Pythagorean  fuzzy deductive system of BCL-algebra

Remark 3.1.

For BCL-algebra, (B; $⊛$ , 0), one can easily prove that the following equations hold, $\forall m$ , $n\in$ B:

Definition 3.2.

A non-empty subset D of BCL-algebra (B; $⊛$ , 0) is called deductive system (DS.) of B if it satisfies, $\forall x$ , $y$ , $z\in$ B:

Example 3.3.

Let B = {0, $p$ , $q$ , $r$ } and define a binary operation $⊛$ on B by the Cayley Table 1 as follows:

In the Table 1, it can be easily seen that B is a BCL-algebra and,

are not deductive systems of B.

Proposition 3.4.

If D is deductive system of B, where (B; $⊛$ , 0) is BCL-algebra, then the following hold true:

Proof.

Suppose D is deductive system of B.

$\Rightarrow x⊛$ 0 = 0 or $x⊛$ 0 = $x$

But by a remark above,

$x⊛$ 0 = 0 $\Rightarrow x$ = 0 and also, $x⊛$ 0 = $x\Rightarrow x$ = 0

$\Rightarrow$ in any case, D = {0} ◻

Definition 3.5.

A fuzzy set ${\eta }_B$ in a BCL-algebra B is called a fuzzy deductive system (fuzzy DS.) of B, if the following axioms are satisfied, for all $x$ , $y$ , $z\in$ B:

Example 3.6.

Suppose B = {0, $q$ , $r$ , $p$ } and the binary operation $⊛$ on B is as given by the Table 1 above:

Define a fuzzy set ${\eta }_B$ : B $\to$ [0. 1] by: ${\eta }_B\left(x\right)$ = $\{\begin{array}{l}0.8,\mathit{\text{if}}x=0,\\ 0.6,\mathit{\text{if}}x=p,q,\\ 0.2,\mathit{\text{if}}x=r.\end{array}$

Then it is easy to check that ${\eta }_B$ is a fuzzy DS. of the BCL-algebra, B.

Definition 3.7.

A Pythagorean fuzzy set $B^P$ = $({\eta }_P$ , ${\tau }_P)$ , where the functions:

${\eta }_P\left(x\right)$ : B $\to$ [0, 1] and ${\tau }_P\left(x\right)$ : B $\to$ [0, 1] define $\text{the degree}$ of $\text{membership}$ and the $\text{degree of non‐membership}$ , respectively in B is called a Pythagorean fuzzy DS. of B if the following axioms are satisfied, for all $x$ , $y$ , $z\in$ B:

Example 3.8.

Let B and the binary operation $⊛$ be as defined as in Table 1 and let the fuzzy sets ${\eta }_B$ : B $\to$ [0. 1] and ${\tau }_B$ : B $\to$ [0. 1] be defined as follows:

Then by routine calculations, it can be easily seen that $B^P$ = $({\eta }_B,{\tau }_B)$ is a $\text{Pythagorean fuzzy}\mathit{DS}.$ of B $(\mathrm{but}\text{not intuitionistic fuzzy}\mathit{DS}.\text{of}B)$ .

Lemma 3.9.

Let $B^P$ = $({\eta }_B$ , ${\tau }_B)$ be Pythagorean fuzzy set in B. If $B^P$ is Pythagorean fuzzy DS. of B, then the following hold, $\forall m$ , $n$ , $u\in$ B:

Proof.

Let $B^P$ = $({\eta }_B$ , ${\tau }_B)$ be Pythagorean fuzzy DS. in B.

Definition 3.10.

For membership fuzzy set; ${\mu }_B$ : B $\to$ [0, 1], and square subtrahend

${\overline{\overline{\mu }}}_B$ : B $\to$ [0, 1] from 1 such that ${\left({\overline{\overline{\mu }}}_B\left(m\right)\right)}^2$ = 1 $-{\left(\mu \left(m\right)\right)}^2$ , we call such fuzzy set, ${\overline{\overline{\mu }}}_B$ ,

the square deviation of ${\mu }_B$ .

Theorem 3.11.

Let ${\eta }_B$ and its square deviation ${\overline{\overline{\eta }}}_B$ be fuzzy sets in B such that

${\left({\eta }_B(m⊛n)\right)}^2$ = ${\left({\eta }_B\left(n\right)\right)}^2$ . Then ${\left({\overline{\overline{\eta }}}_B(m⊛n)\right)}^2$ = ${\left({\overline{\overline{\eta }}}_B\left(n\right)\right)}^2$ , $\forall m$ , $n\in$ B.

Then, $B^P$ = $({\eta }_B$ , ${\overline{\overline{\eta }}}_B)$ is Pythagorean fuzzy DS. of B. iff ${\eta }_B$ and then ${\overline{\overline{\eta }}}_B$ are constants.

Furthermore, $\mathrm{he}\text{accuracy function}{\mathit{\text{a}}}_{\mathit{B}}\left(m\right)$ , the $\text{score function}{s}_B\left(m\right)$ and the $\text{degree of}$

$\text{indeterminacy}{\pi }_B\left(m\right)$ are respectively given as: $\forall m\in$ B:

Proof.

Suppose ${\left({\eta }_B(m⊛n)\right)}^2$ = ${\left({\eta }_B\left(n\right)\right)}^2$

Then ${\left({\overline{\overline{\eta }}}_B(m⊛n)\right)}^2$ = 1 $-{\left({\eta }_B(m⊛n)\right)}^2$ = 1 $-{\left({\eta }_B\left(n\right)\right)}^2$ = ${\left({\overline{\overline{\eta }}}_B\left(n\right)\right)}^2$ and

Let $B^P$ = $({\eta }_B$ , ${\overline{\overline{\eta }}}_B)$ be Pythagorean fuzzy DS.. of B, $a$ nd then ${\left({\overline{\overline{\eta }}}_B(m⊛n)\right)}^2$ = ${\left({\overline{\overline{\eta }}}_B\left(n\right)\right)}^2$ ,

Now we claim to verify that ${\eta }_B$ and ${\overline{\overline{\eta }}}_B$ are constants,

$($ or ${\left({\eta }_B\left(m\right)\right)}^2$ = ${\left({\eta }_B\left(n\right)\right)}^2$ and then ${\left({\overline{\overline{\eta }}}_B\left(m\right)\right)}^2$ = ${\left({\overline{\overline{\eta }}}_B\left(n\right)\right)}^2$ , $\forall m$ , $n\in$ B. $)$

As $B^P$ = $({\eta }_B$ , ${\overline{\overline{\eta }}}_B)$ is a Pythagorean fuzzy DS. of B, ${\eta }_B$ is a fuzzy DS. of B, and hence by

one of the axioms, we have ${\eta }_B\left(0\right)$ = ${\eta }_B(0⊛m)$ , $\forall m\in$ B,

$\Rightarrow {\left({\eta }_B\left(0\right)\right)}^2$ = ${\left({\eta }_B((0⊛m)⊛m)\right)}^2$ = ${\left({\eta }_B\left(m\right)\right)}^2$ , $\forall m\in$ B, and again

${\left({\eta }_B\left(0\right)\right)}^2$ = ${\left({\eta }_B((0⊛n)⊛n)\right)}^2$ = ${\left({\eta }_B\left(n\right)\right)}^2$ , $\forall n\in$ B

$\Rightarrow {\left({\eta }_B\left(0\right)\right)}^2$ = ${\left({\eta }_B\left(m\right)\right)}^2$ = ${\left({\eta }_B\left(n\right)\right)}^2$ , $\forall m$ , $n\in$ B,

Or ${\left({\eta }_B\left(m\right)\right)}^2$ = ${\left({\eta }_B\left(n\right)\right)}^2$ , $\forall m$ , $n\in$ B and hence, ${\eta }_B$ is constant, and analogously, ${\overline{\overline{\eta }}}_B$ is, too.

Conversely, suppose ${\eta }_B$ and ${\overline{\overline{\eta }}}_B$ are constants, or: ${\eta }_B\left(m\right)$ = ${\eta }_B\left(n\right)$ and ${\overline{\overline{\eta }}}_B\left(m\right)$ = ${\overline{\overline{\eta }}}_B\left(n\right)$ , $\forall m$ , $n\in$ B

${\left({\eta }_B(m⊛n)\right)}^2$ = ${\left({\eta }_B\left(n\right)\right)}^2$ and ${\left({\overline{\overline{\eta }}}_B(m⊛n)\right)}^2$ = ${\left({\overline{\overline{\eta }}}_B\left(n\right)\right)}^2$ , and

To prove: $B^P$ = $({\eta }_B$ , ${\overline{\overline{\eta }}}_B)$ is a Pythagorean fuzzy DS. of B,

Therefore, by (i) and (ii) above, $B^P$ = $({\eta }_B$ , ${\overline{\overline{\eta }}}_B)$ is Pythagorean fuzzy DS. of B.

Furthermore:

Proposition 3.12.

Let U be a non-void subset of B such that ${\chi }_U$ is characteristic function and ${\overline{\overline{\chi }}}_U$ is its square deviation. Then $B^P$ = $({\chi }_U$ , ${\overline{\overline{\chi }}}_U)$ is Pythagorean fuzzy DS. of B if and only if U is DS. of B.

Proof.

Let ${\chi }_U$ : U $\to$ [0, 1] be characteristic function and the square deviation ${\overline{\overline{\chi }}}_U$ : U $\to$ [0, 1] be

defined as:

Suppose $B^P$ = $({\chi }_U$ , ${\overline{\overline{\chi }}}_U)\Rightarrow {\left({\chi }_U\right((m⊛n)⊛n\left)\right)}^2\ge {\left({\chi }_U\left(m\right)\right)}^2$ and ${\left({\overline{\overline{\chi }}}_U\right((m⊛n)⊛n\left)\right)}^2\le {\left({\overline{\overline{\chi }}}_U\left(m\right)\right)}^2$

Let $m\in$ U $\Rightarrow {\chi }_U\left(m\right)$ = 1 and ${\overline{\overline{\chi }}}_U\left(m\right)$ = 0

Then ${\chi }_U((m⊛n)⊛n)\ge {\chi }_U\left(m\right)$ = 1 and ${\overline{\overline{\chi }}}_U((m⊛n)⊛n)\le {\overline{\overline{\chi }}}_U\left(m\right)$ = 0

But ${\chi }_U((m⊛n)⊛n)\le$ 1 and ${\overline{\overline{\chi }}}_U((m⊛n)⊛n)\ge$ 0

$\Rightarrow {\chi }_U((m⊛n)⊛n)$ = 1 and ${\overline{\overline{\chi }}}_U((m⊛n)⊛n)$ = 0

$\Rightarrow (m⊛n)⊛n)\in$ U and the other axiom can be checked similarly.

$\Rightarrow$ U is DS. of B.

Conversely, suppose U is DS. of B.

We claim that $B^P$ = $({\chi }_U$ , ${\overline{\overline{\chi }}}_U)$ is Pythagorean fuzzy DS. of B, which means we need to show:

(i) Now we prove this case by taking the following three cases:

Case (1) Let $m\in$ U $(\Rightarrow (m⊛n)⊛n\in U$ by hypothesis $)$

$\Rightarrow {\chi }_U((m⊛n)⊛n)$ = ${\chi }_U\left(m\right)$ = 1 $\Rightarrow {\chi }_U((m⊛n)⊛n)\ge {\chi }_U\left(m\right)$ and

$\Rightarrow {\overline{\overline{\chi }}}_U((m⊛n)⊛n)$ = ${\overline{\overline{\chi }}}_U\left(m\right)$ = 0 $\Rightarrow {\overline{\overline{\chi }}}_U((m⊛n)⊛n)\le {\overline{\overline{\chi }}}_U\left(m\right)$

Case (2) Let $m\notin$ U and $(m⊛n)⊛n\in U$

$\Rightarrow {\chi }_U((m⊛n)⊛n)$ = 1 and ${\chi }_U\left(m\right)$ = 0 $\Rightarrow {\chi }_U((m⊛n)⊛n)\ge {\chi }_U\left(m\right)$ and

${\overline{\overline{\chi }}}_U((m⊛n)⊛n)$ = 0 and ${\overline{\overline{\chi }}}_U\left(m\right)$ = 1 $\Rightarrow {\overline{\overline{\chi }}}_U((m⊛n)⊛n)\le {\overline{\overline{\chi }}}_U\left(m\right)$

The above steps hold for squaring (for Pythagorean) each term (and the steps hereunder also hold)

Case (3) Let $m\notin$ U $(m⊛n)⊛n\notin U$

$\Rightarrow {\chi }_U((m⊛n)⊛n)$ = ${\chi }_U\left(m\right)$ = 0 $\Rightarrow {\chi }_U((m⊛n)⊛n)\ge {\chi }_U\left(m\right)$ and

$\Rightarrow {\overline{\overline{\chi }}}_U((m⊛n)⊛n)$ = ${\overline{\overline{\chi }}}_U\left(m\right)$ = 1 $\Rightarrow {\overline{\overline{\chi }}}_U((m⊛n)⊛n)\le {\overline{\overline{\chi }}}_U\left(m\right)$

(ii) By following similar steps for this as (i) above, where here the cases are:

Case (1) For $m$ , $n\in$ U $(\Rightarrow m⊛(n⊛u)\in U$ by hypothesis $)$

Case (2) For $(m\notin$ U or $n\notin$ U $)$ and $(m⊛n)⊛n\in U$

Case (3) For $(m\notin$ U or $n\notin$ U $)$ and $(m⊛n)⊛n\notin U$

we arrive at the result:

${\chi }_U(m⊛(n⊛u)\ge min\{{\chi }_U\left(m\right),{\chi }_U\left(n\right)\}$ and ${\overline{\overline{\chi }}}_U(m⊛(n⊛u)\le max\{{\overline{\overline{\chi }}}_U\left(m\right),{\chi }_U\left(n\right)\}$

Therefore, $B^P$ is Pythagorean fuzzy DS. of B. ◻

Theorem 3.13.

The intersection, $B_1\cap B_2$ , of any two Pythagorean fuzzy DS.s, $B_1$ and $B_2$ , of B is also a Pythagorean fuzzy DS. of B.

Proof.

Let $B_1$ = $({\eta }_{B_1}$ , ${\tau }_{B_1})$ and $B_2$ =( ${\eta }_{B_2}$ , ${\tau }_{B_2})$ be any two Pythagorean fuzzy DSs. of B.

Then we need to prove that $B_1\cap B_2$ is a Pythagorean fuzzy DS. of B. Let $m$ , $n$ , $u\in$ B.

(i) ${\left({\eta }_{B_1\cap B_2}((m⊛n)⊛n)\right)}^2$ = min $\{{\left({\eta }_{B_1}((m⊛n)⊛n)\right)}^2$ , ${\left({\eta }_{B_2}((m⊛n)⊛n)\right)}^2$ }

$\ge$ min $\{{\left({\eta }_{B_1}\left(m\right)\right)}^2$ , ${\left({\eta }_{B_2}\left(m\right)\right)}^2\}$ = ${\left({\eta }_{B_1\cap B_2}\left(m\right)\right)}^2$ and

${\left({\tau }_{B_1\cap B_2}((m⊛n)⊛n)\right)}^2$ = max $\{{\left({\tau }_{B_1}((m⊛n)⊛n)\right)}^2$ , ${\left({\tau }_{B_2}((m⊛n)⊛n)\right)}^2\}$

$\le$ max $\{{\left({\tau }_{B_1}\left(m\right)\right)}^2,{\left({\tau }_{B_2}\left(m\right)\right)}^2\}$ = ${\left({\tau }_{B_1\cap B_2}\left(m\right)\right)}^2$

(ii) ${\left({\eta }_{B_1\cap B_2}(m⊛(n⊛u))\right)}^2$ = min $\{{\left({\eta }_{B_1}(m⊛(n⊛u))\right)}^2$ , ${\left({\eta }_{B_2}(m⊛(n⊛u))\right)}^2\}$

$\ge$ min $\{$ min $\{{\left({\eta }_{B_1}\left(m\right)\right)}^2$ , ${\left({\eta }_{B_1}\left(n\right)\right)}^2\}$ , min $\{{\left({\eta }_{B_2}\right(\left(m\right)\left)\right)}^2$ , ${\left({\eta }_{B_2}\right(n\left)\right)}^2\left\}\right\}$

= min $\{$ min $\{{\left({\eta }_{B_1}\left(m\right)\right)}^2$ , ${\left({\eta }_{B_2}\right(\left(m\right)\left)\right)}^2$ , min $\left\{{\left({\eta }_{B_1}\left(n\right)\right)}^2\right\}$ , ${\left({\eta }_{B_2}\right(n\left)\right)}^2\left\}\right\}$

= min $\{{\left({\eta }_{B_1\cap B_2}\left(m\right)\right)}^2$ , ${\left({\eta }_{B_1\cap B_2}\right(n\left)\right)}^2\}$ and

Similarly, we have: ${\left({\tau }_{B_1\cap B_2}(m⊛(n⊛u))\right)}^2\le$ max $\{{\left({\eta }_{B_1\cap B_2}\right(m\left)\right)}^2$ , ${\left({\eta }_{B_1\cap B_2}\left(n\right)\right)}^2\}$ and

Hence by (i) and (ii) above, $B_1\cap B_2$ is a Pythagorean fuzzy DS. of B.

The above theorem can also be generalized to any set of Pythagorean fuzzy DS.s as in the following corollary. ◻

Corollary 3.14.

The intersection, ${\bigcap }_{i\in I}{B}_i$ , of any family of Pythagorean fuzzy DSs.,

$\{B_i:i\in I\}$ , in B is also a Pythagorean fuzzy DS. of B, where,

Remark 3.15.

The union of any two Pythagorean fuzzy $\mathit{DS}.s$ of B is not necessarily Pythagorean fuzzy $\mathit{DS}.$ of B.

Example 3.16.

Let (B; $⊛$ , 0) be a BCL-algebra, where B = {p, q, r, 0} and let “ $⊛$ ” be as defined in the Table 1 at the bottom, and define two Pythagorean fuzzy DSs.

$B_1$ = $({\left({\eta }_{B_1}\right)}^2,{\left({\tau }_{B_1}\right)}^2)$ and $B_2$ = $({\left({\eta }_{B_2}\right)}^2,{\left({\tau }_{B_2}\right)}^2)$ as follows:

It is easy to check that $B_1$ and $B_2$ are Pythagorean fuzzy $\mathit{DS}.s$ of B but to show that their union is not necessarily a Pythagorean fuzzy $\mathit{DS}.$ of B, we justify it as follows using the above pairs of Pythagorean fuzzy DS.s of B which are $B_1$ and $B_2$ , where $($ B; $⊛$ , 0 $)$ is as defined in Table at the bottom:

Take $m$ = $r$ , $n$ = $p$ and $u$ = $r$

$\Rightarrow (q⊛(p⊛r))$ = $(q⊛p))$ = $r$ Then

(i) ${\left({\eta }_{B_1\cup B_2}(q⊛(p⊛r))\right)}^2$ = ${\left({\eta }_{B_1\cup B_2}\right(r\left)\right)}^2$ = 0.2

$\ge$ min $\{{\left({\eta }_{B_1\cup B_2}\right(p\left)\right)}^2,{\left({\eta }_{B_1\cup B_2}\right(q\left)\right)}^2\}$

= min{0.5, 0.5 $\}$ = 0.5 is not true.

Thus, by the above justifications, the union of any two Pythagorean fuzzy $\mathit{DS}.s$ of B is not necessarily a Pythagorean fuzzy $\mathit{DS}.$ of B.

Theorem 3.17.

Let $\eta ,$ be a fuzzy set such that ${\eta }_B$ is a membership function and ${\overline{\overline{\eta }}}_B$ is its square deviation in B. Suppose ${\left({\eta }_B^b\right)}^2$ = $\{$ x $\in$ B: ${\left({\eta }_B\left(x\right)\right)}^2\ge {\left({\eta }_B\left(b\right)\right)}^2$ $\forall b\in$ B $\}$ and then

Then $B^P$ = $({\eta }_B$ , ${\overline{\overline{\eta }}}_B)$ is Pythagorean fuzzy DS. of B iff $B_b^P$ = $({\eta }_B^b$ , ${\overline{\overline{\eta }}}_B^b)$ is Pythagorean fuzzy DS. of B.

Proof.

Suppose $B^P$ = $({\eta }_B$ , ${\overline{\overline{\eta }}}_B)$ is Pythagorean fuzzy DS. of B, or $\forall m$ , $n$ , $u$ , $\in$ B:

We need to show that $B_b^P$ = $({\eta }_B^b$ , ${\overline{\overline{\eta }}}_B^b)$ is Pythagorean fuzzy DS. of B.