Study hoop algebras by fuzzy (n-fold) obstinate filters

Definition 2.1

[3] A hoop algebra or hoop is an algebra ( A , ⊙ , → , 1 ) of type ( 2 , 2 , 0 ) such that, for all x , y , z ∈ A :

1. ( A , ⊙ , 1 ) is a commutative monoid,

2. x → x = 1 ,

3. ( x ⊙ y ) → z = x → ( y → z ) ,

4. x ⊙ ( x → y ) = y ⊙ ( y → x ) .

Definition 2.2

[4] Let A be a bounded hoop, and for any x , y ∈ A , we define x ∨ y = ( ( x → y ) → y ) ∧ ( ( y → x ) → x ) . If ∨ is the join operation on A , then A is called a ∨ -hoop.

Proposition 2.3

[8,9] Let A be a bounded hoop. Then the following properties hold, for all x , y , z ∈ A :

1. ( A , ≤ ) is a meet-semilattice with x ∧ y = x ⊙ ( x → y ) ,

2. x ≤ ( x → y ) → y ,

3. 1 → x = x , x ⊙ y ≤ x , y ,

4. if x ≤ y , then z → x ≤ z → y and y → z ≤ x → z ,

5. x ∨ y → z = ( x → z ) ∧ ( y → z ) , x ≤ y → x .

Definition 2.4

[5,11] Let F be a nonempty subset of A such that 1 ∈ F . Then for any x , y , z ∈ A :

1. F is called a filter, if x , x → y ∈ F , then y ∈ F .

2. F is called an implicative filter of A , if x → ( ( y → z ) → y ) ∈ F and x ∈ F , then y ∈ F .

3. F is called a fantastic filter of A , if z → ( y → x ) ∈ F and z ∈ F , then ( ( x → y ) → y ) → x ∈ F .

4. F is called an obstinate filter of A , if F is a proper filter and for any x , y ∉ F , x → y ∈ F and y → x ∈ F .

5. A proper filter F of a ∨ -hoop A is called a prime filter of A , if x ∨ y ∈ F implies x ∈ F or y ∈ F .

6. F is called an n-fold implicative filter of A , if x → ( ( y n → z ) → y ) ∈ F and x ∈ F , then y ∈ F .

7. F is called an n-fold obstinate filter of A , if F is a proper filter and for any x , y ∉ F , x n → y ∈ F and y n → x ∈ F .

Proposition 2.5

[11] Every obstinate filter of A is a (prime, implicative, positive implicative, fantastic) filter of A.

Definition 2.6

[7] Let A and B be two bounded hoops. A map f : A → B is called a hoop homomorphism if and only if for all x , y ∈ A , f ( 0 ) = 0 , f ( 1 ) = 1 , f ( x ⊙ y ) = f ( x ) ⊙ f ( y ) , and f ( x → y ) = f ( x ) → f ( y ) .

Definition 2.7

[4] Let μ be a fuzzy subset of A and μ ( x ) ≤ μ ( 1 ) , for all x ∈ A . Then μ is called:

1. fuzzy filter if μ ( x → y ) ∧ μ ( x ) ≤ μ ( y ) .

2. fuzzy implicative filter if μ ( x → ( ( y → z ) → y ) ) ∧ μ ( x ) ≤ μ ( y ) .

3. fuzzy positive implicative filter if μ ( x → ( y → z ) ) ∧ μ ( x → y ) ≤ μ ( x → z ) .

4. fuzzy fantastic filter if μ ( z → ( x → y ) ) ∧ μ ( z ) ≤ μ ( ( ( y → x ) → x ) → y ) .

Theorem 2.8

[4,10]

1. μ is a fuzzy implicative filter if μ ( x ′ → x ) ≤ μ ( x ) .

2. Let α ∈ [ 0 , 1 ] . Then μ be a fuzzy (positive implicative, fantastic, implicative) filter if and only if μ α ≠ ∅ is a (positive implicative, fantastic, implicative) filter.

3. A is an n-fold implicative bounded hoop if and only if { 1 } is an n-fold implicative filter.

4. Let μ be a fuzzy filter of A and x ≤ y . Then μ ( x ) ≤ μ ( y ) .

Theorem 2.9

[2] Let μ be a fuzzy filters of A, and we define a relation ∼ μ μ ( 1 ) on A as follows:

Then ∼ μ μ ( 1 ) is a congruence relation on A. Also A ∕ μ = ( A ∕ μ , ∧ , ⊙ , → , μ 1 ) is a hoop algebra.

Definition 2.10

[4] Let μ be a fuzzy filter of A and A ∕ μ = { μ x ∣ x ∈ A } . For all μ x , μ y ∈ A ∕ μ , define μ x → μ y = μ x → y , μ x ⊙ μ y = μ x ⊙ y , and μ x : A → [ 0 , 1 ] , which is defined by μ x ( y ) = μ ( x → y ) ∧ μ ( y → x ) .

Definition 2.11

[7,10]

1. A bounded hoop A is called an n-fold implicative bounded hoop if ( x n → 0 ) → x = x for any x ∈ A .

2. A bounded hoop A is called an n-fold obstinate hoop if x n = 0 for any x ≠ 1 .

Definition 3.1

A fuzzy filter μ of A is called a fuzzy obstinate filter if for any x , y ∈ A ,

Example 3.2

(i) Let ( A = { 0 , a , b , 1 } , ≤ ) be a chain. Define the operations ⊙ and → on A as follows:

Then ( A , ⊙ , → , 1 , 0 ) is a bounded hoop algebra.

(1) Let α , β ∈ [ 0 , 1 ] that β < 1 ∕ 2 < α and μ be a fuzzy subset on A such that μ ( b ) = α = μ ( 1 ) , μ ( 0 ) = μ ( a ) = β . It is clear that μ is a fuzzy filter but not a fuzzy obstinate filter. Because, ( 1 − μ ( a ) ) ∧ ( 1 − μ ( 0 ) ) > μ ( a → 0 ) ∧ μ ( 0 → x ) .

(2) Let λ be a fuzzy filter such that λ ( 0 ) = 2 ∕ 3 , λ ( a ) = λ ( b ) = 4 ∕ 5 , λ ( 1 ) = 1 . It is clear that λ is a fuzzy obstinate filter of A .

(ii) Let ( A = { 0 , a , b , c , 1 } , ≤ ) be a poset with 0 < c < a , b < 1 , but a and b are incomparable. Define the operations ⊙ and → on A as follows:

Then ( A , ⊙ , → , 1 , 0 ) is a bounded hoop algebra. Let μ be a fuzzy filter such that μ ( 0 ) = 1 ∕ 2 , μ ( c ) = 4 ∕ 7 = μ ( a ) , μ ( b ) = 5 ∕ 7 , and μ ( 1 ) = 6 ∕ 7 . It is clear that μ is a fuzzy obstinate filter.

Proposition 3.3

Any nonempty subset F of A is an obstinate filter if and only if the characteristic function χ F is a fuzzy obstinate filter.

Proof

Assume that F is an obstinate filter of A . We show that χ F is a fuzzy obstinate filter.

Let x , y ∈ A . We show that

Case one: If x ∈ F or y ∈ F , then ( 1 − χ F ( x ) ) ∧ ( 1 − χ F ( y ) ) = 0 and

Case two: If x ∉ F and y ∉ F , then ( 1 − χ F ( x ) ) ∧ ( 1 − χ F ( y ) ) = 1 . Since F is an obstinate filter and obtain x → y ∈ F and y → x ∈ F , then χ F ( x → y ) ∧ χ F ( y → x ) = 1 . Hence,

Conversely, if χ F is a fuzzy obstinate filter, then we show that F is an obstinate filter of A . Let x , y ∉ F . Then χ F ( x ) = 0 = χ F ( y ) . Since χ F is a fuzzy obstinate filter, thus

We obtain χ F ( x → y ) = χ F ( y → x ) = 1 . Therefore, x → y ∈ F and y → x ∈ F .□

Theorem 3.4

Let μ be a fuzzy obstinate filter of A for any α ∈ [ 0 , 1 ∕ 2 ] . Then level subset μ α = { x ∈ A ∣ μ ( x ) ≥ α } is an obstinate filter of A .

Proof

Let μ be a fuzzy obstinate filter of A . If α = 0 , then μ α = A and μ α is an obstinate filter of A .

Let α ∈ ( 0 , 1 ∕ 2 ] and x , y ∉ μ α . Then μ ( x ) < α and 1 − α < 1 − μ ( x ) . Also μ ( y ) < α and 1 − α < 1 − μ ( y ) . Since μ is a fuzzy obstinate filter, thus

Since α ∈ ( 0 , 1 2 ] , we obtain α < μ ( x → y ) and α < μ ( y → x ) . Hence,

Therefore, level subset μ α is an obstinate filter of A .□

Example 3.5

1. In Example 3.2(i) part(2), let α = 1 ∕ 3 . Then λ 1 ∕ 3 = A and λ 1 ∕ 3 is an obstinate filter.

2. In Example 3.2(ii), let α = 3 ∕ 4 . Then μ 3 ∕ 4 = { 1 , a , b } and 0 = b → a ∉ μ 3 ∕ 4 . Hence, μ 3 ∕ 4 is not an obstinate filter.

3. In Example 3.2(i), let μ ( a ) = μ ( 0 ) = 1 ∕ 3 , μ ( b ) = μ ( 1 ) = 2 ∕ 3 , and α = 1 ∕ 3 . Then μ α = { 0 , a , b , 1 } is an obstinate filter, but μ is not fuzzy obstinate filter. Because ( 1 − μ ( a ) ) ∧ ( 1 − μ ( 0 ) ) > μ ( a → 0 ) ∧ μ ( 0 → a ) .

Theorem 3.6

Let μ be a fuzzy filter of A. Then μ is a fuzzy obstinate filter if and only if for any x ∈ A , 1 − μ ( x ) ≤ μ ( x ′ ) .

Proof

Let for any x ∈ A , 1 − μ ( x ) ≤ μ ( x ′ ) . Then 1 − μ ( y ) ≤ μ ( y ′ ) . Hence, 1 − μ ( x ) ≤ μ ( x → 0 ) and 1 − μ ( y ) ≤ μ ( y → 0 ) . By Proposition 2.3(iv), x → 0 ≤ x → y and y → 0 ≤ y → x . Since μ is a fuzzy filter, μ ( x → 0 ) ≤ μ ( x → y ) and μ ( y → 0 ) ≤ μ ( y → x ) . So 1 − μ ( x ) ≤ μ ( x → y ) and 1 − μ ( y ) ≤ μ ( y → x ) . Hence,

Then μ is a fuzzy obstinate filter of A .

Conversely, let μ be a fuzzy obstinate filter of A . Then for any x , y ∈ A , ( 1 − μ ( x ) ) ∧ ( 1 − μ ( y ) ) ≤ μ ( x → y ) ∧ μ ( y → x ) . If y = 0 , then 1 − μ ( x ) ≤ μ ( x → 0 ) . Therefore, 1 − μ ( x ) ≤ μ ( x ′ ) .□

Example 3.7

In Example 3.2(ii), we have 1 − μ ( 0 ) ≤ μ ( 0 ′ ) , 1 − μ ( a ) ≤ μ ( a ′ ) , 1 − μ ( b ) ≤ μ ( b ′ ) , 1 − μ ( c ) ≤ μ ( c ′ ) , and 1 − μ ( 1 ) ≤ μ ( 1 ′ ) . Hence, for any x ∈ A , 1 − μ ( x ) ≤ μ ( x ′ ) .

Definition 3.8

Let A and B be two hoop algebras and μ be a fuzzy subset of A and λ be a fuzzy subset of B such that f : A → B be a hoop homomorphism. Then image μ under f denoted by f ( μ ) is a fuzzy subset of B that for any y ∈ B :

The preimage of λ under f denoted by f − 1 ( λ ) is a fuzzy set of A denoted by for any x ∈ A , f − 1 ( λ ) ( x ) = λ ( f ( x ) ) .

Definition 3.9

[1] Fuzzy subset μ of A has sup property if for any nonempty subset Y of A , there exists y 0 ∈ Y such that μ ( y 0 ) = sup y ∈ Y μ ( y ) .

Example 3.10

In Example 3.2(i), let Y 1 = { 0 , a } , a ∈ Y 1 such that μ ( a ) = sup y ∈ Y 1 μ ( y ) . Also Y 2 = { 0 , b } , b ∈ Y 2 such that μ ( b ) = sup y ∈ Y 2 μ ( y ) .

Proposition 3.11

1. Let f : A → B be an onto hoop homomorphism. Then preimage of a fuzzy obstinate filter λ under f is a fuzzy obstinate filter of A.

2. Let f : A → B be an onto hoop homomorphism and μ be a fuzzy obstinate filter of A with sup property. Then f ( μ ) is a fuzzy obstinate filter of B.

Proof

1. It is clear.

2. We show that

   ( 1 − f ( μ ) ( y 1 ) ) ∧ ( 1 − f ( μ ) ( y 2 ) ) ≤ f ( μ ) ( y 1 → y 2 ) ∧ f ( μ ) ( y 2 → y 1 ) .

Also, ( 1 − μ ( x 1 ) ) ∧ ( 1 − μ ( x 2 ) ) = ( 1 − f ( μ ) ( y 1 ) ) ∧ ( 1 − f ( μ ) ( y 2 ) ) . Therefore, f ( μ ) is a fuzzy obstinate filter of B .□

Theorem 3.12

Let μ , λ be two fuzzy filters of A, which satisfy μ ⊆ λ , μ ( 1 ) = λ ( 1 ) . If μ is a fuzzy obstinate filter, then λ is a fuzzy obstinate filter of A.

Proof

Let μ be a fuzzy obstinate filter and λ is not fuzzy obstinate filter of A . Then by Theorem 3.6, 1 − μ ( x ) ≤ μ ( x ′ ) . Also μ ( x ) ≤ λ ( x ) and μ ( x ′ ) ≤ λ ( x ′ ) . Hence, 1 − λ ( x ) ≤ 1 − μ ( x ) ≤ μ ( x ′ ) ≤ λ ( x ′ ) . Therefore, λ is a fuzzy obstinate filter.□

Definition 3.13

Let μ be a nonconstant fuzzy filter of ∨ -hoop A . Then μ is called a fuzzy prime filter of A if for each α ∈ [ 0 , 1 ] , μ α = ∅ or μ α is a prime filter of A .

Example 3.14

In Example 3.2(ii), let μ ( a ) = μ ( c ) = μ ( 0 ) = 1 ∕ 5 , μ ( b ) = μ ( 1 ) = 4 ∕ 5 . Then for α = 3 ∕ 5 , μ α is a prime filter.

Proposition 3.15

Let μ be a nonconstant fuzzy filter of ∨ -hoop A . Then μ is a fuzzy prime filter of A if and only if μ ( x ∨ y ) ≤ μ ( x ) ∨ μ ( y ) , for any x , y ∈ A .

Proof

Let μ be a fuzzy prime filter of A and for any x , y ∈ A , α = μ ( x ∨ y ) . Then μ α is a filter of A and x ∨ y ∈ μ α . If μ α = A , then x , y ∈ μ α , and thus, μ ( x ∨ y ) ≤ μ ( x ) ∨ μ ( y ) . If μ α ≠ A , since μ α is a prime filter, then x ∨ y ∈ μ α implies x ∈ μ α or y ∈ μ α . Hence, α ≤ μ ( x ) or α ≤ μ ( y ) . Thus, μ ( x ∨ y ) ≤ μ ( x ) ∨ μ ( y ) .

Conversely, for any α ∈ [ 0 , 1 ] , if μ α ≠ ∅ , then μ α is a filter. If μ α ≠ A and x ∨ y ∈ μ α , then α ≤ μ ( x ∨ y ) . So α ≤ μ ( x ) ∨ μ ( y ) . Hence, x ∈ μ α or y ∈ μ α . Therefore, μ is a fuzzy prime filter.□

Corollary 3.16

Let μ be a nonconstant fuzzy filter of ∨ -hoop A . Then μ is a fuzzy prime filter of A if and only if μ ( x ∨ y ) = μ ( x ) ∨ μ ( y ) , for any x , y ∈ A .

Proof

Since fuzzy filter μ is order-preserving, and x , y ≤ x ∨ y , hence μ ( x ) ∨ μ ( y ) ≤ μ ( x ∨ y ) .□

Proposition 3.17

1. Let μ be a nonconstant fuzzy prime filter of ∨ -hoop A. Then μ μ ( 1 ) is a prime filter of A.

2. If A is a prelinear ∨ -hoop and μ μ ( 1 ) is a prime filter of A, then μ is a fuzzy prime filter of A.

Proof

1. Let μ be a nonconstant fuzzy filter. Then μ ( 0 ) < μ ( 1 ) and 0 ∉ μ μ ( 1 ) . Hence, μ μ ( 1 ) is a prime filter of A .

2. Since ( x → y ) ∨ ( y → x ) = 1 ∈ μ μ ( 1 ) , then x → y ∈ μ μ ( 1 ) or y → x ∈ μ μ ( 1 ) for any x , y ∈ A . By Proposition 2.3(v), ( x ∨ y ) → y = x → y ∈ μ μ ( 1 ) or ( x ∨ y ) → x = y → x ∈ μ μ ( 1 ) . Hence, μ ( ( x ∨ y ) → y ) = μ ( 1 ) or μ ( ( x ∨ y ) → x ) = μ ( 1 ) . We obtain μ ( y ) ≥ μ ( ( x ∨ y ) → y ) ∧ μ ( x ∨ y ) = μ ( x ∨ y ) or μ ( x ) ≥ μ ( ( x ∨ y ) → x ) ∧ μ ( x ∨ y ) = μ ( x ∨ y ) . So μ ( x ) ∨ μ ( y ) ≥ μ ( x ∨ y ) . Therefore, by Proposition 3.15, μ is a fuzzy prime filter.□

Corollary 3.18

1. Let A be a prelinear ∨ -hoop. Then μ μ ( 1 ) is a prime filter of A if and only if μ is a fuzzy prime filter of A.

2. Let A be a prelinear ∨ -hoop. Then a nonempty subset F is a prime filter of A if and only if χ F is a fuzzy prime filter of A.

Theorem 3.19

1. Let μ be a fuzzy prime filter of ∨ -hoop A. Then μ ( x → y ) = μ ( 1 ) or μ ( y → x ) = μ ( 1 ) .

2. Let A be a prelinear ∨ -hoop and μ ( x → y ) = μ ( 1 ) or μ ( y → x ) = μ ( 1 ) . Then μ is a fuzzy prime filter of A .

Proof

1. By Proposition 3.17(i), μ μ ( 1 ) is a prime filter, then x → y ∈ μ μ ( 1 ) or y → x ∈ μ μ ( 1 ) for any x , y ∈ A . Hence, μ ( x → y ) = μ ( 1 ) or μ ( y → x ) = μ ( 1 ) .

2. Let A be a prelinear ∨ -hoop and μ ( x → y ) = μ ( 1 ) or μ ( y → x ) = μ ( 1 ) . Then x → y ∈ μ μ ( 1 ) or y → x ∈ μ μ ( 1 ) . Hence, μ μ ( 1 ) is a prime filter. By Proposition 3.17(ii), μ is a fuzzy prime filter.□

Proposition 3.20

Let A be a ∨ -hoop and μ be a fuzzy prime filter of A and α ∈ [ 0 , μ ( 1 ) ) . Then μ ∨ α is a fuzzy prime filter of A, where ( μ ∨ α ) ( x ) = μ ( x ) ∨ α , for any x ∈ A .

Proof

Let x , y , z ∈ A and x ⊙ y ≤ z . Then μ ( x ) ∧ μ ( y ) ≤ μ ( z ) and ( μ ∨ α ) ( z ) ≥ ( μ ( x ) ∧ μ ( y ) ) ∨ α = ( μ ∨ α ) ( x ) ∧ ( μ ∨ α ) ( y ) . Hence, μ ∨ α is a fuzzy filter of A . Since α < μ ( 1 ) , then ( μ ∨ α ) ( 1 ) = μ ( 1 ) ∨ α = μ ( 1 ) ≠ ( μ ∨ α ) ( 0 ) , so μ ∨ α is a nonconstant fuzzy filter. Then ( μ ∨ α ) ( x ∨ y ) = μ ( x ∨ y ) ∨ α = μ ( x ) ∨ μ ( y ) ∨ α = ( μ ∨ α ) ( x ) ∨ ( μ ∨ α ) ( y ) . Therefore, by Corollary 3.16, μ ∨ α is a fuzzy prime filter.□

Theorem 3.21

Let μ be a nonconstant fuzzy filter of ∨ -hoop A. Then there is a fuzzy prime filter λ of A such that μ ≤ λ .

Proof

If μ is a nonconstant fuzzy filter, then we have μ μ ( 1 ) is a proper filter of A . So there is a prime filter F such that μ μ ( 1 ) ⊆ F . By Corollary 3.18 (ii), χ F is a fuzzy prime filter. Let λ = χ F ∨ α and α = ∨ x ∈ A − F μ ( x ) . Then λ is a fuzzy prime filter and μ ≤ λ .□

Theorem 3.22

Let μ and λ be two fuzzy filters of A, which satisfy μ ⊆ λ , μ ( 1 ) = λ ( 1 ) . If μ is a fuzzy prime filter, then λ is a fuzzy prime filter of A.

Proof

Let μ be a fuzzy prime filter. Then μ ( x → y ) = μ ( 1 ) or μ ( y → x ) = μ ( 1 ) for any x , y ∈ A . If μ ( x → y ) = μ ( 1 ) , by μ ≤ λ and μ ( 1 ) = λ ( 1 ) , then λ ( x → y ) = λ ( 1 ) . Similarly, if μ ( y → x ) = μ ( 1 ) , then λ ( y → x ) = λ ( 1 ) . Therefore, λ is a fuzzy prime filter.□

Proposition 3.23

Every fuzzy obstinate filter μ of A such that μ ( x ) < 1 ∕ 2 is a fuzzy implicative filter, for any x ∈ A .

Proof

Let μ be a fuzzy obstinate filter and x ∈ A such that μ ( x ) < 1 ∕ 2 . Since μ is a fuzzy filter, then

By hypothesis and Theorem 3.6, μ ( ( x ) ′ → x ) ∧ ( 1 − μ ( x ) ) ≤ μ ( x ) . If μ ( ( x ) ′ → x ) ∧ ( 1 − μ ( x ) ) = 1 − μ ( x ) , then μ ( x ) ≥ 1 ∕ 2 , which is the hypothesis. So μ ( ( x ) ′ → x ) ∧ ( 1 − μ ( x ) ) = μ ( ( x ) ′ → x ) and μ ( ( x ) ′ → x ) ≤ μ ( x ) . Therefore, by Theorem 2.8(i), μ is a fuzzy implicative filter.□

Corollary 3.24

Let A be a ∨ -hoop with (DNP) and μ be a fuzzy obstinate filter of A and α ∈ [ 0 , 1 ∕ 2 ] . Then A ∕ μ is a Boolean algebra.

Proposition 3.25

Let μ be a fuzzy obstinate filter of A and α ∈ [ 0 , 1 2 ] . Then: