Pythagorean fuzzy LiuB – subalgebra of LiuB - algebra

Definition 2.1.1.

¹⁴ A BCI-algebra is an algebra (X; *, 0) of (2, 0) kind such that conditions below are fulfilled; ∀k, m, n ∈ X.:

Definition 2.1.2.

¹⁵ Suppose X is a BCI-algebra and the set I is non-empty subset of X. I is known as subalgebra of X if, for every m, n in X.

Definition 2.1.3.

^15,16 A fuzzy set μ in a BCK-algebra X is known as fuzzy subalgebra of X if; for every m, n in X:

Definition 2.2.1.

^8,9,17 Let ∅ ≠ X. A fuzzy subset A ⊆ X is defined as: A = {⟨m, μ_A (m)⟩: m ∈ X}, such that μ_A (m): X → [0, 1] given by the degree of membership and a complement of μ_A, symbolized as ${\overline{\mu }}_A$ , is the fuzzy set in X given by ${\overline{\mu }}_A\left(m\right)=1-{\mu }_A\left(m\right)$ for every m ∈ X, and as a result ${\mu }_A\left(m\right)=1-{\overline{\mu }}_A\left(m\right)$ .

Definition 2.2.2.

¹² An intuitionistic fuzzy set I in the non-void set X is an object where: {m, μ_I (m), υ_I (m): m∈X}, such that the function μ_I (m): X →[0, 1] and υ_I (m): X → [0, 1] define the membership degree and the non-membership degree, respectively where: $0\le {\mu }_I\left(m\right)+{\nu }_I\left(m\right)\le 1$ is fulfilled.

Definition 2.2.3.

¹⁷ A Pythagorean fuzzy set P in X (non-void set) is an object such that: P = {m, η_P (m), τ_P (m): m ∈X}, for a mappings η_p (m): X →[0, 1] and τ_P (m): X→[0, 1] define membership degree & non-membership degree, respectively such that: $0\le {\left({\eta }_P\left(m\right)\right)}^2+{\left({\tau }_P\left(m\right)\right)}^2\le 1$ is fulfilled.

Definition 2.2.4.

¹⁸ Suppose ${\left({\eta }_P\left(m\right)\right)}^2+{\left({\nu }_P\left(m\right)\right)}^2\le 1$ , then there is a degree of indeterminacy of $m\in X\text{to}P\text{given as}{\pi }_P\left(m\right)=\sqrt{1-({\left({\eta }_P\left(m\right)\right)}^2+{\left({\nu }_P\left(m\right)\right)}^2)}\Rightarrow {\left({\pi }_P\left(m\right)\right)}^2=1-({\left({\eta }_P\left(m\right)\right)}^2+{\left({\nu }_P\left(m\right)\right)}^2);{\pi }_P\left(m\right)\in [0,1]$

Definition 2.2.5.

¹⁸ Let P be Pythagorean fuzzy set on X. Then, the score function, s of P is given by: $s_P\left(m\right)={\left({\eta }_P\left(m\right)\right)}^2-{\left({\nu }_P\left(m\right)\right)}^2$ , where $s_P\left(m\right)$ ∈ [-1, 1].

Definition 2.2.6.

¹⁸ For Pythagorean fuzzy set P on X, the accuracy function, a, of P is given by: $a_P\left(m\right)={\left({\eta }_P\left(m\right)\right)}^2+{\left({\nu }_P\left(m\right)\right)}^2$ for a(P) ∈ [0, 1].

Definition 2.2.7.

⁸ The Pythagorean fuzzy set in a BCK-algebra B is known as a Pythagorean fuzzy subalgebra of B if for every m, n ∈ B, the axiom below is fulfilled:

Definition 2.3.1.

⁴ An algebra (B; ⊛, 0) of (2, 0) kind is known as BCL-algebra iff for every m, n, w ∈ B:

Example 2.3.2.

⁵ Let B = {0, 1, 2, 3} and define a binary operation ⊛ on B by the Table 1, below. Then from Table 1, it is simple to verify that B is BCL-algebra.

Definition 2.3.3.

⁵ Suppose (B; ⊛, 0) is a BCL-algebra. A binary relation “≤” on B in which m, n∈ B iff m ⊛ n = 0 for every m, n ∈ B is known as the BCL-ordering and “≤” is partial ordering on B.

Definition 2.3.4.

⁵ Suppose m ≤ n iff m ⊛ n = 0, the definition 2.3.1 for BCL-algebra could be given by:

Definition 2.3.5.

^4,6 An algebra (B; ⊛, 1) of (2, 0) kind is known as BCL⁺-algebra iff for every m, n, w ∈ B:

Definition 2.3.6.

⁶ A Liu-algebra is a tetrad (L; ⊛, ⊙, 1) in which L is a non-void set; ⊛ and ⊙ are two binary operations defined on L, 1 is a constant in L in which for every m, n, w ∈ L such that the statements given below are true:

Definition 2.3.7.

⁶ Suppose K is non-void subset of Liu-algebra (L; ⊛, ⊙, 1).

Thus K is known as subalgebra of L iff m, n ∈ K ⇒ m ⊛ n ∈ K and m ⊙ n ∈ K.

Example 2.3.8.

⁶ Let L = {p, q, r, 1} and two binary operations ⊛ and ⊙ on L be as defined in Table 2 below:

From Table 2, it can easily be justified that (X, ⊛, ⊙, 1) is Liu-algebra, and {1, p}, {1, q}, {1, p, q}, {1, p, r}, {1} L are subalgebras of L but {p, q}, {1, q, r}, {p, q, r} are not.

Proposition 3.1.1.

If (B; ⊛, 0) is a BCL-Algebra, hence all statements hereunder are true, for every m, n ∈ B:

(1) m = 0 (and hence 0 is the least element),

(2) m ⊛ 0 = 0 ⇒ m = 0.

(3) m ⊛ n = m ⇒ m = 0,

(4) m ⊛ n = n ⇒ m = 0,

Proof.

Suppose B is a BCL-Algebra and m ∈ B

Definition 3.2.1.

Liu^B-algebra (LBA.) is an algebra (L; ⊛, ⊙, 0) in which L is a non-void set, ⊛ and ⊙ are two binary operations defined on L, 0 is constant in L, where for every m, n, w∈ L, the axioms hereunder are true:

Proposition 3.2.2.

Let (L; ⊛, ⊙, 0) be a LBA., then the statements hereunder are true for every m, n ∈ L:

Proof.

Suppose (L; ⊛, ⊙, 0) is a LBA. and m, n ∈ L:

Theorem 3.2.3.

Let (L; ⊛, ⊙, 0) be LBA. and m, n, w ∈ L. So the statements below are true:

Proof.

Let (L; ⊛, ⊙, 0) be LBA. and let m, n, w ∈ L:

Definition 3.2.4.

Suppose N is a non-void subset of L. Then N is known as LBS. of L iff: for all m, n ∈ N; m ⊛ n ∈ N and m ⊙ n ∈ N.

Example 3.2.5.

Assume L = {0, m, n, w} and two binary operations ⊛ and ⊙ on L be defined by the Table 3 below:

Then it easy to check that (L, ⊛, ⊙, 0) is LBA. and we have the following as well:

Lemma 3.2.6.

Suppose (L, ⊛, ⊙, 0) is LBA., and N is LBS. of L. Then 0 ∈ N

Proof.

Let m ∈ N ⇒ m ∈ L ⇒ m ∗ m = 0 ∈ N □

Definition 3.2.7.

A fuzzy set ${\eta }_L$ in LBA. L is known as a fuzzy LBS. of L if the statements

hereunder are fulfilled for every m, n ∈ L:

Example 3.2.8.

Assume L = {0, m, n, w} and let the binary operations ⊛ and ⊙ on L be as defined in Table 3 below and the fuzzy set ${\eta }_L$ in L be defined as follows:

Definition 3.3.1.

A Pythagorean fuzzy set L^B = $({\eta }_L,{\tau }_L)$ in a non-void set L in which the functions η_L: L → [0, 1] and τ_L: L → [0, 1] define membership degree and non-membership degree respectively in L is known as a Pythagorean fuzzy Liu ^B-subalgebra (LBS.) of L if the two pairs of statements hereunder are fulfilled for every m, n ∈ L:

Example 3.3.2.

Suppose L = {0, m, n, w} and two binary operations ⊛ and ⊙ on L are given in the Table 3 above:

Let the Pythagorean fuzzy set L^B = $({\eta }_L,{\tau }_L)$ such that

η_L: L → [0, 1] and τ_L: L → [0, 1] given by:

Thus we could simply verify that L ^B is Pythagorean fuzzy LBS. of L.

Lemma 3.3.3.

If $L^B=({\eta }_L,{\tau }_L)$ is a Pythagorean fuzzy LBS. of L, then ∀ m ∈ L; then (η_L (0))² ≥ (η_L(m))² and (τ_L (0))² ≤ (τ_L(m))².

Proof.

For m ∈ L, we have: m ⊛ m = 0. Then (η_L (0))² = (η_L (m ⊛ m))² ≥ min{(η_L(m))², (η_L (m))²} = (η_L (m))² and (τ_L (0))² = (τ_L (m ⊛ m))² ≤ max{(τ_L(m))², (τ_L (m))²} = (τ_L(m))². Hence (η_L (0))² ≥ (η_L (m))² and (τ_L (0))² ≤ (τ_L(m))² □

Definition 3.3.4.

For a membership fuzzy set; μ_L: L → [0, 1], let ${\overline{\overline{\mu }}}_L$ : L → [0, 1]; where ( ${\overline{\overline{\mu }}}_L$ (x))² = 1 − (μ _L(x))². Then we call such ${\overline{\overline{\mu }}}_L$ the square deviation (of μ _L).

Theorem 3.3.5.

Let M be a non-void subset of L and L ^B = $({\chi }_M,{\overline{\overline{\chi }}}_M)$ , where ${\chi }_M$ is characteristic function and the square deviation, ( ${\overline{\overline{\chi }}}_M$ (k))² = 1 − ( ${\chi }_M$ (k))², then L ^B is Pythagorean fuzzy LBS. of L iff M is LBS. of L.

Proof.

Suppose ${\chi }_M$ : M → [0, 1] is characteristic function defined as:

Then characteristic function is LBS. of L.

Suppose L^B is Pythagorean fuzzy LBS. of L.

For k, q ∈ M we have (χ _L (k ⊛ q))² ≥ min{(χ _L(k))², (χ _L (q))²} = 1

⇒ (χ _M (k ⊛ q))² ≥ 1 (and (χ _M (k ⊛ q))² ≤ 1 by definition of χ _M) and ( ${\overline{\overline{\chi }}}_M$ (k ⊛ q))² ≤ 1)

Similarly, k ⊙ q ∈ M. Therefore, M is LBS. of L.

Conversely, suppose M is LBS. of L. Since (χ _M (0))² = 1 ⇒ 0 ∈ M ⇒ M ≠ ∅

Now let k, q ∈ M ⇒ k ⊛ q∈ M and k ⊙ q∈ M ⇒ (χ_L (k ⊛ q))² = 1 ≥ min {(χ_L(k))², (χ_L (q))²} and ( ${\overline{\overline{\mathrm{\chi }}}}_M$ (k ⊛ q))² = 0 ≤ max {( ${\overline{\overline{\mathrm{\chi }}}}_M$ (k))², ( ${\overline{\overline{\mathrm{\chi }}}}_M$ (q))²}

Similarly, (χ_L (k ⊙ q))² = 1 ≥ min {(χ_L(k))², (χ_L (q))²} and ${\overline{\overline{\mathrm{\chi }}}}_M$ (k ⊙ q))² = 0 ≤ max {( ${\overline{\overline{\mathrm{\chi }}}}_M$ (k))², ( ${\overline{\overline{\mathrm{\chi }}}}_M$ (q))²}

Thus, L^B = $({\chi }_M,{\overline{\overline{\chi }}}_M)$ is Pythagorean fuzzy LBS. of L, similarly true for k in M, q not in M or k, q not in M. □

Theorem 3.3.6.

The intersection of any two Pythagorean fuzzy LBS., L₁ ^B and L₂ ^B of L is also a Pythagorean fuzzy LBS. of L.

Proof.

Let L₁^B = $({\eta }_{L_1^B},{\mathrm{\tau }}_{L_1^B})$ & L₂^B = $({\eta }_{L_2^B},{\mathrm{\tau }}_{L_2^B})$ be any two Pythagorean fuzzy LBS. of L.

Then we need to prove: $L_{\mathrm{T}}^B$ = $({\eta }_{L_1^B}\cap {\eta }_{L_2^B},{\mathrm{\tau }}_{L_1^B}\cap {\mathrm{\tau }}_{L_2^B})$ is Pythagorean fuzzy LBS. of L.

For $m,n\in L,\text{and\hspace{0.17em}for\hspace{0.17em}the}\text{two binary\hspace{0.17em}operations ∗ and • of}L,\text{we\hspace{0.17em}have}$ :

Thus, intersection of Pythagorean fuzzy LBSs. of L is Pythagorean fuzzy LBS. of L. □

Corollary 3.3.7.

The intersection, $L_i^B=(\underset{i\in I}{\cap }\left({\eta }_{L_i^B}\right),\underset{i\in I}{\cap }\left({\tau }_{L_i^B}\right))$ of a family of Pythagorean LBS in L is also Pythagorean fuzzy LBS of L, where

Remark 3.3.8.

The union of two Pythagorean fuzzy LBS. of a LBA. L may not necessarily be Pythagorean fuzzy LBS. of L.

Example 3.3.9.

Assume L = {0, m, n, w} and two binary operations ⊛ and ⊙ on L as defined in Tables 3 above. Then it is easy to show that (L, ⊛, ⊙, 0) is LBA.

Define two Pythagorean LBS. of LBA., L₁ ^B = $({\eta }_{L_1^B},{\mathrm{\tau }}_{L_1^B})$ and L₂ ^B = $({\eta }_{L_2^B},{\mathrm{\tau }}_{L_2^B})$ as follows:

Proposition 3.3.10.

Let M be a non-empty subset of LBA. L and ${\eta }_L$ be a fuzzy set in a LBA. L such that

Then M is LBS. of L iff $({\eta }_L,{\overline{\overline{\eta }}}_L)$ is a Pythagorean fuzzy LBS. of L and let ${\overline{\overline{\eta }}}_L\left(m\right)$ = ${\tau }_L$ (m) .

Proof.

For M is LBS. of L, we prove that $({\eta }_L,{\tau }_L)$ is a Pythagorean fuzzy LBS. of L.

Case (i): Let m, n ∈ M ⇒ m ⊛ n ∈ M

Then ( ${\eta }_L$ (m))² = ( ${\eta }_L$ (n))² = δ² ⇒ ( ${\eta }_L$ (m ⊛ n))² = δ²

⇒ ( ${\eta }_L$ (m ⊛ n))² ≥ δ² = min{( ${\eta }_L$ (m))², ( ${\eta }_L$ (n))²} and similarly

( ${\eta }_L$ (m ⊙ n))² ≥ δ² = min{( ${\eta }_L$ (m))², ( ${\eta }_L$ (n))²} and in a similar way, we can show

( ${\tau }_L$ (m ⊛ n))² ≤ 1 - δ² = max{( ${\tau }_L$ (m))², ( ${\tau }_L$ (n))²} and similarly

( ${\tau }_L$ (m ⊙ n))² ≤ 1 - δ² = max{( ${\tau }_L$ (m))², ( ${\tau }_L$ (n))²}

Case (ii): Let n ∈ M, m ∉ M (or m ∈ M, n ∉ M) ⇒ m ⊛ n ∉ M or m ⊛ n ∈ M

Then ( ${\eta }_L$ (m))² = δ²; ( ${\eta }_L$ (n))² = ε² ⇒ ( ${\eta }_L$ (m ⊛ n))² = δ² or ( ${\eta }_L$ (m ⊛ n))² = ε² (for m ⊛ n ∉ M)

⇒ ( ${\eta }_L$ (m ⊛ n))² ≥ ε² = min{( ${\eta }_L$ (m))², ( ${\eta }_L$ (n))²} and similarly

( ${\eta }_L$ (m ⊙ n))² ≥ ε²= min{( ${\eta }_L$ (m))², ( ${\eta }_L$ (n))²} and in a similar way, we can show

( ${\tau }_L$ (m ⊛ n))² ≤ 1 - ε² = max{( ${\tau }_L$ (m))², ( ${\tau }_L$ (n))²} and similarly

( ${\tau }_L$ (m ⊙ n))² ≤ 1 - ε² = max{( ${\tau }_L$ (m))², ( ${\tau }_L$ (n))²}

Similar deductions could be obtained when m ⊛ n ∈ M.

Therefore, if M is LBS. of L then $({\eta }_L,{\overline{\overline{\eta }}}_L)$ is a Pythagorean fuzzy LBS. of L.

Conversely, suppose $({\eta }_L,{\overline{\overline{\eta }}}_L)$ is a Pythagorean fuzzy LBS. of L and let m, n ∈ M

⇒ ( ${\eta }_L$ (m))² = ( ${\eta }_L$ (n))² = δ² ⇒ ( ${\eta }_L$ (m ⊛ n))² ≥ min{( ${\eta }_L$ (m))², ( ${\eta }_L$ (n))²}= δ² and similarly

( ${\eta }_L$ (m ⊙ n))² ≥ min{( ${\eta }_L$ (m))², ( ${\eta }_L$ (n))²} = δ² and in a similar way, we can show

( ${\tau }_L$ (m ⊛ n))² ≤ 1 - δ² = max{( ${\tau }_L$ (m))², ( ${\tau }_L$ (n))²} and similarly

( ${\tau }_L$ (m ⊙ n))² ≤ 1 - δ² = max{( ${\tau }_L$ (m))², ( ${\tau }_L$ (n))²}

Therefore, M is LBS. of L. □

Definition 3.4.1.

Let η_L^d (m) = η_L (m) − (η_L(m))² and τ_L^d (m) = τ_L (m) − (τ_L(m))², for every m∈L, be membership deviations and non-membership deviations, respectively.

Then we call L^d = (η_L^d, τ_L^d) Pythagorean fuzzy LBS. deviation of L.

Remark 3.4.2.

For η_L in a LBA. L and m in L:

$\begin{array}{l}{\eta }_L\left(m\right)\ge {\left({\eta }_L\left(m\right)\right)}^2\text{and}{\tau }_L\left(m\right)\ge {\left({\tau }_L\left(m\right)\right)}^2\\ \Rightarrow {\eta }_L\left(m\right)-{\left({\eta }_L\left(m\right)\right)}^2\ge 0\text{and}{\tau }_L{({\eta }_L\left(m\right)-\left({\tau }_L\left(m\right)\right))}^2\ge 0\\ \Rightarrow {\eta }_L\left(m\right)(1-{\eta }_L\left(m\right))\ge 0\text{and}{\tau }_L\left(m\right)(1-{\tau }_L\left(m\right))\ge 0\end{array}$

Proposition 3.4.3.

Let ∅ ≠ U ⊆ L such that χ_U is characteristic function and ${\overline{\chi }}_U$ (m) = 1 - χ _U(m) is the complement of χ _U(m). Then L ^B = (χ _U, ${\overline{\chi }}_U$ ) is Pythagorean fuzzy LBS. of L iff U is LBS. of L. Furthermore, the accuracy function a _U, the score function s _U and the degree of indeterminacy π _U are respectively hereunder, for every m∈L:

Proof.

Let χ_U: U → [0, 1] be a characteristic function.

Then the Characteristic function, its complement and its square deviation are give:

We have prove above that L ^B = (χ _U, ${\overline{\chi }}_U$ ) or L ^B = (χ _U, ${\overline{\overline{\chi }}}_U$ ) is a Pythagorean fuzzy LBS. of L.

Now we need to prove that U is LBS. of L:

Hence, U is a LBS. of L, by (i) and (ii) above.

Theorem 3.4.4.

Let η_L and ${\overline{\overline{\eta }}}_L$ (its square deviation) be a fuzzy sets in LBA. L such that (η_L (m ⊛ n))² = (η_L (n))² and (η_L (m ⊙ n))² = (η_L (n))².

Then( ${\overline{\overline{\eta }}}_L$ (m ⊛ n))² = ( ${\overline{\overline{\eta }}}_L$ (n))² and ( ${\overline{\overline{\eta }}}_L$ (m ⊙ n)² = ( ${\overline{\overline{\eta }}}_L$ (n))², ∀m, n∈ L.

Again, L^B = (η _L, $\overline{\overline{\eta }}$ ) is Pythagorean fuzzy LBS. of L iff η _{L and} $\overline{\overline{\eta }}$ are constants.

Furthermore, the accuracy function a_L, the score function s_L and the degree of indeterminacy π_L are respectively given as follows; ∀m ∈ L:

Proof.

Suppose (η_L (m ⊛ n))² = (η_L (n))² and ( ${\overline{\overline{\eta }}}_L$ (m ⊙ n))² = ( ${\overline{\overline{\eta }}}_L$ (n))²

Then ( ${\overline{\overline{\eta }}}_L$ (m ⊛ n))² = 1 − (η_L (m ⊛ n))² = 1 − (η_L (n))² = ( ${\overline{\overline{\eta }}}_L$ (n))² and ( ${\overline{\overline{\eta }}}_L$ (m ⊙ n))² = 1 − (η_L (m ⊙ n))² = 1 − (η_L (n))² = ( ${\overline{\overline{\eta }}}_L$ (n))²

Let L = (η_L, ${\overline{\overline{\eta }}}_L$ ₎ be Pythagorean fuzzy LBS. of L, (n)²

Now we need to verify that η_L and $\overline{\overline{\eta }}$ are constants, (or ∀m, n ∈ L, η_L (m) = η_L (n) and then ${\overline{\overline{\eta }}}_L$ (m) = ${\overline{\overline{\eta }}}_L\left(n\right)$

Since L = (η_L ${\overline{\overline{\eta }}}_L$ ₎ is a Pythagorean fuzzy LBS. of L, η_L is a fuzzy LBS. of L, and hence by some of the axioms, we have (η_L (0))² = (η_L (0⊛ m))² = (η_L(0⊙ m))², ∀m ∈ L, ⇒ (η_L (0))² = (η_L (0⊛ m))² = (η_L (m))² = (η_L (0⊙ m)) ², ∀m ∈ L, and again (η_L (0)) ² = (η_L (0⊛ n))² = (η_L (0⊙ n))² = (η_L (n))², ∀n ∈ L ⇒ (η_L (0))² = (η_L (m))² = (η_L (n))², ∀m, n ∈ L, or (η_L (m))² = (η_L (n)) ², ∀m, n ∈ L and hence, η _L is a constant and analogously, ${\overline{\overline{\eta }}}_L$ is too.

Conversely, suppose η_L and η_L are constants, or:

Then

L = (η_L ${\overline{\overline{\eta }}}_L$ ) is a Pythagorean fuzzy LBS. of L

Therefore, by (i) and (ii) above, η_L is a fuzzy LBS. of L so that L = (η_L ${\overline{\overline{\eta }}}_L$ ₎ is a Pythagorean fuzzy LBS. of L. Furthermore:

Theorem 3.4.5.

$\mathrm{Let}{L}^B=({\eta }_L,{\tau }_L)$ be Pythagorean fuzzy LBS. of L and m, n ∈ L.

If (η_L (m ⊛ n))² = (η_L (n))², (τ_L (m ⊛ n))² = (τ_L (n))² and (η_L (m ⊙ n))² = (η_L (n))², (τ_L (m ⊙ n))² = (τ_L (n))²

Then ∀m, n∈ L, the following hold:

Proof.

Let L = (η_L, τ_L) be a Pythagorean fuzzy LBS. of L;

Since, $\forall m\in L,0=0⊛m=0\odot m$

⇒ (η_L (0))² = (η_L (m))² = (η_L (n))² = (η_L (w))², ∀m, n, w∈ L and similarly,

But 0 ≤ (η_L(m))² + (τ_L(m))² ≤ 1 0 ≤ (η_L(m))² + (τ_L (m))² ≤ 1

(η_L (m))² ≤ 1 – (τ_L(0))², and similarly, (τ_L (m))² ≤ 1 – (η_L(0))²,∀m, n, w∈ L.

Then,∀m∈ L, the following hold:

Lemma 3.4.6.

In a Pythagorean fuzzy LBS. deviation of L, L^d = ( ${\eta }_L^d$ , ${\tau }_L^d$ ) we have; ${\eta }_L^d$ : L → [0, 0.25] and ${\tau }_L^d$ : L → [0, 0.25].

Proof.

We prove this following five cases:

Case (i): For η_L

Case (ii): For η_L

Case (iii) For η_L (m) = 0.5 ⇒ ${\eta }_L^d$ (m) = η_L(m) – (η_L(m))² = 0.5 – 0.25 = 0.25

Case (iv): For η_L (m) = 0.5 + ε, where ε ∈ [0, 0.5]