A topology related to implication and upsets on a bounded BCK-algebra

Definition 2.1

[2,8] A BCK-algebra is an algebra A = ( A , → , 1 ) of type (2, 0) satisfying the following, for all x , y , z ∈ A :

1. x → x = 1 ;

2. if x → y = 1 and y → x = 1 , then x = y ;

3. ( x → y ) → ( ( y → z ) → ( x → z ) ) = 1 ;

4. x → ( y → z ) = y → ( x → z ) ;

5. x → ( y → x ) = 1 .

Definition 2.2

[2,8] A BCK-algebra with the product condition (product BCK-algebra) is a BCK-algebra A = ( A , → , 1 ) satisfying the following condition for all x , y ∈ A , x ⊙ y exists, where x ⊙ y = min { z ∈ A ∣ x ≤ y → z } .

Lemma 2.3

[2,8] Let A be a BCK-algebra. Then, the following hold for all x , y , z ∈ A :

1. x ≤ y iff x → y = 1 ;

2. 1 → x = x ;

3. x ≤ y implies z → x ≤ z → y , y → z ≤ x → z ;

4. x → y ≤ ( y → z ) → ( x → z ) ;

5. x → y ≤ ( z → x ) → ( z → y ) ;

6. x ≤ y → z iff y ≤ x → z ;

7. x ≤ ( x → y ) → y ;

8. x → y = ( ( x → y ) → y ) → y .

Lemma 2.4

[2,8] Let A be a product BCK-algebra. Then, the following hold for all x , y , z ∈ A :

1. x ⊙ y ≤ z iff x ≤ y → z ;

2. ( x ⊙ y ) → z = x → ( y → z ) ;

3. x ≤ y implies x ⊙ z ≤ y ⊙ z ;

4. x ⊙ ( x → y ) ≤ x , y .

Lemma 2.5

[2] Let A be a bounded BCK-algebra. Then, the following hold for all x , y ∈ A :

1. x ≤ y implies y ∗ ≤ x ∗ and x ∗ ∗ ≤ y ∗ ∗ ;

2. x ≤ x ∗ ∗ , x ∗ = x ∗ ∗ ∗ , x ∗ ∗ = x ∗ ∗ ∗ ∗ ;

3. x → y ∗ = y → x ∗ ;

4. ( x → y ∗ ) ∗ ∗ = x → y ∗ = x ∗ ∗ → y ∗ = ( x ∗ ∗ → y ∗ ) ∗ ∗ ;

5. ( x → y ∗ ∗ ) ∗ ∗ = x → y ∗ ∗ = x ∗ ∗ → y ∗ ∗ = ( x ∗ ∗ → y ∗ ∗ ) ∗ ∗ ;

6. x → y ≤ ( x → y ) ∗ ∗ ≤ x ∗ ∗ → y ∗ ∗ .

Lemma 2.6

Let A be a bounded BCK-algebra. Then, the following hold for all x , y , z ∈ A :

1. x ≤ y implies z → n x ≤ z → n y , for some n ∈ N ;

2. x ≤ y implies y → n z ≤ x → n z , for some n ∈ N ;

3. x → m ( y → n z ) = y → n ( x → m z ) , for some m , n ∈ N ;

4. x → y ≤ ( z → n x ) → ( z → n y ) , for some n ∈ N ;

5. ( x → n y ∗ ∗ ) ∗ ∗ = x → n y ∗ ∗ = x ∗ ∗ → n y ∗ ∗ = ( x ∗ ∗ → n y ∗ ∗ ) ∗ ∗ , for some n ∈ N ;

6. x → m ( y → n z ∗ ∗ ) ∗ ∗ = x → m ( y → n z ∗ ∗ ) = y → n ( x → m z ∗ ∗ ) = y → n ( x → m z ∗ ∗ ) ∗ ∗ , for some m , n ∈ N .

Proof

1. This follows from Lemma 2.3 (3) by the induction.

2. This follows from Lemma 2.3 (3) by the induction.

3. This follows from Definition 2.1 (C) by the induction.

4. This follows from Lemma 2.3 (5) by the induction.

5. This follows from Lemma 2.5 (5) by the induction.

6. This follows from (3) and (5) by the induction.□

Lemma 2.7

Let A be a bounded product BCK-algebra. Then, the following hold for all x , y ∈ A :

1. x ∗ ∗ ⊙ y ∗ ∗ ≤ ( x ⊙ y ) ∗ ∗ ;

2. ( x ∗ ∗ ⊙ y ∗ ∗ ) ∗ ∗ = ( x ∗ ∗ ⊙ y ) ∗ ∗ = ( x ⊙ y ∗ ∗ ) ∗ ∗ = ( x ⊙ y ) ∗ ∗ ;

3. ( ( x ∗ ∗ ) 2 ) ∗ ∗ = ( x 2 ) ∗ ∗ and so ( ( x ∗ ∗ ) n ) ∗ ∗ = ( x n ) ∗ ∗ , for some n ∈ N ;

4. ( x ∗ ∗ ) n → y ∗ ∗ = ( ( x ∗ ∗ ) n ) ∗ ∗ → y ∗ ∗ = ( x n ) ∗ ∗ → y ∗ ∗ = x n → y ∗ ∗ , for some n ∈ N .

Proof

(1) By Lemmas 2.4 (1) and 2.5 (5), we have x ∗ ∗ ⊙ y ∗ ∗ ≤ ( x ⊙ y ) ∗ ∗ iff x ∗ ∗ ≤ y ∗ ∗ → ( x ⊙ y ) ∗ ∗ iff y ≤ x ∗ ∗ → ( x ⊙ y ) ∗ ∗ iff y ≤ x → ( x ⊙ y ) ∗ ∗ iff x ⊙ y ≤ ( x ⊙ y ) ∗ ∗ .

(2) We first prove that ( x ∗ ∗ ⊙ y ∗ ∗ ) ∗ ∗ = ( x ⊙ y ) ∗ ∗ . Applying (1) and Lemma 2.5 ((1) and (2)), we have ( x ∗ ∗ ⊙ y ∗ ∗ ) ∗ ∗ ≤ ( x ⊙ y ) ∗ ∗ ∗ ∗ = ( x ⊙ y ) ∗ ∗ . From x ≤ x ∗ ∗ and y ≤ y ∗ ∗ , it follows that x ⊙ y ≤ x ∗ ∗ ⊙ y ∗ ∗ , and so by Lemma 2.5 (1), we obtain ( x ⊙ y ) ∗ ∗ ≤ ( x ∗ ∗ ⊙ y ∗ ∗ ) ∗ ∗ . Hence, ( x ∗ ∗ ⊙ y ∗ ∗ ) ∗ ∗ = ( x ⊙ y ) ∗ ∗ .

Next, since x ⊙ y ≤ x ∗ ∗ ⊙ y ≤ x ∗ ∗ ⊙ y ∗ ∗ , by Lemma 2.5 (1), we obtain ( x ⊙ y ) ∗ ∗ ≤ ( x ∗ ∗ ⊙ y ) ∗ ∗ ≤ ( x ∗ ∗ ⊙ y ∗ ∗ ) ∗ ∗ . So ( x ⊙ y ) ∗ ∗ = ( x ∗ ∗ ⊙ y ) ∗ ∗ = ( x ∗ ∗ ⊙ y ∗ ∗ ) ∗ ∗ . Similarly, we can prove that ( x ⊙ y ) ∗ ∗ = ( x ⊙ y ∗ ∗ ) ∗ ∗ = ( x ∗ ∗ ⊙ y ∗ ∗ ) ∗ ∗ .

(3) This follows from (2) by the induction.

(4) By (3) and Lemma 2.5 (5), we have ( x ∗ ∗ ) n → y ∗ ∗ = ( ( x ∗ ∗ ) n ) ∗ ∗ → y ∗ ∗ = ( x n ) ∗ ∗ → y ∗ ∗ = x n → y ∗ ∗ , for some n ∈ N .□

Theorem 2.8

[2] Let A be a bounded BCK-algebra. Then, the following are equivalent:

1. Reg ( A ⁄ D ( A ) ) = A ⁄ D ( A ) ;

2. ( x ∗ ∗ → x ) ∗ ∗ = 1 , for all x ∈ A ;

3. ( x → y ) ∗ ∗ = x ∗ ∗ → y ∗ ∗ , for all x , y ∈ A ;

4. ∗ ∗ is a homomorphism from L onto Reg ( A ) and Reg ( A ) ≅ A ⁄ D ( A ) .

Definition 2.9

[2, 8] A bounded BCK-algebra is called Glivenko if it satisfies one of the equivalent conditions from Theorem 2.8.

Definition 2.10

A bounded BCK-algebra X is called De Morgan if the following condition is satisfied for all x , y ∈ X : ( x ∨ ˜ y ) ∗ = x ∗ ∧ ˜ y ∗ .

Proposition 2.11

Any Glivenko BCK-algebra is a De Morgan BCK-algebra.

Proof

Let A be a Glivenko BCK-algebra. Then, ( x → y ) ∗ ∗ = x ∗ ∗ → y ∗ ∗ , for all x , y ∈ A . So ( x → y ) ∗ = ( x → y ) ∗ ∗ ∗ = ( x ∗ ∗ → y ∗ ∗ ) ∗ = ( y ∗ → x ∗ ) ∗ . Thus, ( x ∨ ˜ y ) ∗ = ( ( x → y ) → y ) ∗ = ( y ∗ → ( x → y ) ∗ ) ∗ = ( y ∗ → ( y ∗ → x ∗ ) ∗ ) ∗ = ( ( x ∗ ∗ → y ∗ ∗ ) → y ∗ ∗ ) ∗ = ( x ∗ ∗ ∨ ˜ y ∗ ∗ ) ∗ = x ∗ ∧ ˜ y ∗ . Therefore, A is a De Morgan BCK-algebra.□

Definition 2.12

[5,6] Let ( X , ≤ ) be an ordered set. Define ↑ : P ( X ) → P ( X ) , by ↑ Y = { x ∈ X ∣ x ≥ y , for some y ∈ Y } , for any subset Y of X . A subset F of X is called an upset if ↑ F = F . Particularly, by ↑ x , we mean ↑ { x } . We denote by U ( X ) the set of all upsets of X . An upset F is called finitely generated if there exists n ∈ N such that F = ↑ { x 1 , x 2 , … , x n } , for some x 1 , x 2 , … , x n ∈ X .

Definition 2.13

[11,12] Let ( X , ⋆ ) be an algebra of type 2 and T be a topology on X . Then, ( X , ⋆ , T ) is called:

1. A left (right) topological algebra, if for all a ∈ X , the map ⋆ : X → X defined by x ↦ a ⋆ x ( x ↦ x ⋆ a ) is continuous;

2. a semitopological algebra, or the operation ⋆ is separately continuous, if ( X , ⋆ , T ) is a left and right topological algebra;

3. a para- ⋆ -topological algebra if ⋆ , as a mapping of X × X to X , is continuous when X × X is endowed with the product topology, or equivalently, if for any x , y ∈ X and any open neighbourhood W of x ⋆ y , there exist two open neighbourhoods U and V of x and y , respectively, such that U ⋆ V ⊆ W .

Theorem 2.14

[5] Let ℬ and ℬ ′ be two bases for two topologies T and T ′ on a given set X, respectively. Then, the following are equivalent:

1. T ′ is finer that T ;

2. for any x ∈ X and any basis element B ∈ ℬ containing x, there is a basis element B ′ ∈ ℬ ′ such that x ∈ B ′ ⊆ B .

Definition 3.1

Let F be an upset of A . Define:

Example 3.2

Consider the bounded BCK-algebra A = {0, a, b, c, d, 1} from [13] and the operation → on A is given in the following table:

Remark 3.3

(1) Zahiri and Borzooei [5] introduced the concept of D y ( F ) , for any upset F of a BL-algebra L , where

y Δ x = ( y → x ) ∗ ∗ and y n Δ x = y Δ ( y n − 1 Δ x ) , for all x , y ∈ A .

Using the Glivenko property in the case of a BL-algebra L , it was proved that y n Δ x = ( y ∗ ∗ ) n → x ∗ ∗ , for x , y ∈ L . Hence, D y ( F ) = { x ∈ L ∣ ( y ∗ ∗ ) n → x ∗ ∗ ∈ F , for some n ∈ N } . And it is shown that any BL-algebra L with such topology is a semitopological BL-algebra.

According to Lemma 2.7(4), we obtain that ( y ∗ ∗ ) n → x ∗ ∗ = y n → x ∗ ∗ . Thus,

(2) Holdon [6] generalized the aforementioned concept to residuated lattices and defined the set D a ( X ) = { x ∈ L ∣ a n ⊖ x ∈ X , for some n ∈ N } in a residuated lattice L , for any upset X of L , where a ⊖ x = a ∗ ∗ → x ∗ ∗ and a n ⊖ x = a ⊖ ( a n − 1 ⊖ x ) .

From Lemma 2.7 (4), it follows that D a ( X ) = { x ∈ L ∣ ( a ∗ ∗ ) n → x ∗ ∗ ∈ X , for some n ∈ N } = { x ∈ L ∣ a n → x ∗ ∗ ∈ X , for some n ∈ N } .

(3) But in the case of a bounded BCK-algebra A , the set D a ( F ) introduced by Zahiri et al. is different from the set D a ( F ) given by Holdon. In general, the former is contained in the latter. In fact, from Lemma 2.5 (5) it follows inductively that a n Δ x = a → n − 1 ( a → x ) ∗ ∗ , for a , x ∈ A . Applying Lemma 2.5 (6), a → n − 1 ( a → x ) ∗ ∗ ≤ a → n − 1 ( a ∗ ∗ → x ∗ ∗ ) = a → n x ∗ ∗ . Thus, a n Δ x ≤ a n ⊖ x , which means that { x ∈ A ∣ a n Δ x ∈ F , for some n ∈ N } ⊆ { x ∈ A ∣ a n ⊖ x ∈ F , for some n ∈ N } , for an upset F of A .

Proposition 3.4

Let F and G be upsets of A. Then, the following hold:

1. D a ∗ ( F ) is an upset of A ;

2. a ∈ D a ∗ ( F ) , F ⊆ D a ∗ ( F ) ;

3. if F ⊆ G , then D a ∗ ( F ) ⊆ D a ∗ ( G ) ;

4. D a ∗ ( D a ∗ ( F ) ) = D a ∗ ( F ) ;

5. if F is an i-filter of A , then so is D a ∗ ( F ) and F ( a ) ⊆ D a ∗ ( F ) ;

6. a ≤ b implies D b ∗ ( F ) ⊆ D a ∗ ( F ) ;

7. if { F α ∣ α ∈ Γ } is a family of upsets of A, then D a ∗ ( ⋃ α ∈ Γ F α ) = ⋃ α ∈ Γ D a ∗ ( F α ) ;

8. if { F 1 , F 2 , … , F n } is a finite set of upsets of A, then D a ∗ ( ⋂ i = 1 n F i ) = ⋂ i = 1 n D a ∗ ( F i ) ;

9. D a ∗ ( D b ∗ ( F ) ) = D b ∗ ( D a ∗ ( F ) ) ;

10. if F and G are i-filters of A, then ⟨ a ∪ F ∪ G ⟩ ⊆ D a ∗ ( F ∨ G ) ;

11. if F and G are i-filters of A, then D a ∗ ( F ) ∩ D a ∗ ( G ) = D a ∗ ( F ∩ G ) and D a ∗ ( F ∨ G ) = D a ∗ ( D a ∗ ( F ) ∨ D a ∗ ( G ) ) ;

12. ( D a ∗ ( ℱ ( A ) ) , ∩ , ⊔ , D a ∗ ( { 1 } ) , A ) is a complete Heyting algebra, where D a ∗ ( F ) ∩ D a ∗ ( G ) = D a ∗ ( F ∩ G ) and ⊔ i ∈ I D a ∗ ( G i ) = D a ∗ ( ∨ i ∈ I G i ) , for F ∈ ℱ ( A ) and { G i ∣ i ∈ I } ⊆ ℱ ( A ) .

Proof

(1) Assume that t ∈ ↑ D a ∗ ( F ) . Then, there is x ∈ D a ∗ ( F ) such that x ≤ t and so a → n x ∗ ∗ ∈ F , for some n ∈ N . From x ≤ t , by Lemma 2.5 (1), we have x ∗ ∗ ≤ t ∗ ∗ . Since F is an upset, by Lemma 2.6 (1), we obtain a → n x ∗ ∗ ≤ a → n t ∗ ∗ ∈ F , i.e., t ∈ D a ∗ ( F ) . Thus, D a ∗ ( F ) is an upset of A .

(2) From a → a ∗ ∗ = 1 ∈ F , we have a ∈ D a ∗ ( F ) . Let x ∈ F . Since F is an upset and x ≤ a → n x ∗ ∗ , for any n ∈ N , we have a → n x ∗ ∗ ∈ F and so x ∈ D a ∗ ( F ) . Hence, F ⊆ D a ∗ ( F ) .

(3) Let F ⊆ G and x ∈ D a ∗ ( F ) . Then, there is n ∈ N such that a → n x ∗ ∗ ∈ F ⊆ G , so x ∈ D a ∗ ( G ) .

(4) By (2), we have D a ∗ ( F ) ⊆ D a ∗ ( D a ∗ ( F ) ) . Let x ∈ D a ∗ ( D a ∗ ( F ) ) . Then, there exists n ∈ N such that a → n x ∗ ∗ ∈ D a ∗ ( F ) , so a → m ( a → n x ∗ ∗ ) ∗ ∗ ∈ F , for some m ∈ N . Applying Lemma 2.6 (5), a → m + n x ∗ ∗ ∈ F , i.e., x ∈ D a ∗ ( F ) .

(5) If F is an i -filter of A , then F is a nonempty upset. Clearly, 1 ∈ D a ∗ ( F ) . Let x , x → y ∈ D a ∗ ( F ) . Then, there are m , n ∈ N such that a → m x ∗ ∗ ∈ F and a → n ( x → y ) ∗ ∗ ∈ F . By Lemmas 2.6 ((3) and (4)) and 2.5 (6), we obtain successively:

Since F is an i -filter of A , it follows that a → m + n y ∗ ∗ ∈ F , i.e., y ∈ D a ∗ ( F ) . Thus, D a ∗ ( F ) is an i -filter of A . Since a → n x ≤ a → n x ∗ ∗ , for some n ∈ N , we have F ( a ) ⊆ D a ∗ ( F ) .

(6) Let a ≤ b and x ∈ D b ∗ ( F ) . Then, there are n ∈ N such that b → n x ∗ ∗ ∈ F . From a ≤ b , by Lemma 2.6 (2), we have b → n x ∗ ∗ ≤ a → n x ∗ ∗ ∈ F , i.e., x ∈ D a ∗ ( F ) . Thus, D b ∗ ( F ) ⊆ D a ∗ ( F ) .

(7) For x ∈ L , we obtain successively: