The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices

Definition 3

[4,8] A nonempty subset I will be called an ideal of L if

• ( I 1 ) If x ≤ y and y ∈ I , then x ∈ I ;

• ( I 2 ) If x , y ∈ I , then x ⊕ y ∈ I .

Remark 1

[8] Every ideal is a lattice ideal in the lattice ( L , ∧ , ∨ , 0 , 1 ) , but the converse is not true. It is easy to see that ideals and lattice ideals coincide iff x ∨ y = x ⊕ y . In Boolean algebras, these notions are equivalent. Moreover, the intersection of any set of ideals is an ideal.

Proposition 1

If I is an ideal of L , then x ∈ I iff x ∗ ∗ ∈ I iff n x ∈ I , for all n ≥ 1 .

Proof

Let I be an ideal of L . Now, we prove that x ∈ I iff x ∗ ∗ ∈ I . Since I is an ideal and 0 , x ∈ I , it follows that x ∗ ∗ = 0 ⊕ x ∈ I . Therefore, x ∗ ∗ ∈ I . Conversely, if x ∗ ∗ ∈ I , by ( c 5 ) , x ≤ x ∗ ∗ ∈ I , we conclude that x ∈ I .

Now, we prove that x ∈ I iff n x ∈ I , for all n ≥ 1 . Since I is an ideal and x ∈ I , it follows that n x ∈ I . Conversely, if n x ∈ I , by ( c 20 ) , x ≤ n x ∈ I , we conclude that x ∈ I . The proof is complete.□

Remark 2

If x , y ∈ L and I ∈ ℐ ( L ) , then x ⊕ y ∈ I iff x ∨ y ∈ I iff x , y ∈ I .

Proposition 2

[8] Let L be a residuated lattice, S ⊆ L a nonempty subset, a ∈ L and I ∈ ℐ ( L ) . Then

1. ( S ] = { x ∈ L : x ≤ s 1 ⊕ ⋯ ⊕ s n , f o r s o m e n ≥ 1 a n d s 1 , … , s n ∈ S } ;

2. ( a ] = { x ∈ L : x ≤ n a f o r s o m e n ≥ 1 } ;

3. I ( a ) = { x ∈ L : x ≤ i ⊕ n a f o r s o m e i ∈ I a n d n ≥ 1 } .

Proposition 3

[4]

1. ( ℐ ( L ) , ⊆ ) is a complete lattice, where for I 1 , I 2 ∈ ℐ ( L ) , I 1 ∨ I 2 = ( I 1 ∪ I 2 ] = { x ∈ L : x ≤ i 1 ⊕ i 2 w i t h i 1 ∈ I 1 a n d i 2 ∈ I 2 } .

2. If Λ is an index set and ( I i ) i ∈ Λ is a family of ideals of L , then the infimum of this family is ∧ i ∈ Λ I i = ∩ i ∈ Λ I i and the supremum is ∨ i ∈ Λ I i = ( ∪ i ∈ Λ I i ] = { x ∈ L : x ≤ x i 1 ⊕ x i 2 ⊕ ⋯ ⊕ x i m , w h e r e i 1 , … , i m ∈ Λ , x i j ∈ I i j , 1 ≤ j ≤ m , f o r s o m e m ≥ 1 } .

Example 1

We consider the set L = { 0 , a , b , c , d , e , f , g , 1 } with 0 < a < b < e < 1 , 0 < a < d < e < 1 , 0 < a < d < g < 1 , 0 < c < d < e < 1 , 0 < c < d < g < 1 , 0 < c < f < g < 1 and elements { a , c } , { b , d } , { d , f } , { e , g } and { b , f } are pairwise incomparable.

Definition 4

1. A proper ideal I ∈ ℐ ( L ) is called prime iff, for all a , b ∈ L , if a ∧ b ∈ I , then a ∈ I or b ∈ I . We denote by Prime ( L ) the set of all prime ideals of L .

2. A proper ideal I is called ∩ -prime iff, for all I 1 , I 2 ∈ ℐ ( L ) if I 1 ∧ I 2 ⊆ I , then I 1 ⊆ I or I 2 ⊆ I .

3. A proper ideal I is called ∩ -irreducible iff, for all I 1 , I 2 ∈ ℐ ( L ) if I 1 ∧ I 2 = I , then I 1 = I or I 2 = I .

4. An ideal M ∈ ℐ ( L ) is called maximal if M is not strictly contained in a proper ideal of L . We denote by Max ( L ) the set of all maximal ideals of L .

Example 2

In this example, we present a residuated lattice with a prime ideal which is not maximal ideal. Let L = { 0 , n , a , b , c , d , 1 } , with 0 < n < a < b < c , d < 1 and elements c and d are incomparable.

Example 3

In this example, we present a residuated lattice with a maximal ideal which is not a prime ideal.

Remark 3

In residuated lattices, the notions of ideals and filters are not dual. Indeed, in the residuated lattice L from Example 3 we have that I = { 0 } is an ideal of L , but F = L \ I = { a , b , c , d , e , f , m , 1 } is not a filter (because a ⊙ d = 0 and 0 ∉ F ). Hence, in residuated lattices, the study of ideals has a proper meaning.

Proposition 4

Any prime ideal is an ∩ -irreducible ( ∩ -prime) ideal, but the converse may not hold.

Proof

Let I 1 , I 2 ∈ ℐ and I ∈ Prime ( L ) be such that I 1 ∩ I 2 = I . By contrary, we assume that I 1 ≠ I and I 2 ≠ I , then there are elements x ∈ I 1 \ I and y ∈ I 2 \ I . Since x ∧ y ∈ I 1 ∩ I 2 = I and I is prime, we conclude that x ∈ I or y ∈ I , which is a contradiction. For the converse, we consider the residuated lattice L from Example 3. It is easy to ascertain that the only proper ideal of L is I 1 = ( 0 ] = { 0 } and I is an ∩ -irreducible ( ∩ -prime) ideal of L . But I is not a prime ideal (because a ∧ b = 0 ∈ I and a , b ∉ I ). The proof is similar for ∩ -prime ideals.□

Remark 4

One of the most important separability properties for the ideal theory in residuated lattices is The Prime Ideal Theorem, that is, for any I ∈ ℐ ( L ) , and S ⊆ L a nonempty ∧ -closed subset of L such that S ∩ I = ∅ , then there is a prime ideal P of L such that I ⊆ P and P ∩ S = ∅ . In this remark, we point that The Prime Ideal Theorem for residuated lattices may not always hold. Indeed, if we consider the residuated lattice L from Example 3, the only proper ideal is I = { 0 } . We consider the set S = { d , e , f , m } to be the ∧ -closed subset of L with the property that S ∩ I = ∅ . But there is no prime ideal P of L such that I ⊆ P and P ∩ S = ∅ .

Remark 5

In conclusion, in the general case of residuated lattices, the concepts of prime ideals and ∩ -irreducible ( ∩ -prime) ideals are not equivalent, Max ( L ) ⊈ Prime ( L ) and The Prime Ideal Theorem may not always hold. We think it will be interesting to find a suitable subvariety of residuated lattices (in which the properties from above hold) in order to study the prime and maximal spectra, the reticulation and stable topology. We consider the variety of De Morgan residuated lattices suitable for two main reasons: first, the concepts of prime ideals and ∩ -irreducible ( ∩ -prime) ideals are equivalent Max ( L ) ⊆ Prime ( L ) and The Prime Ideal Theorem always hold, and, second, it includes important subvarieties of residuated lattices such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and involution residuated lattices (as we will see in the next results).

Example 4

We consider the residuated lattice L from Example 1. It is easy to ascertain that L is an involution residuated lattice, and so L is a De Morgan residuated lattice. The following sets I 1 = { 0 } , I 2 = { 0 , a , b } and I 3 = { 0 , c , f } are proper ideals of L .

Theorem 1

[4] Let L be a De Morgan residuated lattice. For P ∈ ℐ ( L ) , the following conditions are equivalent:

1. P is ∩ -prime;

2. P is ∩ -irreducible;

3. P is prime.

Theorem 2

[4, Theorem 4.14] Let I ∈ ℐ ( L ) , S ⊆ L a nonempty ∧ -closed subset of L such that S ∩ I = ∅ . Then there is a prime ideal P of L such that I ⊆ P and P ∩ S = ∅ .

Theorem 3

[4] Let L be a De Morgan residuated lattice, I ∈ ℐ ( L ) and a ∈ L \ I . Then

1. There is P ∈ Prime ( L ) such that I ⊆ P and a ∉ P ;

2. I is the intersection of those prime ideals which contain I ;

3. ∩ Prime ( L ) = { 0 } .

Theorem 4

[4] Let L be a residuated lattice. If I is a proper ideal of L , then the following conditions are equivalent:

1. I ∈ Max ( L ) ;

2. For any x ∉ I , there exist a ∈ I , n ≥ 1 such that a ⊕ ( n x ) = 1 ;

3. For any x ∈ L , x ∉ I iff ( n x ) ∗ ∈ I , for some n ≥ 1 .

Corollary 1

Let L be a De Morgan residuated lattice. Then

1. Let S be a nonempty ∧ -closed subset of L and I be an ideal of L such that S ∩ I = ∅ . Then there exists a minimal prime ideal Q of L such that I ⊆ Q and S ∩ Q = ∅ .

2. Let S be a nonempty ∧ -closed subset of L such that 0 ∉ S . Then there exists a minimal prime ideal Q such that S ∩ Q = ∅ .

Proof

1. Let I be a proper ideal of L . Consider X = { P ∈ Prime ( L ) : I ⊆ P , S ∩ I = ∅ } . By Theorem 2, we have that X is nonempty and by Zorn Lemma, X has a minimal element.

2. Take I = { 0 } and apply item ( i ) .□

Proposition 5

Let L be a residuated lattice, X , Y be non-empty subsets of L . Then

1. X ⊆ Y ⊆ L implies V ( Y ) ⊆ V ( X ) ⊆ Prime ( L ) ;

2. V ( { 1 } ) = ∅ and V ( ∅ ) = V ( { 0 } ) = Prime ( L ) ;

3. V ( X ) = ∅ iff ( X ] = L ;

4. for L a De Morgan residuated lattice we have: V ( X ) = Prime ( L ) iff X = ∅ or X = { 0 } ;

5. if { X i } i ∈ I is a family of subsets of L , then V ( ∪ i ∈ I X i ) = ∩ i ∈ I V ( X i ) ;

6. V ( ( X ] ) = V ( X ) ;

7. V ( X ) ∪ V ( Y ) = V ( ( X ] ∩ ( Y ] ) ;

8. for L a De Morgan residuated lattice we have: if X , Y ⊆ L , then ( X ] = ( Y ] iff V ( X ) = V ( Y ) ;

9. for L a De Morgan residuated lattice we have: if I , J are ideals of L , then I = J iff V ( I ) = V ( J ) .

Proof

1. Obviously.

2. For any P a prime ideal of L , P is a proper ideal, so 1 ∉ P , that is, P ∉ V ( { 1 } ) . Hence, V ( { 1 } ) = ∅ . It is obvious that V ( ∅ ) = Prime ( L ) . Since 0 is an element of any ideal of L , it follows that 0 is an element of any prime ideal of L , that is, V ( { 0 } ) = Prime ( L ) .

3. " ⇒ . " Suppose that ( X ] ≠ L , that is, ( X ] is a proper ideal of L . Then there is a prime ideal P of L that includes the proper ideal ( X ] . Since X ⊆ ( X ] , it follows that X ⊆ P , so P ∈ V ( X ) . Thus, V ( X ) ≠ ∅ .

   " ⇐ . " If V ( X ) ≠ ∅ , then there is a prime ideal P ∈ V ( X ) . Since P is an ideal including X and ( X ] is the least ideal of L with this property, it follows that L = ( X ] ⊆ P , that is, P = L , a contradiction.

4. " ⇐ . " By (ii).

   " ⇒ . " Suppose that X ≠ ∅ and X ≠ { 0 } . Then there is a ∈ X , a ≠ 0 . Then there is a prime ideal P of L such that a ∉ P . Thus, X ⊈ P , so P ∉ V ( X ) . Therefore, V ( X ) ≠ Prime ( L ) .

5. " ⊆ . " We have that X i ⊆ ∪ i ∈ I X i for all i ∈ I . Applying ( i ) , it follows that V ( ∪ i ∈ I X i ) ⊆ V ( X i ) for all i ∈ I , hence V ( ∪ i ∈ I X i ) ⊆ ∩ i ∈ I V ( X i ) .

   " ⊇ . " If P ∈ ∩ i ∈ I V ( X i ) , then X i ⊆ P for all i ∈ I . We get that ∪ i ∈ I X i ⊆ P , that is, P ∈ V ( ∪ i ∈ I X i ) .

6. " ⊆ . " Let P ∈ V ( X ) , so X ⊆ P . It follows that ( X ] ⊆ P , that is, P ∈ V ( ( X ] ) .

   " ⊇ . " Since X ⊆ ( X ] and applying ( i ) we get that V ( ( X ] ) ⊆ V ( X ) .

7. " ⊆ . " Apply ( i ) .

   " ⊇ . " Let P ∈ V ( ( X ] ∩ ( Y ] ) and suppose that P ∉ V ( X ) ∪ V ( Y ) . Hence, P ∉ V ( X ) = V ( ( X ] ) and P ∉ V ( Y ) = V ( ( Y ] ) , that is, ( X ] ⊈ P and ( Y ] ⊈ P . Thus, there are x ∈ ( X ] , y ∈ ( Y ] such that x , y ∉ P . Since x ∧ y ≤ x , y and ( X ] , ( Y ] are ideals of L , we get that x ∧ y ∈ ( X ] ∩ ( Y ] ⊆ P . Hence, we have obtained x , y ∈ L such that x ∧ y ∈ P and x , y ∉ P , a contradiction with the fact that P is a prime ideal of L .

8. " ⇒ . " Applying ( vi ) , we get that V ( X ) = V ( ( X ] ) = V ( ( Y ] ) = V ( Y ) .

   " ⇐ . " If ( X ] = L , then V ( X ) = ∅ , by (iii) . Thus, V ( Y ) = ∅ , so, applying again (iii) , we get that ( Y ] = L . Therefore, ( X ] = ( Y ] = L . Now, we suppose that ( X ] , ( Y ] are proper ideals of L . Applying Theorem 3 and (vi) , it follows that ( X ] = ∩ { P ∈ Prime ( L ) : P ∈ V ( ( X ] ) } = ∩ { P ∈ Prime ( L ) : P ∈ V ( X ) } = ∩ { P ∈ Prime ( L ) : P ∈ V ( Y ) } = ∩ { P ∈ Prime ( L ) : P ∈ V ( ( Y ] ) } = ( Y ] .

9. Applying (viii) and the fact that, since I , J are ideals of L , we have ( I ] = I and ( J ] = J .□

Proposition 6

Let X , Y be non-empty subsets of L . Then

1. X ⊆ Y ⊆ L implies U ( X ) ⊆ U ( Y ) ⊆ Prime ( L ) ;

2. U ( { 1 } ) = Prime ( L ) and U ( ∅ ) = U ( { 0 } ) = ∅ ;

3. U ( X ) = Prime ( L ) iff ( X ] = L ;

4. for L a De Morgan residuated lattice we have: U ( X ) = ∅ iff X = ∅ or X = { 0 } ;

5. if { X i } i ∈ I is a family of subsets of L , then U ( ∪ i ∈ I X i ) = ∪ i ∈ I U ( X i ) ;

6. U ( ( X ] ) = U ( X ) ;

7. U ( X ) ∪ U ( Y ) = U ( ( X ] ∪ ( Y ] ) ;

8. for L a De Morgan residuated lattice we have: if X , Y ⊆ L , then ( X ] = ( Y ] iff U ( X ) = U ( Y ) ;

9. for L a De Morgan residuated lattice we have: if I , I are ideals of L , then I = J iff U ( I ) = U ( J ) .