Prime Fuzzy Ideals, Completely Prime Fuzzy Ideals of Po-$\Gamma$-Semigroups Based on Fuzzy Points

Abstract

Using fuzzy points the notions of prime fuzzy ideals, weakly prime fuzzy ideals, completely prime fuzzy ideals, and weakly completely prime fuzzy ideals of a po-$\Gamma$-semigroup have been introduced. Some important properties and characterizations of these ideals have been obtained. The relations among various types of primeness have also been investigated.

Author is grateful to University Grants Commission, Govt. of India, for providing research support as SRF.

1 Introduction

Uncertainty is an attribute of information and uncertain data are presented in various domains. The most appropriate theory for dealing with uncertainties was introduced by Zadeh [28] in 1965 by defining fuzzy set which has opened up keen insights and applications in vast range of scientific fields. Rosenfeld [16] pioneered the study of fuzzy algebraic structures by introducing the notions of fuzzy groups and showed that many results in groups can be extended in an elementary manner to develop algebraic concepts. After that Kuroki [12, 14] started the study of fuzzy ideal theory in semigroups. Xie [26] used the notion of fuzzy points to introduce prime fuzzy ideals in semigroups. The notion of $\Gamma$-semigroups was introduced by M.K. Sen [22] as a generalization of semigroups. T.K. Dutta and N.C. Adhikari [5] developed the theory of $\Gamma$-semigroups by introducing the notion of operator semigroups. $\Gamma$-semigroups have also been the object of study of many researchers like Chattopadhyay [1, 8], Chinram et al. [2]. The notion of $\Gamma$-semigroups has been extended to fuzzy setting by S.K. Sardar and S.K. Majumder [17–19]. They have studied fuzzy ideals, fuzzy prime ideals, fuzzy semiprime ideals, and fuzzy ideal extensions in $\Gamma$-semigroups directly as well as via operator semigroups. Sen and Seth [24] introduced the notion of po-$\Gamma$-semigroups. Among the other papers of po-$\Gamma$-semigroups we refer to [7, 25]. Kehayopulu has contributed a lot to the ordered semigroups by using fuzzy notion [9, 10]. In this paper we investigate in po-$\Gamma$-semigroups the validity of various properties of prime fuzzy ideals of semigroups [26], $\Gamma$-semigroups [18, 21] as well as of po-semigroups [10, 27]. We study here prime fuzzy ideals, weakly prime fuzzy ideals, completely prime fuzzy ideals, and weakly completely prime fuzzy ideals in po-$\Gamma$-semigroups by using the notion of fuzzy points.

It is important to mention here as to why different types of prime ideals arise in fuzzy setting in contrast with the crisp setting of semigroups or $\Gamma$-semigroups. When we formulate some fuzzy notions, to check the correctness of the formulation, we always verify whether the level subset criterion and characteristic function criterion are satisfied. Some situations are very nice where translations of crisp notions to fuzzy setting become compatible with the level subset criterion and characteristic function criterion. But in case of prime fuzzy ideals the situation is not so nice. Just by analogy with the definition of prime ideal in crisp algebra (cf. Definition 4.1) if we define prime fuzzy ideal (cf. Definition 4.3) in po-$\Gamma$-semigroups then we see that level subset criterion does not hold (cf. Example 4.13). In order to make the notion compatible with the level subset criterion (cf. Theorem 4.18) the notion of weakly prime fuzzy ideal (cf. Definition 4.17) is introduced.

We organize the paper as follows. In Sect. 2 we recall some preliminary notions of po-$\Gamma$-semigroups as well as of fuzzy subsets. In Sect. 3 we define fuzzy points and their composition in a po-$\Gamma$-semigroup and subsequently characterize composition of two fuzzy points in po-$\Gamma$-semigroups (cf. Theorem 3.2). Also some related properties of fuzzy points are studied in this section. In Sect. 4 prime fuzzy ideals of po-$\Gamma$-semigroups are defined. We then obtain various properties of prime fuzzy ideals (cf. Proposition 4.8, Theorems 4.7, 4.10, Corollary 4.11). An important characterization of prime fuzzy ideals is also obtained (cf. Theorem 4.15). Weakly prime fuzzy ideals of $\Gamma$-semigroups are then defined and studied. It is shown that unlike prime fuzzy ideals they satisfy level subset criterion (cf. Theorem 4.18). Some other important properties of weakly prime fuzzy ideals are also obtained (cf. Theorem 4.23). In Sect. 5 we introduce the notion of completely prime fuzzy ideals (cf. Definition 5.1) and weakly completely prime fuzzy ideals (cf. Definition 5.2) in po-$\Gamma$-semigroups and study their properties (cf. Theorems 5.3, 5.5–5.7).

2 Preliminaries

In this section we recall some elementary notions for their use in the sequel.

Definition 2.1

([23]) Let $S=\{x,y,z,\ldots \}$ and $\Gamma =\{\alpha ,\beta ,\gamma \ldots \}$ be two nonempty sets. Then S is called a $\Gamma$-semigroup if there exists a mapping $S\times \Gamma \times S\rightarrow S$, written as $(a,\alpha ,b)\rightarrow a\alpha b$ satisfying (1) $x\gamma y\in S,(2)$ $(x\beta y)\gamma z=x\beta (y\gamma z)$, for all $x,y,z\in S,\alpha ,\beta ,\gamma \in \Gamma .$

Remark 2.2

Definition 2.1 is the definition of one-sided $\Gamma$-semigroup. It may be noted here that in 1981, Sen [22] introduced the notion of both-sided $\Gamma$-semigroups.

Example 2.3

([22]) Let S be the set of all $2\times 3$ matrices over the set of positive integers and $\Gamma$ be the set of all $3\times 2$ matrices over same set. Then S is a both-sided as well as a one-sided $\Gamma$-semigroup with respect to the usual matrix multiplication.

The following example shows that there exists a one-sided $\Gamma$-semigroup which is not a both-sided $\Gamma$-semigroup.

Example 2.4

([1]) Let S be a set of all negative rational numbers. Obviously, S is not a semigroup under usual product of rational numbers. Let $\Gamma$ = {$-\frac{1}{p}$: p is prime}. Let $a, b, c \in S$ and $\alpha ,\beta \in \Gamma$. Now if $a\alpha b$ is equal to the usual product of rational numbers $a, \alpha , b$ then $a\alpha b \in S$ and $(a\alpha b)\beta c = a\alpha (b\beta c)$. Hence S is a one-sided $\Gamma$-semigroup. It is also clear that it is not a both-sided $\Gamma$-semigroup.

Definition 2.5

([24]) A $\Gamma$-semigroup S is said to be a po-$\Gamma$-semigroup (partially ordered $\Gamma$-semigroup) if (1) $(S,\le )$ and $(\Gamma ,\le )$ are posets, (2) $a\le b$ in S implies that $a\alpha c\le b\alpha c,$ $c\alpha a\le c\alpha b$ in S and $\alpha \le \beta$ in $\Gamma$ implies $a\alpha b\le a\beta b$ in S for all $a,b,c\in S$ and for all $\alpha ,\beta \in \Gamma .$

Example 2.6

([24]) Let S be the set of all isotone mappings from a poset P to another poset Q and $\Gamma$ be the set of all isotone mappings from a poset Q to another poset P. Let $f,g\in S$ and $\alpha \in \Gamma$. Then $f\alpha g$ denotes the usual mapping composition of $f,\alpha$ and g. The relation $\le$ on S defined by $f\le g$ if and only if $f(a)\le g(a)$ for all $a\in P$ is a partial order on S. In a similar fashion $\Gamma$ can be made into a poset. It can be verified that S is a po-$\Gamma$-semigroup.

Remark 2.7

Definition 2.5 is the definition of one-sided po-$\Gamma$-semigroups. It may be noted that the definition of both-sided po-$\Gamma$-semigroups [7] was introduced by T.K. Dutta and N.C. Adhikari. Throughout this paper unless otherwise mentioned S, a po-$\Gamma$-semigroup, is considered to be one sided.

Definition 2.8

A po-$\Gamma$-semigroup S is called a commutative po-$\Gamma$-semigroup if $a\alpha b=b\alpha a$, for all $a,b\in S$ and $\alpha \in \Gamma$.

Definition 2.9

([13]) Let S be a po-$\Gamma$-semigroup. A nonempty subset I of S is said to be a right ideal (left ideal) of S if (1) $I\Gamma S\subseteq I$ (resp. $S\Gamma I\subseteq I),$ (2) $a\in I$ and $b\le a$ imply $b\in I.$ I is said to be an ideal of S if it is a right ideal as well as a left ideal of S.

Definition 2.10

([13]) Let A be a subset of a po-$\Gamma$ semigroup S. Then we define $(A]:=\{x\in S:x\le y$ for some $y\in A\}.$ If A is a singleton $\{a\}$, then for simplification we write (a] instead of $(\{a\}]$.

Proposition 2.11

([11]) Let S be a po-$\Gamma$-semigroup, A and B be two nonempty subsets of S. Then $(A]\Gamma (B]\subseteq (A\Gamma B].$

Moreover, if A and B are any two ideals (left, right or both sided) of S, then

• (1) $(A]=A, (B]=B,$

• (2) $(A\cap B]=(A]\cap (B],$ and

• (3) $(A\cup B]=(A]\cup (B].$

Definition 2.12

([28]) A fuzzy subset $\mu$ of a nonempty set X is a function $\mu :X\rightarrow [0,1].$

Definition 2.13

([4]) Let $\mu$ be a fuzzy subset of a nonempty set X. Then the set $\mu _{t}=\{x\in X:\mu (x)\ge t\}$ for $t\in [0,1],$ is called the level subset or t-level subset of $\mu .$

Definition 2.14

([20]) Let f and g be two fuzzy subsets of a po-$\Gamma$-semigroup S. Then

$$\begin{aligned} (f\circ g)(x)=\left\{ \begin{array}{l} \underset{x\le y\gamma z}{\sup }\{\min \{f(y),g(z)\}\}\text { if there exist }y,z\in S,\gamma \in \Gamma \text { with }x\le y\gamma z, \\ 0 \text { otherwise. } \end{array} \right. \end{aligned}$$

Definition 2.15

([20]) A nonempty fuzzy subset f of a po-$\Gamma$-semigroup S is called a fuzzy left (right) ideal of S if

• (1) $f(x\alpha y)\ge f(y)$ $(f(x\alpha y)\ge f(x))$, for all $x,y\in S,$ $\alpha \in \Gamma$,

• (2) $b\le a\Rightarrow f(b)\ge f(a)$, for all $a,b\in S.$

f is called a fuzzy ideal if f is both fuzzy left ideal and fuzzy right ideal.

3 Some Results of Fuzzy Points in Po-$\Gamma$-Semigroups

Definition 3.1

([15]) Let S be a po-$\Gamma$-semigroup of S. Let $a\in S$ and $t\in (0,1].$ We define a fuzzy subset $a_{t}$ of S as follows:

$$\begin{aligned} a_{t}(x)=\left\{ \begin{array}{ll} t&{}\text { if }\;x\le a, \\ 0 &{}\text { otherwise} \end{array} \right. \end{aligned}$$

for all $x\in S.$ We call $a_{t}$ a fuzzy point or fuzzy singleton of S.

Theorem 3.2

([15]) Let $a_{t}$ and $b_{r}$ be two fuzzy points of a po-$\Gamma$-semigroup S. Then

$$ a_{t}\circ b_{r}=\underset{\gamma \in \Gamma }{\cup }(a\gamma b)_{t\wedge r}. $$

Remark 3.3

For any fuzzy subset f of a po-$\Gamma$-semigroup S, $f=\underset{a_{t}\subseteq f}{\cup }a_{t}.$

The following lemma follows easily.

Lemma 3.4

Let S be a po-$\Gamma$-semigroup, f,  g,  and h be fuzzy subsets of S. Then $f\circ (g\cup h)=(f\circ g)\cup (f\circ h).$

Definition 3.5

Let S be a po-$\Gamma$-semigroup and $a_{t}$ be a fuzzy point of S. Then the fuzzy ideal generated by $a_{t}$ denoted by $<a_{t}>$, is defined to be the smallest fuzzy ideal containing $a_{t}$ in S.

Proposition 3.6

Let S be a po-$\Gamma$-semigroup and $a_{t}$ be a fuzzy point of S. Then the fuzzy ideal $\,<{}a_{t}{}>$ generated by $a_{t}$ is given by

$$\begin{aligned} <a_{t}>(x)=\left\{ \begin{array}{ll} t,~~~~~~~&{}\text {if }\;x\in \,<{}a{}>, \\ 0,~~~~~~~&{}\text {otherwise, } \end{array} \right. \end{aligned}$$

for any $x\in S,$ where $\,<{}a{}>$ is the ideal of S generated by a.

Proof

Let us consider a fuzzy subset g of S defined by

$$\begin{aligned} g(x)=\left\{ \begin{array}{ll} t,~~~~~~~&{}\text {if }\;x\in \,<{}a{}>, \\ 0,~~~~~~~&{}\text {otherwise, } \end{array} \right. \end{aligned}$$

for any $x\in S,$ where $\,<{}a{}>$ is the ideal of S generated by a. Let $x,y\in S$ and $\gamma \in \Gamma .$ If $x,y\in \,<{}a{}>$, then $x\gamma y\in \,<{}a{}>$. So $g(x\gamma y)=t=g(x)=g(y).$ Again if $x,y\not \in \,<{}a{}>$ but $x\gamma y\in \,<{}a{}>$, then $g(x\gamma y)=t\ge 0=g(x)=g(y).$ If $x,y\not \in \,<{}a{}>$ and $x\gamma y\not \in \,<{}a{}>$, then $g(x\gamma y)=0=g(x)=g(y).$ Again if $x\in \,<{}a{}>$ and $y\not \in \,<{}a{}>$, then $x\gamma y\in \,<{}a{}>$. So $g(x\gamma y)=t=g(x)\ge 0=g(y).$ Let $x\le y$ in S. If $y\in \,<{}a{}>,$ then $x\in \,<{}a{}>$ whence $g(x)=t=g(y)$. Again if $y\not \in \,<{}a{}>,$ then $g(x)\ge 0=g(y)$. So g is a fuzzy ideal.

Let f be a fuzzy ideal of S such that $a_{t}\subseteq f.$ Then $f(a)\ge a_{t}(a)=t.$ Now let $z\in \,<{}a{}>=(\{a\}\cup a\Gamma S\cup S\Gamma a\cup S\Gamma a\Gamma S].$ If $z\le a$, then $f(z)\ge f(a)\ge t.$ If $z\le a\alpha x$ for some $x\in S$ and $\alpha \in \Gamma$, then $f(z)\ge f(a\alpha x)\ge f(a)\ge t$ (cf. Definition 2.15). Similarly, we can show that $f(z)\ge t$ if $z\le y\beta a$ or $z\le x\alpha a\beta y$ for some $x,y\in S$ and $\alpha ,\beta \in \Gamma .$ It follows that $g\subseteq f.$ Since $g(x)\ge a_{t}(x)$, for all $x\in S$, g contains $a_{t}.$ This completes the proof. $\square$

Remark 3.7

From the above result, we notice that $<a_{t}>\,=\,tC_{\,<{}a{}>}$ where $C_{\,<{}a{}>}$ is the characteristic function of $\,<{}a{}>$.

Proposition 3.8

Let S be a po-$\Gamma$-semigroup and $a_{t}$ be a fuzzy point of S. Then

$$\begin{aligned} S\circ a_{t}\circ S(x)=\left\{ \begin{array}{ll} t,~~~~~~&{}\text {if }\;x\in (S\Gamma a\Gamma S], \\ 0,~~~~~~&{}\text {otherwise, } \end{array} \right. \end{aligned}$$

for all $x\in S.$ Moreover, $S\circ a_{t}\circ S$ is a fuzzy ideal of S.

Proof

Let $x\in S.$ If $x\nleq w\alpha z\beta y$ for any $w,z,y\in S$ and $\alpha ,\beta \in \Gamma$, then $x\not \in (S\Gamma a\Gamma S]$ and $S\circ a_{t}\circ S(x)=0.$ Now let $x\le w\alpha z\beta y$ for some $w,z,y\in S$ and $\alpha ,\beta \in \Gamma .$ Then

$$\begin{aligned} S\circ a_{t}\circ S(x)= & {} \underset{x\le p\gamma q}{\vee }\{S\circ a_{t}(p)\wedge S(q)\}\\= & {} \underset{x\le p\gamma q}{\vee }\{S\circ a_{t}(p)\}\\= & {} \underset{x\le s\delta r\gamma q}{\vee }\{S(s)\wedge a_{t}(r)\}\\= & {} \underset{x\le s\delta r\gamma q}{\vee } a_{t}(r). \end{aligned}$$

If there exists one $r=a$, then $a_{t}(r)=t$ whence $S\circ a_{t}\circ S(x)=t$. Thus if $x\in (S\Gamma a\Gamma S]$, then $S\circ a_{t}\circ S(x)=t$, otherwise $S\circ a_{t}\circ S(x)=0.$

In order to prove the last part we see $S\circ (S\circ a_{t}\circ S)\subseteq S\circ a_{t}\circ S$ and $(S\circ a_{t}\circ S)\circ S\subseteq S\circ a_{t}\circ S.$ Let $x\le y$ in S. If $x\in (S\Gamma a\Gamma S]$, then $S\circ a_{t}\circ S(x)=t\ge S\circ a_{t}\circ S(y)$. If $x\not \in (S\Gamma a\Gamma S]$, then $y\not \in (S\Gamma a\Gamma S]$ whence $S\circ a_{t}\circ S(x)=0=S\circ a_{t}\circ S(y)$. Hence $S\circ a_{t}\circ S$ is a fuzzy ideal of S.$\square$

The following result is an easy consequence of the above proposition.

Corollary 3.9

Let S be a po-$\Gamma$-semigroup and $a_{t}$ be a fuzzy point of S. Then

$$\begin{aligned} S\circ a_{t}(x)=\left\{ \begin{array}{ll} t,~~~~~~&{}\text {if }\;x\in (S\Gamma a], \\ 0,~~~~~~&{}\text {otherwise, } \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} a_{t}\circ S(x)=\left\{ \begin{array}{ll} t,~~~~~~&{}\text {if }\;x\in (a\Gamma S], \\ 0,~~~~~~&{}\text {otherwise, } \end{array} \right. \end{aligned}$$

for all $x\in S.$ Moreover, $S\circ a_{t}$ is a fuzzy left ideal of S and $a_{t}\circ S$ is a fuzzy right ideal of S.

Remark 3.10

From Proposition 3.8 and Corollary 3.9 we notice that $S\circ a_{t}\circ S=tC_{(S\Gamma a\Gamma S]}$, $S\circ a_{t}=tC_{(S\Gamma a]}$, and $a_{t}\circ S=tC_{(a\Gamma S]}$.

Proposition 3.11

Let S be a po-$\Gamma$-semigroup and $a_{t}$ be a fuzzy point of S. Then $<a_{t}>\,=\,a_{t}\cup a_{t}\circ S\cup S\circ a_{t}\cup S\circ a_{t}\circ S.$

Proof

By Proposition 3.6, for any $x\in S,$

$$\begin{aligned} <a_{t}>(x)=\left\{ \begin{array}{ll} t,~~~~~~&{}\text { if }\;x\in {}\,<{}a{}>, \\ 0,~~~~~~&{}\text {otherwise. } \end{array} \right. \end{aligned}$$

Let $x\in S$. If $x\not \in \,<a{}>$, then $<a_{t}>(x)=0$. In view of Proposition 2.11, $\,<{}a{}>$ $=$ $(\{a\}\cup S\Gamma a\cup a\Gamma S\cup S\Gamma a\Gamma S]$ $=$ $(\{a\}]$ $\cup$ $(S\Gamma a]$ $\cup$ $(a\Gamma S]$ $\cup$ $(S\Gamma a\Gamma S]$. So $x\not \in (S\Gamma a\Gamma S]$ whence $S\circ a_{t}\circ S(x)=0$; $x\not \in (S\Gamma a]$ whence $S\circ a_{t}(x)=0$; $x\not \in (a\Gamma S]$ whence $a_{t}\circ S(x)=0$; and $x\nleq a$ whence $a_{t}(x)=0.$ Hence $a_{t}\cup a_{t}\circ S\cup S\circ a_{t}\cup S\circ a_{t}\circ S(x)=0.$ If $x\in \,<{}a{}>$, then $<a_{t}>(x)=t.$ Again in view of Proposition 2.11, $\,<{}a{}>$ $=$ $(\{a\}\cup S\Gamma a\cup a\Gamma S\cup S\Gamma a\Gamma S]$ $=$ $(\{a\}]$ $\cup$ $(S\Gamma a]$ $\cup$ $(a\Gamma S]$ $\cup$ $(S\Gamma a\Gamma S]$. Now $x\in (S\Gamma a\Gamma S]$ whence $S\circ a_{t}\circ S(x)=t$; $x\in (S\Gamma a]$ whence $S\circ a_{t}(x)=t$; $x\in (a\Gamma S]$ whence $a_{t}\circ S(x)=t$; and $x\le a$ whence $a_{t}(x)=t.$ Hence $a_{t}\cup a_{t}\circ S\cup S\circ a_{t}\cup S\circ a_{t}\circ S(x)=t.$ Consequently, $<a_{t}>\,=a_{t}\cup a_{t}\circ S\cup S\circ a_{t}\cup S\circ a_{t}\circ S.$ $\square$

We omit the proof of the following Corollary since it is similar to that of Corollary 1 of [21].

Corollary 3.12

Let S be a po-$\Gamma$-semigroup and $a_{t}$ be a fuzzy point of S. Then $<a_{t}>^3\,\subseteq S\circ a_{t}\circ S.$

Though the following proposition is easy to obtain, it is also useful for the development of the paper.

Proposition 3.13

Let S be a po-$\Gamma$-semigroup, A and B be subset of S and $C_{A}$ be the characteristic function of A. Then for any $t,r\in (0,1]$, the following statements hold.

• (i) $tC_{A}\circ rC_{B}= (t\wedge r)C_{(A\Gamma B]}.$

• (ii) $tC_{A}\cap tC_{B}=tC_{A\cap B}.$

• (iii) $tC_{(A]}=\underset{a\in A}{\cup } a_{t}.$

• (iv) $S\circ tC_{A}=tC_{(S\Gamma A]}.$

• (v) A is an ideal (right ideal, left ideal) of S if and only if $tC_{A}$ is a fuzzy ideal (fuzzy right ideal, fuzzy left ideal) of S.

4 Prime Fuzzy Ideals and Weakly Prime Fuzzy Ideals in Po-$\Gamma$-Semigroups

In this section, we deduce various properties and characterizations of prime fuzzy ideals and weakly prime fuzzy ideals of po-$\Gamma$-semigroups.

Definition 4.1

([3]) Let S be a po-$\Gamma$-semigroup. Then an ideal $I({\ne }S)$ of S is called if for any two ideals A and B of S, $A\Gamma B\subseteq I$ implies $A\subseteq I$ or $B\subseteq I.$

Definition 4.2

([3]) Let S be a po-$\Gamma$-semigroup. Then an ideal $I({\ne }S)$ of S is called completely if for any $a,b\in S$, $a\Gamma b\subseteq I$ implies $a\in I$ or $b\in I.$

Definition 4.3

Let S be a po-$\Gamma$-semigroup. Then a fuzzy ideal f of S is called prime fuzzy if f is a nonconstant function and for any two fuzzy ideals g and h of S, $g\circ h\subseteq f$ implies $g\subseteq f$ or $h\subseteq f.$

Example 4.4

Let $S=\mathbb {Z}_{0}^{-}$ and $\Gamma =\mathbb {Z}_{0}^{-}$, where $\mathbb {Z}_{0}^{-}$ denotes the set of all negative integers with 0. Then S is a $\Gamma$-semigroup where $a\gamma b$ denotes the usual multiplication of integers $a,\gamma ,b$ where $a,b\in S$ and $\gamma \in \Gamma$. Again with respect to usual $\le$ of $\mathbb {Z}$, S becomes a po-$\Gamma$-semigroup. Let p be a prime number. Now we define a fuzzy subset f on S by

$$\begin{aligned} f(x)=\left\{ \begin{array}{ll} 1,~~~~~~&{}\text {for }\;x\in (p\mathbb {Z}_{0}^{-}],\\ 0.6,~~~~~~&{}\text{ otherwise. } \end{array} \right. \end{aligned}$$

Then f is a prime fuzzy ideal of S.

Example 4.5

  Let $S=\{a,b,c\}$. Let $\Gamma =\{\alpha ,\beta \}$ be the nonempty set of binary operations on S with the following Cayley tables.

$\alpha$	a	b	c	$\beta$	a	b	c
a	a	b	b	a	b	b	b
b	b	b	b	b	b	b	b
c	c	c	c	c	c	c	c

By a routine verification, we see that S is a po-$\Gamma$-semigroup where the partial orders on S and $\Gamma$ are given by $c\le b\le a$ and $\beta \le \alpha ,$ respectively. Now we define a fuzzy subset $\mu$ on S by $\mu (a)=0.5$, $\mu (b)=1=\mu (c)$. It is easy to observe that $\mu$ is a prime fuzzy ideal of S.

Though the proof of the following theorem is straightforward, it also characterizes a prime fuzzy ideal.

Theorem 4.6

Let S be a commutative po-$\Gamma$-semigroup and f be a fuzzy ideal of S. Then f is prime fuzzy ideal if and only if for any fuzzy subsets g and h of S, $g\circ h\subseteq f$ implies $g\subseteq f$ or $h\subseteq f.$

Theorem 4.7

Let S be a po-$\Gamma$-semigroup and I be an ideal of S. Then I is a prime ideal of S if and only if $C_{I}$, the characteristic function of I, is a prime fuzzy ideal of S.

Proof

Let I be a prime ideal of S. Then $C_{I}$ is a fuzzy ideal of S (cf. Proposition 3.13). Now let f and g be two fuzzy ideals of S such that $f\circ g\subseteq C_{I}$ and $f\nsubseteq C_{I}$. Then there exists a fuzzy point $x_{t}\subseteq f$ ($t> 0$) such that $x_{t}\nsubseteq C_{I}.$ Let $y_{r}\subseteq g$ ($r>0$). Then $<x_{t}>\circ <y_{r}>\,\subseteq f\circ g\subseteq C_{I}.$ Again for all $z\in S,$ in view of Propositions 3.6 and 3.13, we obtain

$$\begin{aligned} <x_{t}>\circ <y_{r}>(z)=\left\{ \begin{array}{l} t\wedge r,~~~~~~\text {if }\;z\in (<x>\Gamma <y>], \\ 0,~~~~~~~~~~~\text {otherwise. } \end{array} \right. \end{aligned}$$

Hence $(<x>\Gamma <y>]\subseteq I.$ Using Proposition 2.11 we see that $(<x>]\Gamma (<y>]\subseteq (<x>\Gamma <y>]$ whence $(<x>]\Gamma (<y>]\subseteq I$. This together with the hypothesis implies that $(<x>]\subseteq I$ or $(<y>]\subseteq I$ (cf. Definition 4.1). As $<x>\,\subseteq (<x>]$ and $<y>\,\subseteq (<y>]$, we have $<x>\,\subseteq I$ or $<y>\,\subseteq I$. Since $x_{t}\nsubseteq C_{I}$, $t=x_{t}(x)> C_{I}(x).$ So $C_{I}(x)=0$ whence $x\not \in I.$ Hence $<x>\,\nsubseteq I$. Consequently, $<y>\,\subseteq I.$ Then $y_{r}\subseteq C_{I}$ and so $g\subseteq C_{I}.$ Hence $C_{I}$ is a prime fuzzy ideal of S.

Conversely, suppose $C_{I}$ is a prime fuzzy ideal of S. Then $C_{I}$ is a fuzzy ideal of S which together with Proposition 3.13 implies that I is an ideal of S. Let A and B be two fuzzy ideals of S such that $A\Gamma B\subseteq I.$ Then $(A\Gamma B]\subseteq I.$ Again $C_{A}$, and $C_{B}$ are fuzzy ideals of S and $C_{A}\circ C_{B}= C_{(A\Gamma B]}\subseteq C_{I}$ (cf. Proposition 3.13). So by hypothesis, $C_{A}\subseteq C_{I}$ or $C_{B}\subseteq C_{I}.$ Hence $A\subseteq I$ or $B\subseteq I.$ Consequently, I is a prime ideal of S. $\square$

Proposition 4.8

Let S be a po-$\Gamma$-semigroup and f be a prime fuzzy ideal of S. Then $|Im f|=2.$

Proof

By Definition 4.3, f is a nonconstant fuzzy ideal. So $|Im f|\ge 2$. Suppose $|Im f|> 2.$ Then there exist $x,y,z\in S$ such that f(x), f(y), f(z) are distinct. Let us assume, without loss of generality, $f(x)<f(y)<f(z).$ Then there exist $r,t\in (0,1)$ such that $f(x)<r<f(y)<t<f(z)\cdots (1)$. Then for all $u\in S,$

$$\begin{aligned} <x_{r}>\circ <y_{t}>(u)=\left\{ \begin{array}{l} r\wedge t,~~~~\text {if }\;u\in (<x>\Gamma <y>], \\ 0,~~~~~~~~~\text {otherwise. } \end{array} \right. \end{aligned}$$

Let $u\in (<x>\Gamma <y>].$ Then $u\le a\gamma b$ where $a\in \,<x>$, $b\in \,<y>$ and $\gamma \in \Gamma$. Since f is a fuzzy ideal of S, $f(u)\ge f(a\gamma b)\ge f(x)\vee f(y) > r\wedge t.$ Therefore $<x_{r}>\circ <y_{t}>\,\subseteq f$ which, by Definition 4.3, implies that $<x_{r}>\,\subseteq f$ or $<y_{t}>\,\subseteq f$. Suppose $<x_{r}>\,\subseteq f.$ Then $f(x)\ge \,<x_{r}>(x)=r$ which contradicts (1). Similarly, $<y_{t}>\,\subseteq f$ contradicts (1). Hence $|Im f|=2.$ $\square$

Theorem 4.9

Let S be a po-$\Gamma$-semigroup and f be a prime fuzzy ideal of S. Then there exists $x_{0}\in S$ such that $f(x_{0})=1.$

Proof

By Proposition 4.8, we have $|Im f|=2.$ Suppose $Im f=\{t,s\}$ such that $t<s$. Let if possible $f(x)<1$, for all $x\in S.$ Then $t<s<1.$ Let $f(x)=t$ and $f(y)=s$ for some $x,y\in S$. Then $f(x)=t<s=f(y)< 1.$ Now we choose $t_{1},t_{2}\in (0,1)$ such that $f(x)<t_{1}<f(y)<t_{2}<1$. Then by the similar argument as applied in the proof of Proposition 4.8, we obtain $<x_{t_{1}}>\circ <y_{t_{2}}>\,\subseteq f.$ Since f is a prime fuzzy ideal of S,  $<x_{t_{1}}>\,\subseteq f$ or $<y_{t_{2}}>\,\subseteq f$ whence $f(x)\ge t_{1}$ or $f(y)\ge t_{2}.$ This contradicts the choices of $t_{1}$ and $t_{2}$. Hence there exists an $x_{0}\in S$ such that $f(x_{0})=1$. $\square$

Theorem 4.10

Let S be a po-$\Gamma$-semigroup and f be a prime fuzzy ideal of S. Then each level subset $f_{t}$ (${\ne }S$), $t\in (0,1]$, if nonempty, is a prime ideal of S.

Proof

Since f is a fuzzy ideal, each level subset $f_{t},$ $t\in (0,1]$, if nonempty, is an ideal of S (cf. Theorem 3.5 [20]). Let $t\in (0,1]$ be such that $f_{t}$ (${\ne }S$) is nonempty. Now let I, J be two ideals of S such that $I\Gamma J\subseteq f_{t}.$ Since $f_{t}$ is an ideal of S, $(I\Gamma J]\subseteq f_{t}.$ Then $tC_{(I\Gamma J]}\subseteq f.$ By Proposition 3.13(v), $g:=tC_{I}$ and $h:=tC_{J}$ are fuzzy ideals of S. Since $g\circ h=tC_{I}\circ tC_{J}=tC_{(I\Gamma J]}$ (cf. Proposition 3.13(i)), $g\circ h\subseteq f.$ Since f is a prime fuzzy ideal, $g\subseteq f$ or $h\subseteq f.$ Hence either $tC_{I}\subseteq f$ or $tC_{J}\subseteq f$ whence we obtain $I\subseteq f_{t}$ or $J\subseteq f_{t}.$ Hence $f_{t}$ is a prime ideal of S.    $\square$

As a consequence of Theorems 4.9 and 4.10, we obtain the following result.

Corollary 4.11

If f is a prime fuzzy ideal of a po-$\Gamma$-semigroup S, then the level subset $f_{1}$ is a prime ideal of S.

Remark 4.12

The converse of Theorem 4.10 is not true which is illustrated in the following example.

Example 4.13

Let S be a po-$\Gamma$-semigroup and A be a prime ideal of S. Let

$$\begin{aligned} f(x)=\left\{ \begin{array}{ll} t,~~~~~~&{}\text {if }\;x\in A, \\ 0,~~~~~~&{}\text {otherwise. } \end{array} \right. \end{aligned}$$

Then f is a fuzzy ideal of S. Here $f_{t_{1}}=A,$ where $0<t_{1}\le t$. Hence each of nonempty level subsets of f is a prime ideal of S. But if $0<t<1$, then f is not a prime fuzzy ideal of S (cf. Theorem 4.9).

Lemma 4.14

Let S be a po-$\Gamma$-semigroup. Then a fuzzy subset f of S satisfying (i) and (ii),

• (i) $|Im f|=2$,

• (ii) $f_{1}$ is an ideal of S,

is a fuzzy ideal of S.

The following result also characterizes a prime fuzzy ideal of a po-$\Gamma$-semigroup.

Theorem 4.15

Let S be a po-$\Gamma$-semigroup. Then a fuzzy subset f of S is a prime fuzzy ideal of S if and only if f satisfies the following conditions:

• (i) $|Im f|=2$.

• (ii) $f_{1}$ is a prime ideal of S.

Proof

The direct implication follows easily from Proposition 4.8, Theorem 4.9 and Corollary 4.11.

To prove the converse, we first observe that f is a fuzzy ideal of S (cf. Lemma 4.14). Then let g and h be two fuzzy ideals of S such that $g\circ h\subseteq f$. If $g\nsubseteq f$ and $h\nsubseteq f$, then there exist $x,y\in S$ such that $g(x)>f(x)$ and $h(y)>f(y).$ Thus $x,y\not \in f_{1}$. We claim that $x\Gamma S\Gamma y\nsubseteq f_{1}$. To establish the claim we suppose the contrary. Then $(S\Gamma x\Gamma S\Gamma S\Gamma y\Gamma S]\subseteq f_{1}$ (as $f_{1}$ is an ideal) and so $(S\Gamma x\Gamma S]\Gamma (S\Gamma y\Gamma S]\subseteq (S\Gamma x\Gamma S\Gamma S\Gamma y\Gamma S]\subseteq f_{1}$ (cf. Proposition 2.11) whence $(S\Gamma x\Gamma S]\subseteq f_{1}$ or $(S\Gamma y\Gamma S]\subseteq f_{1}$ (as $f_{1}$ is a prime ideal). Let us assume without loss of generality $(S\Gamma x\Gamma S]\subseteq f_{1}$. Then $<x>^3\,\subseteq (S\Gamma x\Gamma S]\subseteq f_{1}$ which implies that $x\in \,<x>\,\subseteq f_{1}$, which is a contradiction. Hence $x\Gamma S\Gamma y\nsubseteq f_{1}$. Then there exist $s\in S,$ $\alpha ,\beta \in \Gamma$ such that $x\alpha s\beta y\not \in f_{1}$ which means $f(x\alpha s\beta y)<1$. So $f(x\alpha s\beta y)=t=f(x)=f(y)$, where $Im f=\{t,1\}$. But by using Definitions 2.14 and 2.15, we obtain

$$\begin{aligned} (g\circ h)(x\alpha s\beta y)\ge & {} g(x)\wedge h(s\beta y) \\\ge & {} g(x)\wedge h(y) \\> & {} f(x)\wedge f(y) \\= & {} t. \end{aligned}$$

Hence $g\circ h\nsubseteq f$ which is a contradiction. Hence f is a prime fuzzy ideal of S.    $\square$

Corollary 4.16

Let S be a po-$\Gamma$-semigroup and f be a prime fuzzy ideal of S. Then there exists a prime fuzzy ideal g of S such that f is properly contained in g.

Proof

By Theorem 4.15, there exists $x_{0}\in S$ such that $f(x_{0})=1$ and $Im(f)=\{t,1\}$ for some $t\in [0,1)$. Let g be a fuzzy subset of S defined by $g(x)=1$, if $x\in f_{1}$ and $g(x)=r$, if $x\not \in f_{1},$ where $t<r<1.$ Then by Theorem 4.15, g is a prime fuzzy ideal and $f\subsetneqq g$. $\square$

In Theorem 4.10 we have shown that every nonempty level subset of a prime fuzzy ideal is a prime ideal. But Example 4.13 shows that the converse need not be true. In order to make the level subset criterion to hold, a new type of fuzzy primeness in ideals of a po-$\Gamma$-semigroup can be defined what is called weakly prime fuzzy ideal.

Definition 4.17

Let S be a po-$\Gamma$-semigroup. A nonconstant fuzzy ideal f of S is called a weakly prime fuzzy ideal of S if for all ideals A and B of S and for all $t\in (0,1],$ $tC_{A}\circ tC_{B}\subseteq f$ implies $tC_{A}\subseteq f$ or $tC_{B}\subseteq f.$

Theorem 4.18

Let S be a po-$\Gamma$-semigroup and f be a fuzzy ideal of S. Then f is a weakly prime fuzzy ideal of S if and only if each level subset $f_{t}$ ($\ne S$), $t\in (0,1]$, is a prime ideal of S for $f_{t}\ne {\varnothing }$.

Proof

Let f be a weakly prime fuzzy ideal of S and $t\in (0,1]$ such that $f_{t}\ne {\varnothing }$ and $f_{t}\ne S$. Then f is a fuzzy ideal of S. So $f_{t}$ is an ideal of S (cf. Theorem 3.5 [20]). Let A and B be ideals of S with $A\Gamma B\subseteq f_{t}$. Then $f_{t}$ being an ideal of S, $(A\Gamma B]\subseteq f_{t}$. Therefore, $tC_{(A\Gamma B]}\subseteq f$ which means $tC_{A}\circ tC_{B}\subseteq f$ (cf. Proposition 3.13). Hence by hypothesis, $tC_{A}\subseteq f$ or $tC_{B}\subseteq f$ (cf. Definition 4.17). Hence either $A\subseteq f_{t}$ or $B\subseteq f_{t}$. Consequently, $f_{t}$ is a prime ideal of S.

Conversely, suppose each $f_{t}$ (${\ne }S$) is a prime ideal of S, for all $t\in (0,1]$ with $f_{t}\ne {\varnothing }.$ Let A and B be ideals of S such that $tC_{A}\circ tC_{B}\subseteq f$ where $t\in (0,1]$. Then $tC_{(A\Gamma B]}\subseteq f$ (cf. Proposition 3.13) whence $(A\Gamma B]\subseteq f_{t}$. Now by Proposition 2.11, $(A]\Gamma (B]\subseteq (A\Gamma B]\subseteq f_{t}$ whence $A\Gamma B\subseteq f_{t}$ (as A, B are ideals). Hence by hypothesis either $A\subseteq f_{t}$ or $B\subseteq f_{t}$ whence $tC_{A}\subseteq f$ or $tC_{B}\subseteq f.$ Hence f is a weakly prime fuzzy ideal of S. $\square$

As an easy consequence of Theorems 4.10 and 4.18, we obtain the following corollary.

Corollary 4.19

In a po-$\Gamma$-semigroup S, every prime fuzzy ideal is a weakly prime fuzzy ideal.

That the converse of the above corollary is not always true is illustrated in the following examples.

Example 4.20

The fuzzy ideal of Example 4.13 is a weakly prime fuzzy ideal (cf. Theorem 4.18) but not a prime fuzzy ideal.

Example 4.21

Let $S=\{0,a,b,c\}$. Let $\Gamma =\{\alpha ,\beta ,\gamma \}$ be the nonempty set of binary operations on S with the following Cayley tables.

$\alpha$	0	a	b	c	$\beta$	0	a	b	c	$\gamma$	0	a	b	c
0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a	0	0	0	b	a	0	0	0	b	a	0	0	0	0
b	0	0	0	b	b	0	0	0	b	b	0	0	0	0
c	b	b	b	c	c	b	b	b	b	c	0	0	0	0

By a routine but tedious verification, we see that S is a po-$\Gamma$-semigroup where the partial orders on S and $\Gamma$ are given by $0\le a\le b\le c$ and $\gamma \le \beta \le \alpha ,$ respectively. Now we define a fuzzy subset f on S by $f(0)=f(a)=0.8$, $f(b)=0.3$, and $f(c)=0$. It can be checked that f is a weakly prime fuzzy ideal of S. But f is not a prime fuzzy ideal of S (cf. Theorem 4.15).

Remark 4.22

The above corollary and the example together shows that the notion of weakly prime fuzzy ideal generalizes the notion of prime fuzzy ideal.

The following theorem characterizes weakly prime fuzzy ideals of po-$\Gamma$-semigroups.

Theorem 4.23

Let S be a po-$\Gamma$-semigroup and f be a fuzzy ideal of S. Then the following are equivalent.

• (i) f is a weakly prime fuzzy ideal of S.

• (ii) For any $x,y\in S$ and $r\in (0,1]$, if $x_{r}\circ S\circ y_{r}\subseteq f$, then $x_{r}\subseteq f$ or $y_{r}\subseteq f.$

• (iii) For any $x,y\in S$ and $r\in (0,1]$, if $<x_{r}>\circ <y_{r}>\,\subseteq f$, then $x_{r}\subseteq f$ or $y_{r}\subseteq f.$

• (iv) If A and B are right ideals of S such that $tC_{A}\circ tC_{B}\subseteq f$, then $tC_{A}\subseteq f$ or $tC_{B}\subseteq f.$

• (v) If A and B are left ideals of S such that $tC_{A}\circ tC_{B}\subseteq f$, then $tC_{A}\subseteq f$ or $tC_{B}\subseteq f.$

• (vi) If A is a right ideal of S and B is a left ideal of S such that $tC_{A}\circ tC_{B}\subseteq f$, then $tC_{A}\subseteq f$ or $tC_{B}\subseteq f.$

Proof

$(\text{ i })\Rightarrow (\text{ ii })$.

Let f be a weakly prime fuzzy ideal of S. Let $x,y\in S$ and $r\in (0,1]$ be such that $x_{r}\circ S\circ y_{r}\subseteq f.$ Then by Proposition 3.8, $rC_{(S\Gamma x\Gamma S]}\circ rC_{(S\Gamma y\Gamma S]}$ $=(S\circ x_{r}\circ S)\circ (S\circ y_{r}\circ S)$ $\subseteq S\circ (x_{r}\circ S\circ y_{r})\circ S$ $\subseteq S\circ f\circ S\subseteq f$. Hence by hypothesis, $rC_{(S\Gamma x\Gamma S]}\subseteq f$ or $rC_{(S\Gamma y\Gamma S]}\subseteq f$ whence $S\circ x_{r}\circ S\subseteq f$ or $S\circ y_{r}\circ S\subseteq f.$ If $S\circ x_{r}\circ S\subseteq f$, then $<x_{r}>^3\,\subseteq f$ (cf. Corollary 3.12). Hence $(rC_{<x>})^{3}\,\subseteq f$. Since f is weakly prime fuzzy ideal, this implies that $<x_{r}>\,\subseteq f$ whence $x_{r}\subseteq f$. Similarly, if $S\circ y_{r}\circ S\subseteq f$, then $y_{r}\subseteq f$.

$(\text{ ii })\Rightarrow (\text{ iii })$.

Let $x,y\in S$ and $r\in (0,1]$ be such that $<x_{r}>\circ <y_{r}>\,\subseteq f$. Then since $x_{r}\circ S\subseteq \,<x_{r}>$ and $y_{r}\subseteq \,<y_{r}>$, $x_{r}\circ S\circ y_{r}\subseteq f.$ Hence by (ii), $x_{r}\subseteq f$ or $y_{r}\subseteq f.$

$(\text{ iii })\Rightarrow (\text{ iv })$.

Let A, B be two right ideals of S such that $tC_{A}\circ tC_{B}\subseteq f$ and $tC_{A}\nsubseteq f$. Then there exists $a\in A$ such that $a_{t}\nsubseteq f.$ Now for any $b\in B,$ by Proposition 3.13 and hypothesis, we obtain $<a_{t}>\circ <b_{t}>\,= tC_{\,<{}a{}>}\circ tC_{\,<{}b{}>}$ $= tC_{(\,<{}a{}>\Gamma \,<{}b{}>]}$ $\subseteq$ $tC_{((A\Gamma B)\cup (S\Gamma A\Gamma B)]}$ $=$ $tC_{(A\Gamma B]\cup (S\Gamma A\Gamma B]}$ $=$ $tC_{(A\Gamma B]}\cup tC_{(S\Gamma A\Gamma B]}$ $=$ $tC_{(A\Gamma B]}\cup tC_{(S\Gamma (A]\Gamma (B]]}$ $\subseteq$ $tC_{(A\Gamma B]}\cup tC_{(S\Gamma (A\Gamma B]]}$ $=$ $(tC_{A}\circ tC_{B})\cup (S\circ tC_{(A\Gamma B]})$ $=$ $(tC_{A}\circ tC_{B})\cup (S\circ tC_{A}\circ tC_{B})$ (cf. Proposition 3.13) $\subseteq$ $f\cup (S\circ f)$ $\subseteq f.$ Hence by (iii), $b_{t}\subseteq f$. Consequently, $tC_{B}\subseteq f.$

$(\text{ iii })\Rightarrow (\text{ vi })$.

Let A be a right ideal and B be a left ideal of S such that $tC_{A}\circ tC_{B}\subseteq f$ and $tC_{A}\nsubseteq f$. Then there exists $a\in A$ such that $a_{t}\nsubseteq f.$ Now for any $b\in B,$ $<a_{t}>\circ <b_{t}>\,= tC_{\,<{}a{}>}\circ tC_{\,<{}b{}>}=tC_{(\,<{}a{}>\Gamma \,<{}b{}>]}$ $\subseteq$ $tC_{((A\Gamma B)\cup (A\Gamma B\Gamma S)\cup (S\Gamma A\Gamma B)\cup (S\Gamma A\Gamma B\Gamma S)]}$ $=$ $tC_{(A\Gamma B]\cup (A\Gamma B\Gamma S]\cup (S\Gamma A\Gamma B]\cup (S\Gamma A\Gamma B\Gamma S]}$ $=$ $tC_{(A\Gamma B]}\cup tC_{(A\Gamma B\Gamma S]}\cup tC_{(S\Gamma A\Gamma B]}\cup tC_{(S\Gamma A\Gamma B\Gamma S]}=tC_{(A\Gamma B]}\cup tC_{((A]\Gamma (B]\Gamma S]}\cup tC_{(S\Gamma (A]\Gamma (B]]}\cup tC_{(S\Gamma (A]\Gamma (B]\Gamma S]}$ $\subseteq$ $tC_{(A\Gamma B]}\,\cup tC_{((A\Gamma B]\Gamma S]}\cup tC_{(S\Gamma (A\Gamma B]]}\cup tC_{(S\Gamma (A\Gamma B]\Gamma S]}$ $=$ $(tC_{A}\circ tC_{B})\cup (tC_{A}\circ tC_{B}\circ S)\cup (S\circ tC_{A}\circ tC_{B})\cup (S\circ tC_{A}\circ tC_{B}\circ S)$ $\subseteq$ $f\cup (f\circ S)\cup (S\circ f)\cup (S\circ f\circ S)$ $\subseteq$ f. Hence by (iii), $b_{t}\subseteq f$. Consequently, $tC_{B}\subseteq f.$

$(\text{ iv })\Rightarrow (\text{ i })$, $(\text{ v })\Rightarrow (\text{ i })$, $(\text{ vi })\Rightarrow (\text{ i })$ are obvious and $(\text{ iii })\Rightarrow (\text{ v })$ is similar to $(\text{ iii })\Rightarrow (\text{ iv }).$ $\square$

5 Completely Prime and Weakly Completely Prime Fuzzy Ideals in Po-$\Gamma$-Semigroups

In this section, the notion of completely prime ideals of po-$\Gamma$-semigroups has been generalized in fuzzy setting.

Definition 5.1

Let S be a po-$\Gamma$-semigroup. A nonconstant fuzzy ideal f of S is called a completely prime fuzzy if for any two fuzzy points $x_{t},y_{r}$ of S $(t,r\in (0,1])$, $x_{t}\circ y_{r}\subseteq f$ implies that $x_{t}\subseteq f$ or $y_{r}\subseteq f.$

Definition 5.2

Let S be a po-$\Gamma$-semigroup. A nonconstant fuzzy ideal f of S is called a weakly completely prime fuzzy if for any fuzzy points $x_{t},y_{t}$ of S $(t\in (0,1])$, $x_{t}\circ y_{t}\subseteq f$ implies that $x_{t}\subseteq f$ or $y_{t}\subseteq f.$

Theorem 5.3

Let S be a po-$\Gamma$-semigroup and f be a fuzzy ideal of S. Then f is completely prime fuzzy ideal if and only if for any fuzzy subsets f and g of S, $g\,\circ \,h\subseteq f$ implies $g\subseteq f$ or $h\subseteq f.$

Proof

Let f be a completely prime fuzzy ideal and g, h be two fuzzy subsets such that $g\circ h\subseteq f$ and $g\nsubseteq f$. Then there exists an $x_{t}\subseteq g$ such that $x_{t}\nsubseteq f.$ Since $g\circ h\subseteq f,$ $x_{t}\circ y_{r}\subseteq f$, for all $y_{r}\subseteq h.$ So $y_{r}\subseteq f$, for all $y_{r}\subseteq h.$ Hence $h\subseteq f.$

The converse follows easily. $\square$

Definitions 5.1, 5.2 and Theorems 5.3 and 4.6 together give rise to the following result.

Corollary 5.4

Let S be a po-$\Gamma$-semigroup and f be a completely prime fuzzy ideal of S. Then f is a prime fuzzy ideal and a weakly completely prime fuzzy ideal of S. Further if S is commutative, then f is a prime fuzzy ideal if and only if f is a completely prime fuzzy ideal.

The following theorem characterizes a completely prime fuzzy ideal.

Theorem 5.5

Let S be a po-$\Gamma$-semigroup and f be a fuzzy subset of S. Then f is a completely prime fuzzy ideal of S if and only if f satisfies the following conditions:

• (1) $|Im f|=2$.

• (2) $f_{1}$ is a completely prime ideal of S.

Proof

Let f be a completely prime fuzzy ideal of S. Then by Corollary 5.4, f is a prime fuzzy ideal of S. So by Theorem 4.15, $f_{1}$ is a prime ideal and $|Im f|=2$. Let $x,y\in S$ such that $x\Gamma y\in f_{1}.$ Then $f(x\gamma y)=1$, for all $\gamma \in \Gamma$. So the fuzzy point $(x\gamma y)_{1}\subseteq f$, for all $\gamma \in \Gamma$ whence $\underset{\gamma \in \Gamma }{\cup }(x\gamma y)_{1}\subseteq f$. Therefore $x_{1}\circ y_{1}\subseteq f$ (cf. Theorem 3.2). Since f is a completely prime fuzzy ideal, $x_{1}\subseteq f$ or $y_{1}\subseteq f$ whence $x\in f_{1}$ or $y\in f_{1}$. Hence $f_{1}$ is a completely prime ideal of S.

Conversely, suppose the given conditions hold, i.e., $Im(f)=\{t,1\}$ $(t<1)$ and $f_{1}$ is a completely prime ideal. Then $f_{1}$ is a prime ideal of S. So by Theorem 4.15, f is a prime fuzzy ideal. Hence f is a nonconstant fuzzy ideal of S. Let $x_{r}$ and $y_{s}$ ($r,s>0$) be two fuzzy points of S such that $x_{r}\circ y_{s}\subseteq f$. If possible let $x_{r}\nsubseteq f$ and $y_{s}\nsubseteq f$, then $f(x)<r$ and $f(y)<s.$ So $f(x)=f(y)=t$. Thus $x,y\not \in f_{1}$, which implies $x\Gamma y\nsubseteq f_{1}$ as $f_{1}$ is completely prime. So there exists $\gamma \in \Gamma$ such that $x\gamma y\not \in f_{1}$ whence $f(x\gamma y)=t$. Now $(x_{r}\circ y_{s})(x\gamma y)=(\underset{\beta \in \Gamma }{\cup }(x\beta y)_{r\wedge s})(x\gamma y)=r\wedge s>t=f(x\gamma y)$. This is a contradiction to $x_{r}\circ y_{s}\subseteq f$. Hence f is a completely prime fuzzy ideal of S. $\square$

The following theorem characterizes a weakly completely prime fuzzy ideal.

Theorem 5.6

Let S be a po-$\Gamma$-semigroup and f be a fuzzy ideal of S. Then f is weakly completely prime fuzzy ideal if and only if $\underset{\gamma \in \Gamma }{\inf }f(x\gamma y)=\max \{f(x),f(y)\}$, for all $x,y\in S.$

Proof

Let f be a weakly completely prime fuzzy ideal and $x,y\in S$. Since f is a fuzzy ideal, $f(x\gamma y)\ge \max \{f(x),f(y)\}$, for all $\gamma \in \Gamma .$ So $\underset{\gamma \in \Gamma }{\inf }f(x\gamma y)\ge \max \{f(x),f(y)\}$. Now let $\underset{\gamma \in \Gamma }{\inf }f(x\gamma y)=t$ where $t\in [0,1].$ If $t=0$, then $\underset{\gamma \in \Gamma }{\inf }f(x\gamma y)\le \max \{f(x),f(y)\}$. Otherwise, $f(x\gamma y)\ge t$, for all $\gamma \in \Gamma$, i.e., $(x\gamma y)_{t}\subseteq f$, for all $\gamma \in \Gamma$. So $\underset{\gamma \in \Gamma }{\cup }(x\gamma y)_{t}\subseteq f$ which means $x_{t}\circ y_{t}\subseteq f$ (cf. Theorem 3.2). Since f is a weakly completely prime fuzzy ideal, $x_{t}\subseteq f$ or $y_{t}\subseteq f$, i.e., $f(x)\ge t$ or $f(y)\ge t$. So $\max \{f(x),f(y)\}\ge t=\underset{\gamma \in \Gamma }{\inf }f(x\gamma y)$. Hence $\underset{\gamma \in \Gamma }{\inf }f(x\gamma y)=\max \{f(x),f(y)\}$.

Conversely, suppose the condition holds, let $x_{t}$ and $y_{t}$ be two fuzzy points of S such that $x_{t}\circ y_{t}\subseteq f$ where $t\in (0,1]$. Then $\underset{\gamma \in \Gamma }{\cup }(x\gamma y)_{t}\subseteq f$ (cf. Theorem 3.2), i.e., $(x\gamma y)_{t}\subseteq f$, for all $\gamma \in \Gamma$, i.e., $f(x\gamma y)\ge t$, for all $\gamma \in \Gamma$ which implies $\underset{\gamma \in \Gamma }{\inf }f(x\gamma y)\ge t$. So by the hypothesis $\max \{f(x),f(y)\}\ge t.$ Then $f(x)\ge t$ or $f(y)\ge t$, i.e., $x_{t}\subseteq f$ or $y_{t}\subseteq f$. Hence f is a weakly completely prime fuzzy ideal of S. $\square$

Theorem 5.7

Let S be a po-$\Gamma$-semigroup and f be a fuzzy ideal of S. Then f is weakly completely prime fuzzy ideal if and only if each $f_{t}$, $t\in (0,1]$, is a completely prime ideal of S for $f_{t}\ne \emptyset .$

Proof

Let f be a weakly completely prime fuzzy ideal of S, $x,y\in S$ and $t\in (0,1]$ such that $f_{t}\ne \emptyset$. Let $x\Gamma y\subseteq f_{t}$. Then $f(x\gamma y)\ge t$, for all $\gamma \in \Gamma$ which means $\underset{\gamma \in \Gamma }{\inf }f(x\gamma y)\ge t.$ So $\max \{f(x),f(y)\}\ge t$ (cf. Theorem 5.6) which implies $f(x)\ge t$ or $f(y)\ge t$, i.e., $x\in f_{t}$ or $y\in f_{t}$. Hence $f_{t}$ is a completely prime ideal of S.

Conversely, suppose each $f_{t}$, $t\in (0,1]$, is a completely prime ideal of S for $f_{t}\ne \emptyset .$ Let $x,y\in S$. Since f is a fuzzy ideal, $f(x\gamma y)\ge \max \{f(x),f(y)\}$, for all $\gamma \in \Gamma .$ So $\underset{\gamma \in \Gamma }{\inf }f(x\gamma y)\ge \max \{f(x),f(y)\}$. Now let $\underset{\gamma \in \Gamma }{\inf }f(x\gamma y)=t$ where $t\in [0,1].$ If $t=0$, then $\underset{\gamma \in \Gamma }{\inf }f(x\gamma y)\le \max \{f(x),f(y)\}$. Otherwise, $f(x\gamma y)\ge t$, for all $\gamma \in \Gamma$, i.e., $x\gamma y \in f_{t}$, for all $\gamma \in \Gamma$, i.e., $x\Gamma y \in f_{t}$. Since $f_{t}$ is a completely prime fuzzy ideal, $x\in f_{t}$ or $y\in f_{t}$, i.e., $f(x)\ge t$ or $f(y)\ge t$. So $\max \{f(x),f(y)\}\ge t=\underset{\gamma \in \Gamma }{\inf }f(x\gamma y)$. Hence $\underset{\gamma \in \Gamma }{\inf }f(x\gamma y)=\max \{f(x),f(y)\}$ whence f is a weakly completely prime fuzzy ideal (cf. Theorem 5.6). $\square$

By Theorems 4.18 and 5.7 we have the following result.

Corollary 5.8

Let S be a po-$\Gamma$-semigroup and f be a weakly completely prime fuzzy ideal of S. Then f is a weakly prime fuzzy ideal of S.

Remark 5.9

Since in a both-sided po-$\Gamma$-semigroup the notions of prime ideals and completely prime ideals coincide (cf. Theorem 2.9 [18]), in view of Theorems 4.18 and 5.7 the notions of weakly completely prime fuzzy ideals and weakly prime fuzzy ideals coincide. Hence the above corollary is meaningless in a both-sided po-$\Gamma$-semigroup.

Remark 5.10

The proofs of results on completely prime and weakly completely prime fuzzy ideals in po-$\Gamma$-semigroups indicate that these are also true in $\Gamma$-semigroups without partial order.

To conclude this section, we give the following interrelations among various fuzzy primeness studied in this paper.

6 Concluding Remark

Theorem 4.23 is analogous to Theorem 3.4 [6]. The said theorem of [6] plays an important role in radical theory of $\Gamma$-semigroups. So Theorem 4.23 may help to study radical theory in po-$\Gamma$-semigroups via fuzzy subsets. This possibility of study of radical theory in po-$\Gamma$-semigroups is also indicated in the work of radical theory in po-semigroups via weakly prime fuzzy ideals by Xie and Tang [27].