The Notions of Fuzzy Set on Pseudo Quasi Ordered Residuated Systems

Definition 2.1.

¹ A residuated relational system is defined as a structure $\mathfrak{K}\mathfrak{=}(K,·,\to ,1,R)$ , where $(K,·,\to ,1)$ is an algebra of type (2, 2, 0) and R is a binary relation on K satisfying the following properties:

We refer the operations ‘ $·$ ’ as multiplication and ‘ $\to$ ’ as its residuum.

Definition 2.2.

¹ A QORS is a residuated relational system $\mathfrak{K}=(K,·,\to ,1,\preccurlyeq )$ , where $\preccurlyeq$ is a quasi-ordered relation defined on the monoid $(K,·)$ .

Proposition 2.3.

¹ Let $\mathfrak{K}$ be a QORS. Then, for any $a,b,c\in K$ ,

Definition 2.4.

² For a non-empty subset J of a QORS, $\mathfrak{K}$ we say that it is a filter in $\mathfrak{K}$ if it satisfies the following conditions:

for all $a,b\in K$ .

Definition 2.5.

⁵ A non-empty subset J of a QORS, $\mathfrak{K}$ is implicative filter in K if the following conditions hold for any $a,b,c\in K$ :

Theorem 2.6.

⁷ Let J be a non-empty subset of a QORS, $\mathfrak{K}$ such that $a\in J$ and $a\preccurlyeq b$ implies $b\in J$ for all $a,b\in K$ . If $(a\to (b\to (b\to c)))\in J$ , then $b\in J$ .

Definition 2.7.

⁷ For a non-empty subset J of a QORS, $\mathfrak{K}$ we say that it is comparative filter in $\mathfrak{K}$ if the following conditions hold for any $a,b,c\in K$ :

Theorem 2.8.

⁷ A comparative filter of a QORS, $\mathfrak{K}$ is a filter of $\mathfrak{K}$ .

Definition 2.9.

⁴ A pseudo quasi-ordered relational system is a structure $\mathcal{P}\mathfrak{K}=(K,·,\to ,⇝,1,\preccurlyeq )$ where an algebra $(K,·,\to ,⇝,1)$ is an algebra of type (2,2,2,0) and $\preccurlyeq$ is a quasi-ordered in K satisfying the following properties:

Definition 2.10.

¹⁰ A fuzzy subset $ϵ$ of a bounded lattice $K$ is said to be a fuzzy ideal of K, if for all $a,b\in K$

Definition 2.11.

¹³ A fuzzy subset $ϵ$ of a bounded lattice K is said to be a fuzzy filter of K, if for all $a,b\in K$

Theorem 3.1.

Let $\mathfrak{K}=(K,.,\to ,1,\preccurlyeq )$ be a QORS and $ϵ$ be a fuzzy subset on $K$ satisfying (QF). Then, for any $a,b\in K$ , $ϵ\left(a\right)\wedge ϵ\left(b\right)\ge ϵ(a·b)$ .

Proof.

Suppose $\mathfrak{K}$ be a QORS and $ϵ$ be a fuzzy subset of $K$ satisfying (QF). Clearly, $a·b\preccurlyeq a$ and $a·b\preccurlyeq b$ for any $a,b\in K$ . Thus, from the assumption it follows that $ϵ\left(a\right)\ge ϵ(a·b)$ and $ϵ\left(b\right)\ge ϵ(a·b)$ so that $ϵ\left(a\right)\wedge ϵ\left(b\right)\ge ϵ(a·b)$ .      □

Theorem 3.2.

Let $\mathfrak{K}$ be a QORS and $ϵ$ be a fuzzy subset of $K$ satisfying (QF). Then, $ϵ(a\to b)\ge ϵ\left(a\right)$ for all $a,b\in K$ .

Proof.

Let ϵ be a fuzzy subset of $K$ and $a,b\in K$ such that $a\preccurlyeq b$ . Then, as $a·a\preccurlyeq a$ , we have $a·a\preccurlyeq b$ and hence $a\preccurlyeq a\to b$ . Therefore, $ϵ(a\to b)\ge ϵ\left(a\right)$ .      □

Theorem 3.3.

Let ϵ be a fuzzy subset of a QORS, $\mathfrak{K}$ satisfying (QF) and $a\preccurlyeq b\to c$ for all $a,b,c\in K$ . Then, $ϵ\left(c\right)\ge ϵ(a·b)$ iff $ϵ\left(b\right)\ge ϵ\left(a\right)$ .

Proof.

Let ϵ be a fuzzy subset on $K$ and $a,b,c\in K$ such that $a\preccurlyeq b$ and $a\preccurlyeq b\to c$ . So, $a·b\preccurlyeq c$ and hence $ϵ\left(c\right)\ge ϵ(a·b)$ . Conversely, let $a\preccurlyeq b$ and $a\preccurlyeq b\to c$ implies that $ϵ\left(c\right)\ge ϵ(a·b)$ . Obviously, $a·1\preccurlyeq b$ so that $a\preccurlyeq 1\to b$ . Consequently, $ϵ\left(b\right)\ge ϵ(a·1)=ϵ\left(a\right)$ .      □

Remark 3.4.

For a fuzzy subset ϵ of a QORS, $\mathfrak{K}$ , $ϵ\left(1\right)\ge ϵ\left(a\right)$ for all $a\in K$ .

Definition 3.5.

A fuzzy subset ϵ of a QORS, $\mathfrak{K}$ satisfying (QF) is said to be a fuzzy filter of $K$ whenever $ϵ\left(1\right)=1$ and $ϵ\left(b\right)\ge ϵ\left(a\right)\wedge ϵ(a\to b)$ , for every $a,b\in K$ .

Example 3.6.

Let $K=\{1,2,3,4\}$ and the operations ‘·’ and ‘→’ be defined on $K$ as follows:

Then, $\mathfrak{K}$ = (K, . , →, 1, ≼) is a QORS where ‘ $\preccurlyeq$ ’ is defined by:

Now, define a fuzzy subset $ϵ$ of $K$ by:

Hence, ϵ is a fuzzy filter of $K$ .

Theorem 3.7.

The family $\mathfrak{F}\left(K\right)$ of all fuzzy filters in a QORS, $\mathfrak{K}$ forms a complete lattice.

Proof.

Let ${\left\{{ϵ}_i\right\}}_{i\in △}$ be a family of all fuzzy filters in a QORS, $\mathfrak{K}$ . Clearly, ${\left\{{ϵ}_i\right\}}_{i\in △}$ is a partially ordered set under set inclusion where $△$ is an indexed set. Let $a,b\in K$ such that $a\preccurlyeq b$ . Since ${ϵ}_i$ is a fuzzy filter of $\mathfrak{K}$ , ${ϵ}_i\left(b\right)\ge {ϵ}_i\left(a\right)$ for all $i\in △$ . Thus, ${\bigcap }_{i\in △}{ϵ}_i\left(b\right)\ge {\bigcap }_{i\in △}{ϵ}_i\left(a\right)$ so that condition 1 of Definition 3.5 is satisfied.

Clearly, for any $a,b\in K$ ,

From the fact that ${ϵ}_i$ is a fuzzy filter for each $i\in △$ , we have

This shows that condition 2 of Definition 3.5 is satisfied. Hence, ${\bigcap }_{i\in △}{ϵ}_i$ is a fuzzy filter of $\mathfrak{K}$ . Now, consider fuzzy filters of $K$ which contains ${\bigcup }_{i\in △}{ϵ}_i$ and denote the family of such filters by $\mathfrak{G}$ .

Clearly, $\bigcap \mathfrak{G}$ is the smallest fuzzy filter which contains ${\bigcup }_{i\in △}{ϵ}_i$ . Taking

makes the algebra ( $\mathfrak{K}$ , $⨅$ , $⨆$ ), a complete lattice.      □

Corollary 3.8.

Let $\mathfrak{K}$ be a QORS and $ϵ$ be a fuzzy subset of $K$ . Then, there exists a minimal fuzzy filter of $K$ which contains $ϵ$ .

Definition 3.9.

A fuzzy subset $ϵ$ of $K$ satisfying (QF) is said to be a 2-fuzzy filter in $K$ if $ϵ(b·c)\ge ϵ(a\to c)\wedge ϵ((a\to b)\to c)$ for all $a,b,c\in K$ .

Theorem 3.10.

Let $\mathfrak{K}$ be a QORS. Then, every 2-fuzzy filter of $K$ is a fuzzy filter of $K$ .

Proof.

Let $ϵ$ be a 2-fuzzy filter of a QORS, $\mathfrak{K}$ . For any $a,b\in K$ , we show that $ϵ\left(b\right)\ge ϵ\left(a\right)\wedge ϵ(a\to b)$ . From Theorem 3.2, $ϵ(a\to b)\ge ϵ\left(a\right)$ . So, as $a\to b\preccurlyeq 1$ and $ϵ\left(1\right)\ge ϵ(a\to b)$ , then $ϵ((a\to b)\to 1)\ge ϵ(a\to b)$ . Also, as $a\preccurlyeq 1$ and $ϵ\left(1\right)\ge ϵ\left(a\right),$ then $ϵ(a\to 1)\ge ϵ\left(a\right)$ . Thus,

By the condition of 2-fuzzy filters, we get

Therefore, $ϵ$ is a fuzzy filter of $K$ .      □

Definition 4.1.

A fuzzy subset $ϵ$ of a QORS, $\mathfrak{K}$ satisfying (QF) is said to be an implicative fuzzy filter of $K$ if $ϵ(a\to c)\ge ϵ(a\to (b\to c))\wedge ϵ(a\to b)$ for all $a,b,c\in K$ .

Example 4.2.

Let $K=\{a,b,c,1\}$ and the operations ‘·’ and ‘→’ be defined on K as follows:

Then, $\mathfrak{K}$ $=(K,.,\to ,1,\preccurlyeq )$ is a QORS where ‘ $\preccurlyeq$ ’ is defined by:

Now, define a fuzzy subset $ϵ$ by:

It is simple to show that $ϵ$ is an implicative fuzzy filter of $K$ .

Theorem 4.3.

Let $ϵ$ be an implicative fuzzy filter of a QORS, $\mathfrak{K}$ . Then, for any $a,b\in K,ϵ(a\to b)\ge ϵ(a\to (a\to b))$ .

Proof.

Let $ϵ$ be an implicative fuzzy filter of a QORS, $\mathfrak{K}$ . Let $a,b\in K$ , then taking $b=a$ in the hypothesis of Definition 4.1, we get

Since $1·a\preccurlyeq a$ so that $1\preccurlyeq (a\to a)$ and we have $ϵ(a\to a)\ge ϵ\left(1\right)$ . Thus,

This proves the theorem.      □

Lemma 4.4.

Let $ϵ$ be a fuzzy subset on a QORS, $\mathfrak{K}$ satisfying (QF). Then, $ϵ(1\to a)=ϵ\left(a\right)$ for all $a\in K$ .

Proof.

Let $ϵ$ be a fuzzy subset of a QORS, $\mathfrak{K}$ and $a\in K$ . We have $a\preccurlyeq 1\to a$ and $1\to a\preccurlyeq a$ . Thus by hypothesis $ϵ\left(a\right)\le ϵ(1\to a)$ and $ϵ(1\to a)\le ϵ\left(a\right)$ so that $ϵ\left(a\right)=ϵ(1\to a)$ .      □

Theorem 4.5.

Every implicative fuzzy filter of a QORS, $\mathfrak{K}$ is a fuzzy filter of $K$ .

Proof.

Let ϵ be an implicative fuzzy filter of a QORS, $\mathfrak{K}$ and $a,b\in K$ such that $a\preccurlyeq b.$ From Lemma 4.4, we have $ϵ\left(b\right)=ϵ(1\to b)$ and from the hypothesis of Definition 4.1, we get

Hence,

Therefore, $ϵ$ is a fuzzy filter of $K$ .      □

Lemma 4.6.

Let $ϵ$ be a fuzzy subset of a QORS, $\mathfrak{K}$ and $a,b,c\in K$ . Then, (QF) holds iff for $a\preccurlyeq b\to c,ϵ\left(c\right)\ge ϵ(a·b)$ .

Proof.

The forward proof is trivial. Conversely, let $a,b,c\in K$ such that if $a\preccurlyeq b\to c$ , then $ϵ\left(c\right)\ge ϵ(a·b)$ . Let $a,b\in K$ such that $a\preccurlyeq b.$ This is the same as $a·1\preccurlyeq b$ . Thus, $a\preccurlyeq 1\to b$ . Then, by the hypothesis we obtain that $ϵ\left(b\right)\ge ϵ(a·1)=ϵ\left(a\right)$ .      □

Lemma 4.7.

Let $ϵ$ be a fuzzy subset of a QORS, $K$ satisfying (QF). Then, for any $a,b,c\in K,ϵ(a\to (b\to c))=ϵ(b\to (a\to c))$ .

Proof.

Let $ϵ$ be a fuzzy subset of a QORS, $\mathfrak{K}$ satisfying (QF). Then, for any $a,b,c\in K$ ,

Thus,

Similarly,

Therefore,

Theorem 4.8.

Let $ϵ$ be a fuzzy subset of a QORS, $\mathfrak{K}$ satisfying (QF). If $ϵ(b\to c)\ge ϵ\left(a\right)\wedge ϵ(a\to (b\to (b\to c)))$ for any $a,b,c\in K$ , then ϵ is an implicative fuzzy filter of $\mathfrak{K}$ .

Theorem 4.9.

Let $ϵ$ be a fuzzy subset of a QORS, $\mathfrak{K}$ satisfying (QF) and let $a,b\in K$ such that

Then, $ϵ$ is an implicative fuzzy filter of $K$ .

Proof.

Let $ϵ$ be a fuzzy subset of a QORS, $\mathfrak{K}$ satisfying the given conditions and let $a,b,c\in K$ . Using (2), we have

Also, by (1) we have

Thus,

Therefore, by Theorem 4.8, $ϵ$ is an implicative fuzzy filter of $K$ .      □

Theorem 4.10.

A fuzzy subset $ϵ$ of a QORS, $\mathfrak{K}$ is an implicative fuzzy filter iff ${ϵ}_{\alpha }$ is an implicative filter of $K$ , for all $\alpha \in [0,1].$

Proof.

Suppose $ϵ$ be an implicative fuzzy filter of a QORS, $\mathfrak{K}$ . Let $a\to (b\to c)\in {ϵ}_{\alpha }$ and $\to b\in {ϵ}_{\alpha }$ , for any $a,b,c\in K$ and for all $\alpha \in [0,1]$ . So, $ϵ(a\to (b\to c))\ge \alpha$ and $ϵ(a\to b)\ge \alpha$ . Thus,

in order that $a\to c\in {ϵ}_{\alpha }$ . Hence, ${ϵ}_{\alpha }$ is an implicative filter of $K$ . Now, let ${ϵ}_{\alpha }$ be an implicative filter of $K$ and let

Thus, $ϵ(a\to (b\to c))\ge \alpha$ and $ϵ(a\to b)\ge \alpha$ so that

Therefore, $ϵ$ is an implicative fuzzy filter of $K$ .      □

Definition 4.11.

A fuzzy subset $ϵ$ of a QORS, $\mathfrak{K}$ satisfying (QF) is said to be a comparative fuzzy filter of $K$ if $ϵ\left(b\right)\ge ϵ\left(a\right)\wedge ϵ(a\to ((b\to c)\to b))$ for all $a,b,c\in K$ .

Example 4.12.

Let $K=\{a,b,c,d,1\}$ and the operations ‘^.’ and ‘→’ be defined on $K$ as follows:

Then, $\mathfrak{K}=(K,.,\to ,1,\preccurlyeq )$ is a QORS where ‘ $\preccurlyeq$ ’ is defined by:

Now, define a fuzzy subset $ϵ$ of $K$ by:

Thus, $ϵ$ is a comparative fuzzy filter of $K$ .

Theorem 4.13.

Let $ϵ$ be a comparative fuzzy filter of a QORS, $\mathfrak{K}$ . Then, for any $b,c\in K,ϵ\left(b\right)\ge ϵ((b\to c)\to b).$

Proof.

Let $ϵ$ be a comparative fuzzy filter of a QORS, $\mathfrak{K}$ and $b,c\in K$ . Then, from the assumption of Definition 4.11

From Lemma 4.4, we obtain that

     □

Remark 4.14.

If $ϵ$ is a comparative fuzzy filter of a QORS, $\mathfrak{K}$ and $a,b,c\in K$ , then $ϵ\left(b\right)\ge ϵ\left(a\right)\wedge ϵ((b\to c)\to (a\to b)).$

Theorem 4.15.

A comparative fuzzy filter $ϵ$ of a QORS, $\mathfrak{K}$ is a fuzzy filter of $K$ .

Proof.

Let $ϵ$ be a comparative fuzzy filter of a QORS, $\mathfrak{K}$ and $a,b,c\in K$ . Then,

Since $a\to b\preccurlyeq (b\to c)\to (a\to c)$ , we get

Thus,

Theorem 4.16.

Let $ϵ$ be a fuzzy filter of $K$ . Then $ϵ$ is a comparative fuzzy filter of $K$ iff $ϵ\left(b\right)\ge ϵ((b\to c)\to b)$ for all $a,b,c\in K$ .

Proof.

The proof of the forward direction follows from Theorem 4.13. Now, let $ϵ$ be a fuzzy filter of $K$ , and $ϵ\left(b\right)\ge ϵ((b\to c)\to b)$ for all $b,c\in K$ . By the given condition of Definition 3.5, we have $ϵ((b\to c)\to b)\ge ϵ\left(a\right)\wedge ϵ(a\to ((b\to c)\to b))$ for all $a,b,c\in K$ . Thus,

So, $ϵ$ is a comparative fuzzy filter of $K$ .      □

Theorem 4.17.

Let $\mathfrak{K}$ be a QORS such that $a\to b=((a\to b)\to b)\to b$ for all $a,b\in K$ . If $ϵ$ is a comparative fuzzy filter of $K$ , then $ϵ((b\to a)\to a)\ge ϵ((a\to b)\to b)$ , for all $a,b\in K$ .

Proof.

Suppose $ϵ$ is a comparative fuzzy filter of $K$ and $a,b\in K$ . Then, from Theorem 4.16, we obtain that

Since

Theorem 4.18.

Let $ϵ$ be an implicative fuzzy filter of a QORS, $\mathfrak{K}$ such that $ϵ((b\to a)\to a)\ge ϵ((a\to b)\to b)$ for all $a,b\in K$ . Then, $ϵ$ is a comparative fuzzy filter of $K$ .

Proof.

Let $ϵ$ be an implicative fuzzy filter of $K$ and $a,b\in K$ such that

Clearly, $ϵ$ is a fuzzy filter of $K$ , i.e., $ϵ\left(b\right)\ge ϵ\left(a\right)\wedge ϵ(a\to b)$ . Proving $ϵ\left(b\right)\ge ϵ(b\to a)\to b,$ completes the proof of the theorem. Now, consider the fact that, $a\to b\preccurlyeq (b\to c)\to (a\to c)$ for all $c\in K$ and apply it on $(a\to b)\to a,$ we obtain that

Substituting $a\to b$ in place of $a$ in Theorem 4.3 gives

Applying the condition of Definition 3.5 we get

Also, from the fact that $b·a\preccurlyeq b$ we get $b\preccurlyeq a\to b.$ Thus, $ϵ(a\to b)\ge$ $ϵ\left(b\right).$ Following this we have $ϵ\left(b\right)\ge ϵ((a\to b)\to a)$ . This proves the theorem.      □

Theorem 4.19.

A comparative fuzzy filter of a QORS, $\mathfrak{K}$ is an implica- tive fuzzy filter.

Proof.

Suppose ϵ be a comparative fuzzy filter of K. Since

From the fact that b · a $\preccurlyeq$ b implies b $\preccurlyeq$ a → b and (a → c).a $\preccurlyeq$ a → c implies a → c $\preccurlyeq \preccurlyeq$ a → (a → c) we obtain that

So, considering the condition a → b = ((a → b) → b) → b and applying Theorem 4.13 we obtain the following:

Thus,

Therefore, ϵ is an implicative fuzzy filter of $K$ .      □

Theorem 4.20.

The family ${\mathfrak{F}}_c\left(K\right)$ of all comparative fuzzy filters in a QORS, $\mathfrak{K}$ forms a complete lattice.

Proof.

The proof follows from Theorem 3.7 and Theorem 4.15.      □

Corollary 4.21.

Let $\mathfrak{K}$ be a QORS. For any fuzzy subset ϵ of $K$ , there is a unique minimum comparative fuzzy filter in $K$ that contains $ϵ$ .

Theorem 4.22.

A fuzzy subset $ϵ$ of a QORS, $\mathfrak{K}$ is a comparative fuzzy filter iff ${ϵ}_{\alpha }$ for all $\alpha \in [0,1],$ is a comparative filter of $K$ .

Proof.

Suppose $ϵ$ is a comparative fuzzy filter of $K$ . Let $a\to ((b\to c)\to b)\in {ϵ}_{\alpha }$ and $a\in {ϵ}_{\alpha }$ for all $a,b,c\in K$ .

Since

This implies that

Since ${ϵ}_{\alpha }$ is a comparative filter, $b\in {ϵ}_{\alpha }$ i.e.,

Therefore, $ϵ$ is a comparative fuzzy filter of $K$ .      □

Definition 4.23.

A fuzzy filter $ϵ$ of a QORS, $\mathfrak{K}$ satisfying the condition $ϵ((a\to b)\to b)\ge ϵ\left(c\right)\wedge ϵ(c\to ((b\to a)\to a))$ for all $a,b,c\in K$ , is said to be a normal fuzzy filter of $K$ .

A QORS, $\mathfrak{K}$ satisfying the condition $(a\to b)\to b\equiv \preccurlyeq (b\to a)\to a$ for all $a,b\in K$ , is said to be strong QORS.¹⁴

Theorem 4.24.

Let $\mathfrak{K}$ be a strong QORS and $ϵ$ be a fuzzy filter of $K$ . Then $ϵ$ is a normal fuzzy filter of $K$ .

Proof.

Suppose $\mathfrak{K}$ be a strong QORS and $ϵ$ be a fuzzy filter of $K$ . Thus, from the condition of Definition 3.5 and for $a,b,c\in K$ as $(a\to b)\to b=(b\to a)\to a$ , we have

Therefore, $ϵ$ is a normal fuzzy filter of $K$ .      □

Theorem 4.25.

A fuzzy filter $ϵ$ of a QORS, $\mathfrak{K}$ is a normal fuzzy filter of $K$ if and only if $ϵ((a\to b)\to b)\ge ϵ((b\to a)\to a).$

Proof.

Suppose $ϵ$ be a normal fuzzy filter of $K$ . From Lemma 4.4,

$ϵ((a\to b)\to b)\ge ϵ(1\to ((b\to a)\to a))=ϵ((b\to a)\to a)),$ for all $a,b\in K.$

Conversely, suppose that

$ϵ((a\to b)\to b)\ge ϵ((b\to a)\to a),$ for all $a,b\in K.$ Since $ϵ$ is a fuzzy filter of $K$ from Definition 3.5, we have

Thus,

So, $ϵ$ is a normal fuzzy filter of $K$ .      □

Theorem 4.26.

A comparative fuzzy filter of a QORS, $\mathfrak{K}$ is a normal fuzzy filter of $K$ .

Proof.

Suppose $ϵ$ be a comparative fuzzy filter of $K$ . For any $a,b,c\in K$ , we have $a·(b\to a)\preccurlyeq a$ so that $a\preccurlyeq (b\to a)\to a.$ Thus,

Also, as $b\preccurlyeq (b\to a)\to a,$ we have

which shows that

Since $ϵ$ is comparative fuzzy filter,

So, as $ϵ$ is a fuzzy filter we have

Thus,

Hence, $ϵ$ is normal fuzzy filter of $K$ .      □

Theorem 4.27.

A normal fuzzy filter of a QORS, $\mathfrak{K}$ is a comparative fuzzy filter if it is an implicative fuzzy filter of $K$ .

Proof.

Suppose $ϵ$ be both a normal fuzzy filter and an implicative fuzzy filter of a QORS $\mathfrak{K}$ . Clearly, for any $a,b\in K$ ; if $a\preccurlyeq b$ , then $ϵ\left(a\right)\le ϵ\left(b\right)$ . Since $ϵ$ is a fuzzy filter of $K$ , to show that ϵ is a comparative fuzzy filter of $K$ by Theorem 4.16 it suffices to show that $ϵ\left(b\right)\ge ϵ((b\to c)\to b)$ for all $b,c\in K$ . From the condition of Definition 3.5, we have