Spectra and reticulation of semihoops

Definition 2.1

[14] An algebra ( A , ⊙ , → , ∧ , 1 ) of type ( 2 , 2 , 2 , 0 ) is called a semihoop if it satisfies the following conditions:

1. ( A , ∧ , 1 ) is a ∧ -semilattice with upper bound 1;

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

3. ( x ⊙ y ) → z = x → ( y → z ) , for all x , y , z ∈ A (Galois connect).

Example 2.2

[1] Let A = { 0 , a , b , 1 } be a chain. We define ⊙ and → on A as follows:

Then ( A , ⊙ , → , ∧ , 1 ) is a semihoop, where x ∧ y = min { x , y } , for all x , y ∈ A .

Example 2.3

[1] Let A = { 0 , a , b , c , 1 } . Define ⊙ and → as follows:

Then ( A , ⊙ , → , ∧ , 1 ) is a semihoop, where x ∧ y = x ⊙ y , for all x , y ∈ A .

Proposition 2.4

[14] Let A be a semihoop, then the following hold, for any x , y , z ∈ A :

1. x ⊙ y ≤ z if and only if x ≤ y → z ;

2. x ⊙ y ≤ x , y ;

3. 1 → x = x , x → 1 = 1 ;

4. x n ≤ x , for all n ∈ N + ;

5. x ⊙ ( x → y ) ≤ y ;

6. if x ≤ y , then x ⊙ z ≤ y ⊙ z , y → z ≤ x → z and z → x ≤ z → y .

Definition 2.5

[1] Let A be a semihoop and F be a non-empty subset of A . Then F is said be a filter of A , if it satisfies:

1. x ⊙ y ∈ F , for any x , y ∈ F ;

2. x ≤ y and x ∈ F imply y ∈ F , for any x , y ∈ A .

Lemma 2.6

(MP rule) Let A be a semihoop and F be a non-empty subset of A . Then F ∈ ℱ ( A ) if and only if 1 ∈ F and if x , x → y ∈ F , for some x , y ∈ A , then y ∈ F .

Lemma 2.7

[1] Let A be a semihoop and x ∈ A . Then ⟨ x ⟩ = { a ∈ A ∣ x n ≤ a , ∃ n ∈ N + } .

Corollary 2.8

[1] Let A be a semihoop, F ∈ ℱ ( A ) , and x ∈ A . Then ⟨ F ∪ { x } ⟩ = { a ∈ A ∣ y ⊙ x n ≤ a , ∃ n ∈ N + , y ∈ F } .

Definition 2.9

[1] A proper filter F of semihoop A is called a maximal filter of A , if it is not properly contained in any other proper filters of A .

Lemma 2.10

[1] Let A be a semihoop and F be a proper filter of A . Then the following conditions are equivalent:

1. F ∈ Max ( A ) .

2. if x ∉ F , then ⟨ F ∪ { x } ⟩ = A .

Definition 2.11

[1] A proper filter F of semihoop A is called prime filter of A , if any H , G ∈ ℱ ( A ) such that H ∩ G ⊆ F , then H ⊆ F or H ⊆ F .

Lemma 2.12

[1] Let A be a semihoop and for any x , y ∈ A , we define: x ∨ y = [ ( x → y ) → y ] ∧ [ ( y → x ) → x ] . Then the following conditions are equivalent:

1. ∨ is an associative operation on A ;

2. x ≤ y imply x ∨ z ≤ y ∨ z , for all x , y , z ∈ A ;

3. x ∨ ( y ∧ z ) ≤ ( x ∨ y ) ∧ ( x ∨ z ) , for all x , y , z ∈ A ;

4. ∨ is the join operation on A .

Lemma 2.13

[1] A semihoop is a ∨ -semihoop if it satisfies one of the equivalent conditions of Lemma 2.12.

Definition 2.14

[1] Let A be a ∨ -semihoop and F be a proper filter of A . Then the following conditions are equivalent:

1. F is a prime filter.

2. if x ∨ y , for some x , y ∈ A , then x ∈ F or y ∈ F .

Definition 2.15

[15] In a bounded semihoop A , the binary operation ⊕ is defined by x ⊕ y = x ′ → y , for any x , y ∈ A .

Lemma 2.16

[15] Let A be a bounded semihoop, then the following hold, for any x , y , z ∈ A :

1. if x ≤ y , then x ⊕ z ≤ y ⊕ z ;

2. x , y ≤ x ⊕ y ;

3. x ⊕ x ′ = 1 ;

4. x ⊕ y = 1 if and only if x ′ ≤ y .

Proposition 3.1

Let A be a ∨ -semihoop. Then A satisfies x ⊙ ( y ∨ z ) = ( x ⊙ y ) ∨ ( x ⊙ z ) , for any x , y , z ∈ A .

Proof

For any x , y , z ∈ A . Clearly, y , z ≤ y ∨ z , that is, x ⊙ y , x ⊙ z ≤ x ⊙ ( y ∨ z ) . Then ( x ⊙ y ) ∨ ( x ⊙ z ) ≤ x ⊙ ( y ∨ z ) . Conversely, since x ⊙ y , x ⊙ z ≤ ( x ⊙ y ) ∨ ( x ⊙ z ) , by Gaolis Connect, we obtain y , z ≤ x → [ ( x ⊙ y ) ∨ ( x ⊙ z ) ] , that is, y ∨ z ≤ x → [ ( x ⊙ y ) ∨ ( x ⊙ z ) ] . Then, we obtain x ⊙ ( y ∨ z ) ≤ ( x ⊙ y ) ∨ ( x ⊙ z ) .□

Proposition 3.2

Let A be a ∨ -semihoop and F be a proper filter of A. Suppose that x , y ∈ A , and x , y ∉ F , then ⟨ F ∪ { x } ⟩ ∩ ⟨ F ∪ { y } ⟩ = ⟨ F ∪ { x ∨ y } ⟩ .

Proof

Let a ∈ ⟨ F ∪ { x } ⟩ ∩ ⟨ F ∪ { y } ⟩ . Hence, by Corollary 2.8 , there exist f 1 , f 2 ∈ F , n 1 , n 2 ∈ Z + such that a ≥ f 1 ⊙ x n 1 , a ≥ f 2 ⊙ y n 2 . Take f = f 1 ⊕ f 2 , n = m a x { n 1 , n 2 } , then we obtain a ≥ f ⊙ x n , a ≥ f ⊙ y n . With Proposition 3.1, a = a ∨ a ≥ f ⊙ ( x n ∨ y n ) = ( f ⊙ x n ) ∨ ( f ⊙ y n ) . Since ( x ∨ y ) n × n ≤ x n ∨ y n , then we obtain a ≥ f ⊙ ( x ∨ y ) n × n . It follows that a ∈ ⟨ F ∪ { x ∨ y } ⟩ . Conversely, suppose that b ∈ ⟨ F ∪ { x ∨ y } ⟩ , then there exist t ∈ F , s ∈ Z + such that b ≥ t ⊙ ( x ∨ y ) s . Obviously, b ≥ t ⊙ x s and b ≥ t ⊙ y s . It follows that b ∈ ⟨ F ∪ { x } ⟩ and b ∈ ⟨ F ∪ { y } ⟩ , that is, b ∈ ⟨ F ∪ { x } ⟩ ∩ ⟨ F ∪ { y } ⟩ .□

Proposition 3.3

Let A be a ∨ -semihoop and M ∈ Max ( A ) , then M is a prime filter of A .

Proof

Let A be a ∨ -semihoop and M ∈ Max ( A ) . Suppose that H , G ∈ ℱ ( A ) such that H ∩ G ⊆ M . If H , G ⊈ M , then there exist x ∈ H , y ∈ G such that x , y ∉ M . By Lemma 2.10, we have ⟨ M ∪ { x } ⟩ = ⟨ M ∪ { y } ⟩ = A . Moreover, since x , y ≤ x ∨ y , so we have x ∨ y ∈ H ∩ G ⊆ M . By Proposition 3.2 , A = ⟨ M ∪ { x } ⟩ ∩ ⟨ M ∪ { y } ⟩ = ⟨ M ∪ { x ∨ y } ⟩ = M , which is a contradiction. Therefore, M is a prime filter.□

Lemma 3.4

Let A be a semihoop, then the union of any chain of proper filters of A is a proper filter of A.

Proof

Let F = ∪ { F i ∣ F 1 ⊆ F 2 ⊆ ⋯ ⊆ F s } , 1 ≤ i ≤ s . It is the union of any proper filters of A . Since any F i is a proper filter of A , then { 1 } ∈ F i ⇒ { 1 } ⊆ F ≠ ∅ . Suppose that x ∈ F , y ∈ A , and x ≤ y . Since x ∈ F , there exists t such that x ∈ F t . Hence, y ∈ F t , that is, y ∈ F . Finally, suppose that x , y ∈ F , then there exist 1 ≤ m and n ≤ s such that x ∈ F m , y ∈ F n . Without losing generality, assume F m ⊆ F n . If F is a chain, then x , y ∈ F n . Since F n is proper, then x ⊙ y ∈ F n , and we obtain x ⊙ y ∈ F . In summary, F is a proper filter of A .□

Proposition 3.5

Let A be a ∨ -semihoop and a ∈ A , a ≠ 1 . Then there exists a prime filter P of A such that a ∉ P .

Proof

Suppose that a ∈ A and a ≠ 1 , let ℱ ( a ) = { F ∣ F ∈ ℱ ( A ) } , F be a proper filter of A , and a ∉ F . Since a ≠ 1 , then { 1 } ∈ ℱ ( a ) ≠ ∅ . Applying Lemma 3.4 and Zorn’s lemma on ( ℱ ( a ) , ⊆ ) , we obtain a maximal element P on ( ℱ ( a ) , ⊆ ) . That means P is a maxmial filter of A with a ∉ P . By Proposition 3.3, then P is a prime filter of A .□

Proposition 3.6

Let A be a ∨ -semihoop and F be a proper filter of A, then F = ∩ { P ∈ Spec ( A ) ∣ F ⊆ P } .

Proof

Obviously, F ⊆ ∩ { P ∈ Spec ( A ) ∣ F ⊆ P } . Suppose that a ∉ F , a ≠ 1 . By Proposition 3.5 , there exists a prime filter P such that a ∉ P and F ⊆ P . It follows that ∩ { P ∈ Spec ( A ) ∣ F ⊆ P } ⊆ F .□

Lemma 3.7

If A is a ∨ -semihoop, then every proper filter F of A is contained in a maximal filter.

Corollary 3.8

If A is a ∨ -semihoop, then every proper filter F of A is contained in a prime filter.

Proof

It directly follows Propositions 3.5 and 3.6.□

Corollary 3.9

Any semihoop has a maximal, prime filter.

Proposition 3.10

Let A be a ∨ -semihoop. The following propositions hold:

1. if X ⊆ Y ⊆ A , then E ( Y ) ⊆ E ( X ) ⊆ Spec ( A ) ;

2. E ( { 0 } ) = ∅ , E ( ∅ ) = E ( { 1 } ) = Spec ( A ) ;

3. E ( X ) = E ( ⟨ X ⟩ ) , for any X ⊆ A ;

4. E ( X ) = ∅ ⇔ ⟨ X ⟩ = A ;

5. E ( X ) = Spec ( A ) ⇔ X = ∅ or X = { 1 } ;

6. if { X i } i ∈ I is a family of subsets of A , then E ( ∪ i ∈ I X i ) = ∩ i ∈ I E ( X i ) ;

7. if X , Y ⊆ A , then E ( X ) ∪ E ( Y ) = E ( ⟨ X ⟩ ∩ ⟨ Y ⟩ ) ;

8. if X , Y ⊆ A , then ⟨ X ⟩ = ⟨ Y ⟩ if and only if E ( X ) = E ( Y ) ;

9. if F , G are filters of A , then F = G if and only if E ( F ) = E ( G ) .

Proof

( 1 ) , ( 2 ) , and ( 3 ) are obviously established.

(4) ( ⇒ ) Let E ( X ) = ∅ . Suppose that ⟨ X ⟩ ≠ A , namely ⟨ X ⟩ is a proper filter of A . Applying Corollary 3.8, there exists a prime filter P such that X ⊆ ⟨ X ⟩ ⊆ P , that is, P ∈ E ( X ) ≠ ∅ , which is a contradiction. Therefore, ⟨ X ⟩ = A .

( ⇐ ) By ( 2 ) and ( 3 ) , we have E ( X ) = E ( ⟨ X ⟩ ) = E ( A ) = ∅ .

(5) Let E ( X ) = Spec ( A ) . Suppose that X ≠ ∅ and X ≠ { 1 } . It means that there exists a ≠ 1 ∈ X , by Proposition 3.5, there exists a prime filter P such that a ∉ P , which is a contradiction. Conversely, it directly follows from ( 2 ) .

(6) Clearly, X i ⊆ ∪ i ∈ I X i , by ( 1 ) , it follows that E ( ∪ i ∈ I X i ) ⊆ E ( X i ) . Thus, E ( ∪ i ∈ I X i ) ⊆ ∩ i ∈ I E ( X i ) . Conversely, suppose that P ∈ ∩ i ∈ I E ( X i ) , then X i ⊆ P for any i ∈ I . Therefore, we obtain ∪ i ∈ I X i ⊆ P , that is, P ∈ E ( ∪ i ∈ I X i ) .

(7) First, ⟨ X ⟩ ∩ ⟨ Y ⟩ ⊆ ⟨ X ⟩ , ⟨ Y ⟩ , by ( 1 ) and ( 3 ) , then we obtain E ( X ) ∪ E ( Y ) ⊆ E ( ⟨ X ⟩ ∩ ⟨ Y ⟩ ) . Conversely, suppose that P ∈ E ( ⟨ X ⟩ ∩ ⟨ Y ⟩ ) , that is, ⟨ X ⟩ ∩ ⟨ Y ⟩ ⊆ P . If P ∉ E ( X ) ∪ E ( Y ) , then P ∉ E ( X ) = E ( ⟨ X ⟩ ) and P ∉ E ( Y ) = E ( ⟨ Y ⟩ ) . It follows that ⟨ X ⟩ ⊈ P and ⟨ Y ⟩ ⊈ P . Thus, there exist x ∈ ⟨ X ⟩ and y ∈ ⟨ Y ⟩ such that x ∉ P and y ∉ P . Moreover, x , y ≤ x ∨ y , since ⟨ X ⟩ and ⟨ Y ⟩ are filters, so x ∨ y ∈ ⟨ X ⟩ ∩ ⟨ Y ⟩ ⊆ P . Since P ∈ Spec ( A ) , we have x ∈ P or y ∈ P , which is a contradiction. Hence, E ( ⟨ X ⟩ ∩ ⟨ Y ⟩ ) ⊆ E ( X ) ∪ E ( Y ) .

(8) ( ⇒ ) Let ⟨ X ⟩ = ⟨ Y ⟩ . Applying ( 3 ) , it directly follows that E ( X ) = E ( Y ) . ( ⇐ ) Suppose that E ( X ) = E ( Y ) . First, if 0 ∈ ⟨ X ⟩ , then E ( Y ) = E ( X ) = E ( A ) = ∅ . By ( 4 ) , we obtain ⟨ X ⟩ = ⟨ Y ⟩ = A . Moreover, if ⟨ X ⟩ is a proper filter of A , by Proposition 3.6, then ⟨ X ⟩ = ∩ { P ∈ Spec ( A ) ∣ ⟨ X ⟩ ⊆ P } = ∩ { P ∈ Spec ( A ) ∣ P ∈ E ( ⟨ X ⟩ ) = E ( X ) } = ∩ { P ∈ Spec ( A ) ∣ P ∈ E ( Y ) = E ( ⟨ Y ⟩ ) } = ∩ { P ∈ Spec ( A ) ∣ ⟨ Y ⟩ ⊆ P } = ⟨ Y ⟩ .

(9) It follows by ( 8 ) .□

Proposition 3.11

Let A be a ∨ -semihoop. The following propositions hold:

1. if X ⊆ Y ⊆ A , then D ( X ) ⊆ D ( Y ) ⊆ Spec ( A ) ;

2. D ( { 0 } ) = Spec ( A ) , D ( ∅ ) = D ( { 1 } ) = ∅ ;

3. D ( X ) = D ( ⟨ X ⟩ ) , for any X ⊆ A ;

4. D ( X ) = ∅ ⇔ X = ∅ or X = { 1 } ;

5. D ( X ) = Spec ( A ) ⇔ ⟨ X ⟩ = A ;

6. if { X i } i ∈ I is a family of subsets of A, then D ( ∪ i ∈ I X i ) = ∪ i ∈ I D ( X i ) ;

7. if X , Y ⊆ A , then D ( X ) ∪ D ( Y ) = D ( ⟨ X ⟩ ∪ ⟨ Y ⟩ ) ;

8. if X , Y ⊆ A , then ⟨ X ⟩ = ⟨ Y ⟩ if and only if D ( X ) = D ( Y ) ;

9. if F , G are filters of A , then F = G if and only if D ( F ) = D ( G ) .

Proposition 3.12

Let A be a ∨ -semihoop and a , b ∈ A . The following statements hold:

1. D ( a ) = Spec ( A ) if and only if ⟨ a ⟩ = A ;

2. D ( a ) = ∅ if and only if a = 1 ;

3. D ( a ) = D ( b ) if and only if ⟨ a ⟩ = ⟨ b ⟩ ;

4. if a ≤ b , then D ( b ) ⊆ D ( a ) ;

5. D ( a ) ∩ D ( b ) = D ( a ∨ b ) ;

6. D ( a ) ∪ D ( b ) = D ( a ∧ b ) = D ( a ⊙ b ) .

Proof

By Proposition 3.11 ( 4 ) , ( 5 ) , and ( 8 ) , we know ( 1 ) , ( 2 ) , and ( 3 ) hold obviously.

(4) Let a ≤ b . Suppose that P ∈ D ( b ) , then b ∉ P . If P ∉ D ( a ) , then a ∈ P and from a ≤ b , we have b ∈ P , which is a contradiction.

(5) For any prime filter P of A , we have a ∨ b ∉ P iff a ∉ P and b ∉ P . Thus, P ∈ D ( a ∨ b ) iff a ∨ b ∉ P iff a ∉ P and b ∉ P iff P ∈ D ( a ) and P ∈ D ( b ) iff P ∈ D ( a ) ∩ D ( b ) .

(6) Since a ⊙ b ≤ a ∧ b ≤ a , b , by ( 4 ) , it follows that D ( a ⊙ b ) ⊇ D ( a ∧ b ) ⊇ D ( a ) ∪ D ( b ) . Conversely, suppose that P ∈ D ( a ⊙ b ) , that is, a ⊙ b ∉ P . Since P is a filter, we can obtain a ∉ P or b ∉ P . Thus, D ( a ) ∪ D ( b ) ⊇ D ( a ⊙ b ) .□

Proposition 3.13

Let A be a ∨ -semihoop. Then family { D ( a ) } a ∈ A is a basis for the topology of Spec ( A ) .

Proof

Let X ⊆ A . Then D ( X ) = D ( ∪ a ∈ X { a } ) = ∪ a ∈ X D ( a ) by Proposition 3.11 ( 6 ) . Hence, any open subset of Spec ( A ) is the union of subsets from the family { D ( a ) ∣ a ∈ A } .□

Proposition 3.14

Let A be a ∨ -semihoop. Then for any a ∈ A , D ( a ) is compact in Spec ( A ) .

Proof

Obviously, we can obtain any cover of D ( a ) with basic open sets containing a finite cover of D ( a ) . Suppose that D ( a ) = ∪ i ∈ I D ( a i ) , by Proposition 3.11 ( 6 ) , we have D ( a ) = ∪ i ∈ I D ( a i ) = D ( ∪ i ∈ I { a i } ) . According to Proposition 3.11 ( 8 ) , then ⟨ a ⟩ = ⟨ ∪ i ∈ I { a i } ⟩ , so a ∈ ⟨ ∪ i ∈ I { a i } ⟩ . By the definition of generated filter, there are n ≥ 1 and i 1 , … , i n ∈ I such that a i 1 ⊙ ⋯ ⊙ a i n ≤ a . We should prove that D ( a ) = D ( a i 1 ⊙ ⋯ ⊙ a i n ) . Applying Proposition 3.12 ( 4 ) and ( 6 ) , we have D ( a ) ⊆ D ( a i 1 ⊙ ⋯ ⊙ a i n ) = D ( a i 1 ) ∪ ⋯ ∪ D ( a i n ) . The other inclusion is clear, since D ( a i 1 ) ∪ ⋯ ∪ D ( a i n ) ⊆ ∪ i ∈ I D ( a i ) = D ( a ) .□

Proposition 3.15

Spec ( A ) is a T 0 topological space.

Proof

It remains to prove that for any different prime filters P ≠ Q ∈ Spec ( A ) , there exists an open set U such that Q ∈ U , P ∉ U , or Q ∉ U , P ∈ U . Since P ≠ Q , we obtain P ⊈ Q or Q ⊈ P . There is no loss of generality in assuming P ⊈ Q , then there exists a ∈ A such that a ∈ P and a ∉ Q . Take U = D ( a ) . Then Q ∈ U and P ∉ U .□

Proposition 3.16

Let A be a ∨ -semihoop, X , Y ⊆ A , { X i } i ∈ I be a family of subsets of A and a , b ∈ A . Then:

1. if X ⊆ Y ⊆ A , then D Max ( X ) ⊆ D Max ( Y ) ⊆ Max ( A ) ;

2. D Max ( 0 ) = Max ( A ) , D Max ( ∅ ) = D Max ( 1 ) = ∅ ;

3. D Max ( X ) = Max ( A ) if and only if ⟨ X ⟩ = A ;

4. D Max ( ∪ i ∈ I X i ) = ∪ i ∈ I D Max ( X i ) ;

5. D Max ( X ) = D Max ( ⟨ X ⟩ ) ;

6. D Max ( X ) ∪ D Max ( Y ) = D Max ( ⟨ X ⟩ ∪ ⟨ Y ⟩ ) ;

7. D Max ( a ) = Max ( A ) if and only if ⟨ a ⟩ = A ;

8. if a ≤ b , then D Max ( b ) ⊆ D Max ( a ) ;

9. D Max ( a ) ∩ D Max ( b ) = D Max ( a ∨ b ) ;

10. D Max ( a ) ∪ D Max ( b ) = D Max ( a ∧ b ) = D Max ( a ⊙ b ) .

Proof

We only prove ( 3 ) , and other statements are immediate consequences of Proposition 3.11.

(3) Let D Max ( X ) = Max ( A ) . If ⟨ X ⟩ ≠ A , then ⟨ X ⟩ is a proper filter of A , by Lemma 3.7, there exists a maximal filter M of A such that ⟨ X ⟩ ⊆ M . Thus, M ∉ D Max ( X ) . This is a contradiction. Conversely, let ⟨ X ⟩ = A . By Proposition 3.11 ( 5 ) , we have D ( X ) = Spec ( A ) . Therefore, D Max ( X ) = D ( X ) ∩ Max ( A ) = Max ( A ) .□

Proposition 3.17

If A is a ∨ -semihoop with negation, then Max ( A ) is a compact space.

Proof

Since there exists { a i } i ∈ I ⊆ A such that Max ( A ) = ∪ i ∈ I D Max ( a i ) , then by Proposition 3.16 (4), we have Max ( A ) = D Max ( ∪ i ∈ I { a i } ) . And applying Proposition 3.16 ( 3 ) , we obtain A = ⟨ ∪ i ∈ I { a i } ⟩ with negation, so 0 ∈ ∪ i ∈ I { a i } . It follows that there are n ≥ 1 and i 1 , … , i n ∈ I such that a i 1 ⊙ ⋯ ⊙ a i n = 0 . According to Proposition 3.16 ( 2 ) , we obtain Max ( A ) = D Max ( 0 ) = D Max ( a i 1 ⊙ ⋯ ⊙ a i n ) = ∪ j = 1 n D Max ( a i j ) . Therefore, Max ( A ) is compact.□