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. (a)

0 ∈ I

^{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: μ ( m ∗ n ) ≥ min { μ ( m ) , μ ( n ) } ,

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 ≤ μ I ( m ) + ν I ( m ) ≤ 1 is fulfilled.

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 ≤ ( η P ( m ) ) 2 + ( τ P ( m ) ) 2 ≤ 1 is fulfilled.

¹⁸ Suppose ( η P ( m ) ) 2 + ( ν P ( m ) ) 2 ≤ 1 , then there is a degree of indeterminacy of m ∈ X to P given as π P ( m ) = 1 − ( ( η P ( m ) ) 2 + ( ν P ( m ) ) 2 ) ⇒ ( π P ( m ) ) 2 = 1 − ( ( η P ( m ) ) 2 + ( ν P ( m ) ) 2 ) ; π P ( m ) ∈ [ 0 , 1 ]

¹⁸ Let P be Pythagorean fuzzy set on X. Then, the score function, s of P is given by: s P ( m ) = ( η P ( m ) ) 2 − ( ν P ( m ) ) 2 , where s P ( m ) ∈ [-1, 1].

¹⁸ For Pythagorean fuzzy set P on X, the accuracy function, a, of P is given by: a P ( m ) = ( η P ( m ) ) 2 + ( ν P ( m ) ) 2 for a(P) ∈ [0, 1].

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: l P ( m ⊛ n ) ≥ min { l P ( m ) , l P ( n ) } and m P ( m ⊛ n ) ≤ max { m P ( m ) , m P ( n ) } .

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.

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.

⁵ Suppose m ≤ n iff m ⊛ n = 0, the definition 2.3.1 for BCL-algebra could be given by: (1)

m ≤ m

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

m ⊛ m = 1;

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: (1)

(L; ⊛, 1) is BCL ⁺-algebra

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.

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.

Suppose B is a BCL-Algebra and m ∈ B (1)

Here, the claim is to prove 0 ⊛ m = 0.

But before proving this part of the proposition, there is a need to prove: 0 ⊛ (0 ⊛ m) = 0, ∀ m ∈ B: [ ( ( m ⊛ m ) ⊛ m ) ⊛ ( ( m ⊛ m ) ⊛ m ) ] ⊛ ( ( m ⊛ m ) ⊛ m ) = 0 ⇒ 0 ⊛ ( 0 ⊛ m ) = 0 .

Hence 0 ⊛ (0 ⊛ m) = 0, ∀ m ∈ B

Assume, ∀ m ∈ B; 0 ⊛ m ≠ 0 and so, let 0 ⊛ m = n, where n ≠ 0.

But 0 ⊛ (0 ⊛ m) = 0, ∀ m ∈ B

⇒ 0 ⊛ n = 0, ∀ m, n ∈ B and n ≠ 0 arriving at a contradiction to what we assumed.

Thus 0 ⊛ m = 0, ∀m ∈ B

Again, by the definition of binary relation on B:

0 * m = 0 ⇔ 0 ≤ m, ∀ m ∈ B so that it tells us that 0 is the least element of B.

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

0 ⊙ 0 = 0

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

0 = (0 ⊙ m) ⊛ (0 ⊙ m) = 0 ⊙ ( m ⊛ m) = 0 ⊙ 0, ∀ m ∈ L

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

m ≤ n ⇒ w ⊙ m ≤ w ⊙ n

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

m ≤ n ⇒ m ⊛ n = 0 and ( w ⊙ m) ⊛ ( w ⊙ n) = w ⊙ ( m ⊛ n) = w ⊙ 0 = 0 ⇒ ( w ⊙ m) ⊛ ( w ⊙ n) = 0 ⇒ w ⊙ m ≤ w ⊙ n = 0

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.

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: (1)

L, {0}, {0, m}, {0, n}, {0, w} are LBSs. of L.

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

Let m ∈ N ⇒ m ∈ L ⇒ m ∗ m = 0 ∈ N □

A fuzzy set η L in LBA. L is known as a fuzzy LBS. of L if the statements

hereunder are fulfilled for every m, n ∈ L: (i)

η _L ( m ⊛ n) ≥ min{ η _L ( m), η _L ( n)}

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 η L in L be defined as follows: η L ( k ) = { 0.9 , iff k = 0 0.7 , iff k = m 0.4 , iff k = n , w .

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 = ( η L , τ L ) such that

η _L : L → [0, 1] and τ _L : L → [0, 1] given by: η L ( k ) = { 0.9 , iff k = 0 0.4 , iff k = m , n , w . and τ L ( k ) = { 0.3 , iff k = 0 0.7 , iff k = m , n , w

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

If L B = ( η L , τ 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.

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 □

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

Let M be a non-void subset of L and L ^B = ( χ M , χ ¯ ¯ M ) , where χ M is characteristic function and the square deviation, ( χ ¯ ¯ M (k)) ² = 1 − ( χ M (k)) ², then L ^B is Pythagorean fuzzy LBS. of L iff M is LBS. of L.

Suppose χ M : M → [0, 1] is characteristic function defined as: χ M ( k ) = { 1 , iff k ∈ M , 0 , iff k ∉ M and χ ¯ M ( k ) = ( χ ¯ ¯ M ( k ) ) 2 = { 0 , iff k ∈ M , 1 , iff k ∉ M

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 ( χ ¯ ¯ M ( k ⊛ q)) ² ≤ 1) ⇒ ( χ M ( k ⊛ q ) ) 2 = 1 ⇒ k ⊛ q ∈ M

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 ( χ ¯ ¯ M ( k ⊛ q)) ² = 0 ≤ max {( χ ¯ ¯ M ( k)) ², ( χ ¯ ¯ M ( q)) ²}

Similarly, ( χ _L ( k ⊙ q)) ² = 1 ≥ min {( χ _L ( k)) ², ( χ _L ( q)) ²} and χ ¯ ¯ M ( k ⊙ q)) ² = 0 ≤ max {( χ ¯ ¯ M ( k)) ², ( χ ¯ ¯ M ( q)) ²}

Thus, L ^B = ( χ M , χ ¯ ¯ M ) is Pythagorean fuzzy LBS. of L, similarly true for k in M, q not in M or k, q not in M. □

The intersection of any two Pythagorean fuzzy LBS ., L ₁ ^B and L ₂ ^B of L is also a Pythagorean fuzzy LBS. of L.

Let L ₁ ^B = ( η L 1 B , τ L 1 B ) & L ₂ ^B = ( η L 2 B , τ L 2 B ) be any two Pythagorean fuzzy LBS. of L.

Then we need to prove: L T B = ( η L 1 B ∩ η L 2 B , τ L 1 B ∩ τ L 2 B ) is Pythagorean fuzzy LBS. of L.

For m , n ∈ L , and for the two binary operations ∗ and • of L , we have : (i)

( ( η L 1 B ∩ η L 2 B ) ( m ∗ n ) ) 2 = min { ( ( η L 1 B ) ( m ∗ n ) ) 2 , ( ( η L 2 B ) ( m ∗ n ) ) 2 } ≥ min { min { ( ( η L 1 B ) ( m ) ) 2 , ( ( η L 1 B ) ( n ) ) 2 } , min { ( ( η L 2 B ) ( m ) ) 2 , ( ( η L 2 B ) ( n ) ) 2 } } = min { min { ( ( η L 1 B ) ( m ) ) 2 , ( ( η L 2 B ) ( m ) ) 2 } , min { ( ( η L 1 B ) ( n ) ) 2 , ( ( η L 2 B ) ( n ) ) 2 } } = min { ( ( η L 1 B ∩ η L 2 B ) ( m ) ) 2 , ( ( η L 1 B ∩ η L 2 B ) ( n ) ) 2 }

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

The intersection, L i B = ( ∩ i ∈ I ( η L i B ) , ∩ i ∈ I ( τ L i B ) ) of a family of Pythagorean LBS in L is also Pythagorean fuzzy LBS of L, where ∩ i ∈ I ( η L i B ) ( m ) = inf i ∈ I ( η L i B ) ( m ) and ∩ i ∈ I ( τ L i B ) ( m ) = sup i ∈ I ( η L i B ) ( m )

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

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., L ₁ ^B = ( η L 1 B , τ L 1 B ) and L ₂ ^B = ( η L 2 B , τ L 2 B ) as follows: η L 1 B ( k ) = { 0.9 , iff k = 0 , 0.6 , iff k = m , 0.4 , iff k = n , w , and τ L 1 B ( k ) = { 0 , iff k = 0 , 0.5 , iff k = n , 0.3 , iff k = m , w , and η L 2 B ( k ) = { 0.8 , iff k = 0 , 0.7 , iff k = m , n , 0.5 , iff k = w , and τ L 2 B ( k ) = { 0.2 , iff k = 0 , 0.4 , iff k = m , 0.6 , iff k = n , w . Then ( η L 1 B ∪ η L 2 B ) ( k ) = { 0.9 , iff k = 0 , 0.7 , iff k = m , n , 0.5 , iff k = w , and ( τ L 1 B ∪ τ L 2 B ) ( k ) = { 0 , iff k = 0 , 0.3 , iff k = m , w 0.5 , iff k = n . Then Hence as m⊛ n = w; we have: ( ( η L 1 B ∪ η L 2 B ) ( m ⊛ n ) ) 2 = ( ( η L 1 B ∪ η L 2 B ) ( w ) ) 2 = ( 0.5 ) 2 = 0.25 ≥ min { ( ( η L 1 B ∪ η L 2 B ) ( m ) ) 2 , ( ( η L 1 B ∪ η L 2 B ) ( n ) ) 2 } = min { ( 0.7 ) 2 , ( 0.7 ) 2 } = 0.49 which is false describing that the union of two Pythagorean fuzzy LBS. of a LBA. is not necessarily Pythagorean fuzzy LBS. of LBA.

Let M be a non-empty subset of LBA. L and η L be a fuzzy set in a LBA. L such that η L ( m ) = { δ , iff m ∈ M , ε , iff m ∉ M where δ , ε ∈ [ 0 , 1 ]   and δ > ε and then ( η ¯ ¯ L ( m ) ) 2 = { 1 − δ 2 , iff m ∈ M , 1 − ε 2 , iff m ∉ M

Then M is LBS. of L iff ( η L , η ¯ ¯ L ) is a Pythagorean fuzzy LBS. of L and let η ¯ ¯ L ( m ) = τ L ( m) .

For M is LBS. of L, we prove that ( η L , τ L ) is a Pythagorean fuzzy LBS. of L.

Case (i): Let m, n ∈ M ⇒ m ⊛ n ∈ M

Then ( η L ( m)) ² = ( η L ( n)) ² = δ ² ⇒ ( η L ( m ⊛ n)) ² = δ ²

⇒ ( η L ( m ⊛ n)) ² ≥ δ ² = min{( η L ( m)) ², ( η L ( n)) ²} and similarly

( η L ( m ⊙ n)) ² ≥ δ ² = min{( η L ( m)) ², ( η L ( n)) ²} and in a similar way, we can show

( τ L ( m ⊛ n)) ² ≤ 1 - δ ² = max{( τ L ( m)) ², ( τ L ( n)) ²} and similarly

( τ L ( m ⊙ n)) ² ≤ 1 - δ ² = max{( τ L ( m)) ², ( τ L ( n)) ²}

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

Then ( η L ( m)) ² = δ ²; ( η L ( n)) ² = ε ² ⇒ ( η L ( m ⊛ n)) ² = δ ² or ( η L ( m ⊛ n)) ² = ε ² (for m ⊛ n ∉ M)

⇒ ( η L ( m ⊛ n)) ² ≥ ε ² = min{( η L ( m)) ², ( η L ( n)) ²} and similarly

( η L ( m ⊙ n)) ² ≥ ε ²= min{( η L ( m)) ², ( η L ( n)) ²} and in a similar way, we can show

( τ L ( m ⊛ n)) ² ≤ 1 - ε ² = max{( τ L ( m)) ², ( τ L ( n)) ²} and similarly

( τ L ( m ⊙ n)) ² ≤ 1 - ε ² = max{( τ L ( m)) ², ( τ L ( n)) ²}

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

Therefore, if M is LBS. of L then ( η L , η ¯ ¯ L ) is a Pythagorean fuzzy LBS. of L.

Conversely, suppose ( η L , η ¯ ¯ L ) is a Pythagorean fuzzy LBS. of L and let m, n ∈ M

⇒ ( η L ( m)) ² = ( η L ( n)) ² = δ ² ⇒ ( η L ( m ⊛ n)) ² ≥ min{( η L ( m)) ², ( η L ( n)) ²}= δ ² and similarly

( η L ( m ⊙ n)) ² ≥ min{( η L ( m)) ², ( η L ( n)) ²} = δ ² and in a similar way, we can show

( τ L ( m ⊛ n)) ² ≤ 1 - δ ² = max{( τ L ( m)) ², ( τ L ( n)) ²} and similarly

( τ L ( m ⊙ n)) ² ≤ 1 - δ ² = max{( τ L ( m)) ², ( τ L ( n)) ²}

Therefore, M is LBS. of L. □

For η _L in a LBA. L and m in L:

η L ( m ) ≥ ( η L ( m ) ) 2 and τ L ( m ) ≥ ( τ L ( m ) ) 2 ⇒ η L ( m ) − ( η L ( m ) ) 2 ≥ 0 and τ L ( η L ( m ) − ( τ L ( m ) ) ) 2 ≥ 0 ⇒ η L ( m ) ( 1 − η L ( m ) ) ≥ 0 and τ L ( m ) ( 1 − τ L ( m ) ) ≥ 0

Let ∅ ≠ U ⊆ L such that χ _U is characteristic function and χ ¯ U ( m) = 1 - χ _U ( m) is the complement of χ _U ( m). Then L ^B = ( χ _U , χ ¯ 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: (a)

a _U ( m) = 1,

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

Then the Characteristic function, its complement and its square deviation are give: χ U ( m ) = { 1 , iff m ∈ U , 0 , iff m ∉ U , χ ¯ U ( m ) = ( χ ¯ ¯ U ( m ) ) 2 = { 0 , iff m ∈ U , 1 , iff m ∉ U .

We have prove above that L ^B = ( χ _U , χ ¯ U ) or L ^B = ( χ _U , χ ¯ ¯ U ) is a Pythagorean fuzzy LBS. of L.

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

Let n, m ∈ U ⇒ χ _U ( n) = χ _U ( m) = 1 and χ ¯ U ( m) = χ ¯ U ( n) = 0 = χ ¯ ¯ U ( m) = χ ¯ ¯ U ( n)

⇒ = χ ¯ ¯ _U ( m ⊛ n) ≥ min{ χ ¯ ¯ _U ( m), χ _U ( n)} = min{1, 1} = 1 and χ ¯ ¯ _U ( m ⊛ n) ≤ max{ χ ( n), χ ( m)} = min{0, 0} = 0

But χ _U ( m ⊛ n) ≤ 1 and χ ¯ ¯ ( m) ≥ 0, ∀ m ∈ L, by the definitions.

Hence χ _U ( m ⊛ n) = 1 and χ ¯ ¯ ( m ⊛ n) = 0 ⇒ m ⊛ n ∈ U

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

The accuracy function:

Let η _L and η ¯ ¯ 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( η ¯ ¯ L ( m ⊛ n)) ² = ( η ¯ ¯ L ( n)) ² and ( η ¯ ¯ L ( m ⊙ n) ² = ( η ¯ ¯ L ( n)) ², ∀ m, n∈ L.

Again, L ^B = ( η _L, η ¯ ¯ ) is Pythagorean fuzzy LBS. of L iff η _{L and} η ¯ ¯ 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: (a)

a _L ( m) = 1

Suppose ( η _L ( m ⊛ n)) ² = ( η _L ( n)) ² and ( η ¯ ¯ L ( m ⊙ n)) ² = ( η ¯ ¯ L ( n)) ²

Then ( η ¯ ¯ L ( m ⊛ n)) ² = 1 − ( η _L ( m ⊛ n)) ² = 1 − ( η _L ( n)) ² = ( η ¯ ¯ L ( n)) ² and ( η ¯ ¯ L ( m ⊙ n)) ² = 1 − ( η _L ( m ⊙ n)) ² = 1 − ( η _L ( n)) ² = ( η ¯ ¯ L ( n)) ²

Let L = ( η _L , η ¯ ¯ L ₎ be Pythagorean fuzzy LBS. of L, ( n) ²

Now we need to verify that η _L and η ¯ ¯ are constants, (or ∀ m, n ∈ L, η _L ( m) = η _L ( n) and then η ¯ ¯ L ( m) = η ¯ ¯ L ( n )

Since L = ( η _L η ¯ ¯ 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, η ¯ ¯ L is too.

Conversely, suppose η _L and η _L are constants, or: ( η L ( m ) ) 2 = ( η L ( n ) ) 2 and η L ( m ) 2 = η L ( n ) 2 , ∀ m , n ∈ L ,

Then ( η L ( m ⊛ n ) ) 2 = ( η L ( n ) ) 2 and ( η ¯ ¯ L ( m ⊛ n ) ) 2 = ( η ¯ ¯ L ( n ) ) 2 , ∀ m , n ∈ L , ( η L ( m ⊙ n ) ) 2 = ( η L ( n ) ) 2 and ( η ¯ ¯ L ( m ⊙ n ) ) 2 = ( η ¯ ¯ L ( n ) ) 2 , ∀ m , n ∈ L ,

L = ( η _L η ¯ ¯ L ) is a Pythagorean fuzzy LBS. of L (i)

( η _L ( m)) ² = ( η _L ( m ⊛ n)) ² = ( η _L ( n)) ² ⇒ ( η _L ( m ⊛ n)) ² ≥ min{( η _L ( m)) ², ( η _L ( n)) ²} and ( η ¯ ¯ L ( m ⊛ n)) ² ≤ max{( η ¯ ¯ L ( m)) ², ( η ¯ ¯ L ( n)) ²}

Therefore, by (i) and (ii) above, η _L is a fuzzy LBS. of L so that L = ( η _L η ¯ ¯ L ₎ is a Pythagorean fuzzy LBS. of L. Furthermore: (a)

a _L ( m) = ( η _L ( m)) ² + ( η ¯ ¯ L ( m)) ² = ( η _L ( m)) ² + (1 − ( η _L ( m)) ²) = 1

Let L B = ( η L , τ 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: (1)

The accuracy function: a _L ( m) ≤ 2 – ( η (0)) ² – ( τ (0)) ²

Let L = ( η _L , τ _L ) be a Pythagorean fuzzy LBS. of L; ( η L ( m ⊛ n ) ) 2 = ( η L ( n ) ) 2 , and ( τ L ( m ⊛ n ) ) 2 = ( τ L ( n ) ) 2 , ( η L ( m ⊙ n ) ) 2 = ( η L ( n ) ) 2 and ( τ L ( m ⊙ n ) ) 2 = ( τ L ( n ) 2 )

Since, ∀ m ∈ L , 0 = 0 ⊛ m = 0 ⊙ m ⇒ ( η L ( 0 ) ) 2 = ( η L ( 0 ⊛ m ) ) 2 = ( η L ( 0 ⊙ m ) ) 2 = ( η L ( m ) ) 2 ⇒ ( η L ( 0 ) ) 2 = ( η L ( m ) ) 2 , ∀ m ∈ L

⇒ ( η _L (0)) ² = ( η _L ( m)) ² = ( η _L ( n)) ² = ( η _L ( w)) ², ∀ m, n, w∈ L and similarly, ( τ L ( 0 ) ) 2 = ( τ L ( m ) ) 2 = ( τ L ( n ) ) 2 = ( τ L ( w ) ) 2 , ∀ m , n , w ∈ L .

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: (1)

The accuracy function:

In a Pythagorean fuzzy LBS. deviation of L, L ^d = ( η L d , τ L d ) we have; η L d : L → [0, 0.25] and τ L d : L → [0, 0.25].

We prove this following five cases:

Case (i): For η _L ( m ) = 0 ⇒ η L d ( m ) = η L ( m ) − ( η L ( m ) ) 2 = 0 − 0 = 0 ∈ [ 0 , 0.25 ] . ⇒ 0 = η L d ( m ) ≤ 0.25 ⇒ η L d ( m ) ∈ [ 0 , 0.25 ]

Case (ii): For η _L ( m ) = 1 ⇒ η L d ( m ) = η L ( m ) − ( η L ( m ) ) 2 = 1 − 1 = 0 ∈ [ 0 , 0.25 ] . ⇒ 0 = η L d ( m ) ≤ 0.25 ⇒ η L d ( m ) ∈ [ 0 , 0.25 ]

Case (iii) For η _L ( m) = 0.5 ⇒ η L d ( m) = η _L ( m) – ( η _L ( m)) ² = 0.5 – 0.25 = 0.25 ⇒ η L d ( m ) = 0.25 ⇒ η L d ( m ) ∈ [ 0 , 0.25 ]

Case (iv): For η _L ( m) = 0.5 + ε, where ε ∈ [0, 0.5] ⇒ ( η ( m ) ) 2 = ( 0.5 + ε ) 2 = 0.25 + ε + ε 2 η L d ( m ) = η L ( m ) – ( η L ( m ) ) 2 = 0.5 + ε – ( 0.25 + ε + ε 2 ) = 0.25 – ε 2 ≤ 0.25 ⇒ η L d ( m ) ∈ [ 0 , 0.25 ]

Case (v): For η _L ( m) = 0.5 – ε, where ε ∈ [0, 0.5] η L d ( m ) = η L ( m ) – ( η L ( m ) ) 2 = 0.5 – ε – ( 0.25 – ε + ε 2 ) = 0.25 – ε 2 ≤ 0.25 ⇒ η L d ( m ) ∈ [ 0 , 0.25 ]

Now, suppose η L d ( m)> 0.25 ⇒ η L d ( m)= η L ( m) − ( η L ( m)) ² >0.25

⇒ − ( η _L ( m)) ² + η L ( m) – 0.25 > 0 and this quadratic inequality has no real solution

so that there is no η _L ( m) satisfying this inequality under the real number system and hence η L d ∈ [0, 0.25]

Therefore, by the above five cases, we have exhaustively shown that η ^d ∈ [0, 0.25].

Following similar steps for τ L d as η L d above, we can easily show that τ ^d ∈ [0, 0.25]. □

If L ^B = ( η _L , τ _L ) is a Pythagorean fuzzy LBS.of L so that the ordered pairs below are each also Pythagorean fuzzy Liu ^B -subalgebras (1)

( η _L, η ¯ ¯ L )

1.

( η _L, τ ¯ ¯ L ) are Pair of LBSs. of L.

Recall that ( η _L , η ¯ ¯ L ) is Pythagorean fuzzy LBS. of L as previously proved above and then ( τ _L , τ ¯ ¯ L ,) is also Pythagorean fuzzy LBS. of L

By the hypothesis, LB = ( η ¯ ¯ L , τ ¯ ¯ L ) is Pythagorean fuzzy LBS. of L