almost semigroup (
\(\mathcal {LA}\)-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 (
\(\mathcal {AG}\)-groupoid). An
\(\mathcal {AG}\)-groupoid is a groupoid
\(\mathcal {S}\) whose elements satisfy the left invertive law
\((ab)c=(cb)a, \forall a,b,c\in \mathcal {S}.\) In an
\(\mathcal {AG}\)-groupoid, the medial law [
12]
\((ab)(cd)=(ac)(bd), \forall a,b,c,d\in \mathcal {S}\) holds. An
\(\mathcal {AG}\)-groupoid may or may not contains a left identity. The left identity of an
\(\mathcal {AG}\)-groupoid allow us to introduce the inverses of elements in an
\(\mathcal {AG}\)-groupoid. If an
\(\mathcal {AG}\)-groupoid contains a left identity, then it is unique [
16]. In an
\(\mathcal {AG}\)-groupoid
\(\mathcal {S}\) with left identity, the paramedial law
\((ab)(cd)=(dc)(ba), \forall a,b,c,d\in \mathcal {S}\) holds. If an
\(\mathcal {AG}\)-groupoid contains a left identity, then by using medial law, we get
\(a(bc)=b(ac), \forall a,b,c\in \mathcal {S}.\) If an
\(\mathcal {AG}\)-groupoid
\(\mathcal {S}\) satisfies
\(a(bc)=b(ac), \forall a,b,c\in \mathcal {S}\) without left identity, then
\(\mathcal {S}\) is called an
\(\mathcal {AG}^{**}\)-groupoid. Several examples and interesting properties of
\(\mathcal {AG}\)-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
\((\in _{\gamma }, \in _{\gamma } \vee q_{\delta })\)-fuzzy sets in ordered
\(\mathcal {AG}\)-groupoids.
\(\mathcal {AG}\)-groupoids are the generalization of the concept of semigroups and it was difficult to handle the results on
\((\in _{\gamma }, \in _{\gamma }\vee q_{\delta })\)-fuzzy sets in ordered
\(\mathcal {AG}\)-groupoids. In this paper we introduce
\((\in _{\gamma }, \in _{\gamma } \vee q_{\delta })\)-fuzzy ideals in an ordered
\(\mathcal {AG}\)-groupoid and introduce some new results which are infect the generalization of the concept of
\((\in _{\gamma }, \in _{\gamma }\vee q_{\delta })\)-fuzzy ideals in an ordered semigroup. We characterize an intra-regular ordered
\(\mathcal {AG}\)-groupoid by the properties of its
\((\in _{\gamma }, \in _{\gamma } \vee q_{\delta })\)-fuzzy ideals.
2 Preliminaries and examples
In this section, we will present some basic definitions needed for next section.
An ordered \(\mathcal {AG}\)-groupoid is the generalization of an ordered semigroup because if an ordered \(\mathcal {AG}\)-groupoid has a right identity then it becomes an ordered semigroup.
Let
\(A\) be a non-empty subset an ordered
\(\mathcal {AG}\)-groupoid
\(G\), then
$$\begin{aligned} \left( A\right] =\left\{ t\in S\;|\;t\le a,\; \text {for}\; \text {some}\;a\in A\right\} . \end{aligned}$$
For
\(A=\{a\},\) we usually written as
\(\left( a\right] .\)
A fuzzy subset
\(f\) of a given set
\(G\) is described as an arbitrary function
\(f:G\longrightarrow [0, 1]\), where
\([0, 1]\) is the usual closed interval of real numbers. For any two fuzzy subsets
\(f\) and
\(g\) of
\(G\),
\(f\subseteq g\) means that,
\(f(x)\le g(x), \forall x\in G\). Let
\(f\) and
\(g\) be any fuzzy subsets of an ordered
\(\mathcal {AG}\)-groupoid
\(G\), then the product
\(f\circ g\) is defined by
$$\begin{aligned} \left( f\circ g\right) (a)= \left\{ \begin{array}{cc} \mathop {\bigvee }\limits _{a\le bc} \left\{ f(b)\wedge g(c)\right\} , &{} \text { if\, there\, exist\, }b, c\in G, \text { such\, that\, }a\le bc \\ 0,&{} \text { otherwise.} \end{array} \right. \end{aligned}$$
Let
\(\mathcal {F}(G)\) denotes the collection of all fuzzy subsets of an ordered
\(\mathcal {AG}\)-groupoid
\(G\), then
\((\mathcal {F}(G),\circ )\) becomes an ordered
\(\mathcal {AG}\)-groupoid [
15].
The characteristic function
\(\chi _{A}\) for a non-empty subset
\(A\) of an ordered
\(\mathcal {AG}\)-groupoid
\(G\) is defined by
$$\begin{aligned} \chi _{A}(x)=\left\{ \begin{array}{cc} 1, &{} \text { if\, }x\in A, \\ 0, &{} \text { if\, }x\notin A. \end{array} \right. \end{aligned}$$
A fuzzy subset
\(f\) of an ordered
\(\mathcal {AG}\)-groupoid
\(G\) of the form
$$\begin{aligned} f(y)=\left\{ \begin{array}{cc} r(\ne 0), &{} if\,y\le x \\ 0, &{} \text { otherwise} \end{array} \right. \end{aligned}$$
is said to be a fuzzy point with support
\(x\) and value
\(r\) and is denoted by
\(x_{r}\), where
\(r\in (0,1].\)
In what follows let \(\gamma , \delta \in [0,1]\) be such that \(\gamma < \delta\). For any \(B\subseteq A,\) we define \(X_{\gamma B}^{\delta }\) be the fuzzy subset of \(X\) by \(X_{\gamma B}^{\delta }(x)\ge \delta\) and \(X_{\gamma B}^{\delta }(x)\le \gamma , \forall x\in B.\) Otherwise, clearly \(X_{\gamma B}^{\delta }\) is the characteristic function of \(B\) if \(\gamma =0\) and \(\delta =1.\)
Now we introduce a new relation on \(\mathcal {F}(G)\), denoted as "\(\subseteq \vee q_{(\gamma , \delta )}\)", as follows.
For any \(f,g \in \mathcal {F}(G),\) by \(f\subseteq \vee q_{(\gamma , \delta )}g,\) we mean that \(x_{r}\in _{\gamma } f \Longrightarrow x_{r}\in _{\gamma } \vee q_{\delta }g, \forall x\in G\) and \(r\in (\gamma , 1].\)
Moreover \(f\) and \(g\) are said to be \((\gamma , \delta )\)-equal, denoted by \(f=_{(\gamma , \delta )}g,\) if \(f\subseteq \vee q_{(\gamma , \delta )}g\) and \(g\subseteq \vee q_{(\gamma ,\delta )}f\).
Proof
\((1):\) Assume that \(A\) and \(B\) are any subset of an ordered \(\mathcal {AG}\)-groupoid \(G.\) Let for any \(x\in G\) such that \(x\in A\subseteq B,\) then by definition, we can write \(\chi _{\gamma B}^{\delta }\ge \delta \rightarrow (i)\). Let \(x_{r}\in _{\gamma }\chi _{\gamma A}^{\delta }\), then \(\chi _{\gamma A}^{\delta } (x)\ge r>\gamma .\) Now two cases arises for Eq. \((i).\)
Case\((a)\): if \(\delta \ge r,\) then \((i)\Rightarrow \chi _{\gamma B}^{\delta }\ge r\), therefore \(x_{r}\in _{\gamma }\chi _{\gamma B}^{\delta }.\)
Case\((b)\): if \(\delta <r\) then \((i)\Rightarrow \chi _{\gamma B}^{\delta }+r>2\delta ,\) therefore \(x_{r}q_{\delta }\chi _{\gamma B}^{\delta }.\)
Hence \(\chi _{\gamma A}^{\delta }\subseteq \vee q_{(\gamma , \delta )} \chi _{\gamma B}^{\delta }.\)
Conversely, let \(\chi _{\gamma A}^{\delta }\subseteq \vee q_{(\gamma , \delta )}\chi _{\gamma B}^{\delta }.\) Let \(x\in A,\) then by definition we can write \(\chi _{\gamma A}^{\delta }\ge \delta .\) Let \(x_{r}\in _{\gamma }\chi _{\gamma A}^{\delta }\subseteq \vee q_{(\gamma , \delta )}\chi _{\gamma B}^{\delta },\) where \(\chi _{\gamma A}^{\delta }\) and \(\chi _{\gamma B}^{\delta }\) are any fuzzy subsets of \(G.\) Thus \(x_{r}\in _{\gamma }\chi _{\gamma A}^{\delta },\)
\(x_{r}\in _{\gamma }\chi _{\gamma B}^{\delta }\) or \(x_{r}q_{\delta }\chi _{\gamma B}^{\delta }.\) As \(x_{r}\in _{\gamma }\chi _{\gamma A}^{\delta }\), then \(\chi _{\gamma A}^{\delta }(x)\ge r>\gamma\) and \(\chi _{\gamma B}^{\delta }(x)\ge r>\gamma\) or \(\chi _{\gamma B}^{\delta }(x)+\delta >2\delta \rightarrow (ii).\) Now here arises two cases for \((ii).\)
Case
\((a)\): if
\(r<\delta ,\) then
$$\begin{aligned} (ii)\Rightarrow \chi _{\gamma B}^{\delta }(x)\ge 2\delta -r>\delta \Longrightarrow \chi _{\gamma B}^{\delta }(x)>\delta \Longrightarrow x_{r}\in _{\gamma }\chi _{\gamma B}^{\delta }. \end{aligned}$$
Case
\((b)\): if
\(r\ge \delta ,\) then
$$\begin{aligned} (ii)\Rightarrow \chi _{\gamma B}^{\delta }(x)\ge r\ge \delta \Longrightarrow \chi _{\gamma B}^{\delta }(x)\ge \delta \Longrightarrow x\in B. \end{aligned}$$
Hence
\(A\subseteq B.\)
\((2):\) Assume that \(\chi _{\gamma A}^{\delta }\) and \(\chi _{\gamma B}^{\delta }\) are any fuzzy subsets of an ordered \(\mathcal {AG}\)-groupoid \(G.\) Let \(x_{r}\in _{\gamma }\chi _{\gamma A}^{\delta }\cap \chi _{\gamma B}^{\delta }\), then \(x_{r}\in _{\gamma }\chi _{\gamma A}^{\delta }\) and \(x_{r}\in _{\gamma }\chi _{\gamma B}^{\delta }\Longrightarrow \chi _{\gamma A}^{\delta }(x)\ge r>\gamma\) and \(\chi _{\gamma B}^{\delta }(x)\ge r>\gamma .\) Let \(x\in A\cap B,\) then by definition \(\chi _{\gamma (A\cap B)}^{\delta }(x)\ge \delta \rightarrow (iii).\) Here arises two cases for \((iii)\).
Case
\((a)\): Let
\(r\le \delta ,\) then
$$\begin{aligned} (iii)\Rightarrow \chi _{\gamma (A\cap B)}^{\delta }(x)\ge \delta \ge r\Rightarrow \chi _{\gamma (A\cap B)}^{\delta }(x)\ge r\Rightarrow x_{r}\in _{\gamma }\chi _{\gamma A\cap B}^{\delta }. \end{aligned}$$
Case
\(\ (b)\): Let
\(r>\delta\) then
$$\begin{aligned} (iii)\Rightarrow \chi _{\gamma A}^{\delta }(x)+r\ge \delta +\delta =2\delta \Rightarrow \chi _{\gamma A}^{\delta }(x)+r>2\delta \Longrightarrow x_{r}q_{\delta }\chi _{\gamma A\cap B}^{\delta }. \end{aligned}$$
Hence
\(\chi _{\gamma A}^{\delta }\cap \chi _{\gamma B}^{\delta }\subseteq q_{(\gamma , \delta )}\chi _{\gamma (A\cap B)}^{\delta }\rightarrow (iv).\)
Assume that for any
\(x\in G,\) there exists a fuzzy subset
\(\chi _{\gamma (A\cap B)}^{\delta }\) of
\(G\) such that
\(x_{r}\in _{\gamma }\chi _{\gamma (A\cap B)}^{\delta }\), then
\(\chi _{\gamma (A\cap B)}^{\delta }(x)\ge r>\gamma .\) Let
\(x\in A\cap B\), then by definition, we can write
\(\chi _{\gamma (A)}^{\delta }(x)\ge \delta\) and
\(\chi _{\gamma (B)}^{\delta }(x)\ge \delta .\) Thus
$$\begin{aligned} \chi _{\gamma (A)}^{\delta }\cap \chi _{\gamma (B)}^{\delta }\ge \delta \cap \delta =\delta \Rightarrow \chi _{\gamma (A)}^{\delta }\cap \chi _{\gamma (B)}^{\delta }\ge \delta \rightarrow (v). \end{aligned}$$
Here arises two cases for
\((v)\).
Case
\((a)\): if
\(r\le \delta ,\) then
$$\begin{aligned} (v)\Rightarrow \chi _{\gamma (A)}^{\delta }\cap \chi _{\gamma (B)}^{\delta }\ge \delta \ge r\Longrightarrow \chi _{\gamma (A)}^{\delta }\cap \chi _{\gamma (B)}^{\delta }\ge r\Longrightarrow x_{r}\in _{\gamma }\chi _{\gamma (A)}^{\delta }\cap \chi _{\gamma (B)}^{\delta }. \end{aligned}$$
Case
\((b)\): if
\(r>\delta ,\) then
$$\begin{aligned} (v)\Rightarrow \chi _{\gamma (A)}^{\delta }\cap \chi _{\gamma (B)}^{\delta }+r\ge \delta +\delta >2\delta \Longrightarrow \chi _{\gamma (A)}^{\delta }\cap \chi _{\gamma (B)}^{\delta }+r>2\delta . \end{aligned}$$
Hence
\(\chi _{\gamma (A\cap B)}^{\delta }\subseteq \vee q_{(\gamma ,\delta )}\chi _{\gamma A}^{\delta }\cap \chi _{\gamma B}^{\delta }\rightarrow (vi). \)From
\((iv)\) and
\((vi),\) we get
\(\chi _{\gamma A}^{\delta }\cap \chi _{\gamma B}^{\delta }=_{(\gamma ,\delta )}\chi _{\gamma (A\cap B)}^{\delta }.\)
\((3):\) Assume that
\(\chi _{\gamma A}^{\delta }\) and
\(\chi _{\gamma B}^{\delta }\) are any fuzzy subsets of
\(G.\) Let
\(x\in G\), then
$$\begin{aligned} x_{r}\in _{\gamma }\chi _{\gamma A}^{\delta }\circ \chi _{\gamma B}^{\delta }\Longrightarrow \chi _{\gamma A}^{\delta }\circ \chi _{\gamma B}^{\delta }\ge r>\gamma . \end{aligned}$$
Now we have to show that
\(x_{r}\in _{\gamma }\chi _{\gamma (AB]}^{\delta }\) or
\(q_{\delta }\chi _{\gamma (AB]}^{\delta }.\) Let
\(x\in (AB],\) then
\(\chi _{\gamma (AB]}^{\delta }(x)\ge \delta \rightarrow (vii).\) Now here arises two cases for
\((vii)\).
Case
\((a)\): if
\(r\le \delta ,\) then
$$\begin{aligned} (vii)\Rightarrow \chi _{\gamma (AB]}^{\delta }(x)\ge \delta \ge r\Longrightarrow \chi _{\gamma (AB]}^{\delta }(x)\ge r\Rightarrow x_{r}\in _{\gamma }\chi _{\gamma (AB]}^{\delta }. \end{aligned}$$
Case
\((b)\): if
\(r>\delta ,\) then
$$\begin{aligned} (vii)\Rightarrow \chi _{\gamma (AB]}^{\delta }(x)+r\ge \delta +\delta >2\delta \Rightarrow \chi _{\gamma (AB]}^{\delta }(x)+r>2\delta \Rightarrow x_{r}q_{\delta }\chi _{\gamma (AB]}^{\delta }. \end{aligned}$$
Hence
\(\chi _{\gamma A}^{\delta }\circ \chi _{\gamma B}^{\delta }\subseteq q_{(\gamma , \delta )}\chi _{\gamma (AB]}^{\delta }\rightarrow (viii).\)
Assume that for any
\(x\in G\) there exists
\(r\in (\gamma , 1]\) and
\(\gamma ,\delta \in [0,1]\) such that
\(x_{r}\in _{\gamma }\chi _{\gamma (AB]}^{\delta }\Longrightarrow \chi _{\gamma (AB]}^{\delta }(x)\ge r>\gamma .\) Let
\(x\in (AB],\) then
\(x\le ab\) for
\(a\in A,\)
\(b\in B,\) and if
\(a\in A,\) then by definition, we can write
\(\chi _{\gamma A}^{\delta }(x)\ge \delta\) and similarly for
\(b\in B\Longrightarrow \chi _{\gamma B}^{\delta }(x)\ge \delta .\) Thus we can write
$$\begin{aligned} \chi _{\gamma A}^{\delta }\circ \chi _{\gamma B}^{\delta }(x)&= \vee _{x\le ab}\{ \chi _{\gamma A}^{\delta }(a)\cap \chi _{\gamma B}^{\delta }(b)\} \\&\ge \vee _{x\le ab}\{ \delta \cap \delta \}=\delta . \end{aligned}$$
Thus
\(\chi _{\gamma A}^{\delta }\circ \chi _{\gamma B}^{\delta }(x)\ge \delta \rightarrow (ix).\) Here arises two cases for
\((ix)\).
Case
\((a)\): if
\(r\le \delta ,\) then
$$\begin{aligned} (ix)\Rightarrow \chi _{\gamma A}^{\delta }\circ \chi _{\gamma B}^{\delta }(a)\ge \delta \ge r\Longrightarrow \chi _{\gamma A}^{\delta }\circ \chi _{\gamma B}^{\delta }(a)\ge r\Rightarrow x_{r}\in _{\gamma }\chi _{\gamma A}^{\delta }\circ \chi _{\gamma B}^{\delta }. \end{aligned}$$
Case
\((b)\): if
\(r>\delta ,\) then
$$\begin{aligned} (ix)\Rightarrow \chi _{\gamma A}^{\delta }\circ \chi _{\gamma B}^{\delta }(a)+r\ge \delta +\delta >2\delta \Longrightarrow \chi _{\gamma A}^{\delta }\circ \chi _{\gamma B}^{\delta }(a)+r>2\delta .So\vee q_{(\gamma , \delta )}\chi _{\gamma A}^{\delta }\circ \chi _{\gamma B}^{\delta }. \end{aligned}$$
Hence
\(\chi _{\gamma (AB]}^{\delta }\subseteq \vee q_{(\gamma , \delta )} \chi _{\gamma A}^{\delta }\circ \chi _{\gamma B}^{\delta }\rightarrow (x).\) From
\((viii)\) and
\((x),\) we get
\(\chi _{\gamma A}^{\delta }\circ \chi _{\gamma B}^{\delta }=_{(\gamma , \delta )}\chi _{\gamma (AB]}^{\delta }.\)
\(\square\)
4 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 AG-groupoids. We introduced \((\in _{\gamma }, \in _{\gamma }\vee q_{\delta })\)-fuzzy (left, right, bi-) ideals of an ordered Abel Grassman’s groupoids and characterized intra-regular ordered \(\mathcal {AG}\)-groupoids in terms of these generalized fuzzy ideals.
In our future study, may be the following topics should be considered:
1.
Characterization of completely regular ordered \(\mathcal {AG}\)-groupoids in terms of \((\in _{\gamma }, \in _{\gamma }\vee q_{\delta })\)-fuzzy ideals.
2.
On \((\in , \in \vee q_{k})\)-fuzzy soft ideals of ordered \(\mathcal {AG}\)-groupoids.
3.
A study of \((\in _{\gamma }, \in _{\gamma }\vee q_{\delta })\)-fuzzy soft ideals in ordered \(\mathcal {AG}\)-groupoids.