Generalized Lower and Upper Approximations in Quantales

Definition 2.1. A quantale is a complete lattice Q with an associative binary operation ∘ satisfying

An element e ∈ Q is called a left (right) unit if and only if e∘a = a (a∘e = a), e is called a unit if it is both a right and left unit.

A quantale Q is called a commutative quantale if a∘b = b∘a for all a, b ∈ Q.

A quantale Q is called an idempotent quantale if a∘a = a for all a ∈ Q.

A subset S of Q is called a subquantale of Q if it is closed under ∘ and arbitrary sups.

In a quantale Q, we denote the top element of Q by 1 and the bottom by 0. For A, B⊆Q, we write A∘B to denote the set {a∘b∣a ∈ A, b ∈ B}, A∨B to denote {a∨b∣a ∈ A, b ∈ B} and $$.

Definition 2.2. Let Q be a quantale, a subset ∅ ≠ I⊆Q is called a left (right) ideal of Q if

• (1)

  a, b ∈ I implies a∨b ∈ I,

• (2)

  a ∈ I, b ∈ Q and b ≤ a imply b ∈ I for all,

• (3)

  a ∈ Q and x ∈ I imply a∘x ∈ I (x∘a ∈ I).

A subset I⊆Q is called an ideal if it is both a left and a right ideal.

Let X be a subset of Q, we write ↓X = {y ∈ Q∣y ≤ x for some x ∈ X}, X is a lower set if and only if X = ↓X. It is obvious that an ideal I is a directed lower set. For every (left, right) ideal I of Q, it is easy to see that 0 ∈ I.

An ideal of Q is called a prime ideal if a∘b ∈ I implies a ∈ I or b ∈ I for all a, b ∈ Q.

An ideal I of Q is called a semi-prime ideal if a∘a ∈ I implies a ∈ I for all a ∈ Q.

An ideal I of Q (I ≠ Q) is called a primary ideal if for all a, b ∈ Q, a∘b ∈ I and a ∉ I imply bⁿ ∈ I for some n > 0. ($$).

Definition 2.3. A nonempty subset M⊆Q is called an m-system of Q, if for all a, b ∈ M, ↓(a∘1∘b)∩M ≠ ∅.

A nonempty subset S⊆Q is called a multiplicative set of Q, if a∘b ∈ S for all a, b ∈ S.

Every ideal of Q is both an m-system and a multiplicative set.

Definition 2.4. Let Q be a quantale, an equivalence relation θ on L is called a congruence on Q if for all a, b, c, d, a_i, b_i ∈ Q  (i ∈ I), we have

• (1)

  aθb, cθd⇒(a∘c)θ(b∘d),

• (2)

  $$.

Definition 2.5. Let Q be a quantale, a congruence θ on Q is called a complete congruence, if

• (1)

  [a] _θ∘[b] _θ = [a∘b] _θ for all a, b ∈ Q,

• (2)

  $$ for all a_i ∈ Q  (i ∈ I).

Definition 2.6. Let (Q₁, ∘₁) and (Q₂, ∘₂) be two quantales. A map f : Q₁ → Q₂ is said to be a homomorphism if

• (1)

  f(a  ∘₁  b) = f(a)∘₂f(b) for all a, b ∈ Q₁;

• (2)

  $$ for all a_i ∈ Q₁  (i ∈ I).

Definition 2.7. Let U and W be two nonempty universes. Let T be a set-valued mapping given by T : U → P(W), where P(W) denotes the set of all subsets of W. Then the triple (U, W, T) is referred to as a generalized approximation space. For any set A⊆W, the generalized lower and upper approximations, T₋(A) and T⁻(A), are defined by

The pair (T₋(A), T⁻(A)) is referred to as a generalized rough set.

Theorem 2.8. Let U, W be nonempty universes and T : U → P^*(W) be a set-valued mapping, where P^*(W) denotes the set of all nonempty subsets of W. If A⊆W, then T₋(A)⊆T⁻(A).

Theorem 2.9. Let (U, W, T) be a generalized approximation space, its lower and upper approximation operators satisfy the following properties. For all A, B ∈ P(W),

• (1)

  T₋(A) = (T⁻(A^c)) ^c, T⁻(A) = (T₋(A^c)) ^c,

• (2)

  T₋(W) = U, T⁻(∅) = ∅,

• (3)

  T₋(A∩B) = T₋(A)∩T₋(B), T⁻(A ∪ B) = T⁻(A) ∪ T⁻(B),

• (4)

  A⊆B⇒T₋(A)⊆T₋(B), T⁻(A)⊆T⁻(B),

• (5)

  T₋(A ∪ B)⊇T₋(A) ∪ T₋(B), T⁻(A∩B)⊆T⁻(A)∩T⁻(B),

Theorem 3.1. Let T : Q₁ → P(Q₂) be a set-valued mapping and ∅ ≠ A, B⊆Q₂. Then

• (1)

  T⁻(A) ∪ T⁻(B)⊆T⁻(A∨B), if 0 ∈ A∩B,

• (2)

  T⁻(A) ∪ T⁻(B)⊆T⁻(A  ∘₂  B), if e ∈ A∩B,

• (3)

  T₋(A)∩T₋(B)⊆T₋(A∨B),

• (4)

  T₋(A)∩T₋(B)⊆T₋(A∨B), if Q₂ is an idempotent quantale,

• (5)

  T₋(A) ∪ T₋(B)⊆T₋(A∨B), if 0 ∈ A∩B,

• (6)

  T₋(A) ∪ T₋(B)⊆T₋(A  ∘₂  B), if e ∈ A∩B.

Proof. (1) Suppose that a ∈ A, we have a = a∨0 ∈ A∨B for 0 ∈ B. So A⊆A∨B. Similarly, B⊆A∨B. So A ∪ B⊆A∨B. By Theorem 2.9, we have T⁻(A) ∪ T⁻(B) = T⁻(A ∪ B)⊆T⁻(A∨B).

(2) Suppose that a ∈ A, we have a = a  ∘₂  e ∈ A  ∘₂  B for e ∈ B. So A⊆A  ∘₂  B. Similarly, B⊆A  ∘₂  B. So A ∪ B⊆A  ∘₂  B. By Theorem 2.9, we have T⁻(A) ∪ T⁻(B) = T⁻(A ∪ B)⊆T⁻(A  ∘₂  B).

(3) It is obvious that A∩B⊆A∨B. By Theorem 2.9, we have T₋(A)  ∩  T₋(B) = T₋(A  ∩  B)⊆T₋(A∨B).

(4) Since Q₂ is an idempotent quantale, we have A∩B⊆A∘B. By Theorem 2.9, we have T₋(A)∩T₋(B) = T₋(A∩B)⊆T₋(A  ∘₂  B).

(5) and (6) The proofs are similar to (1) and (2), respectively.

Definition 3.2. A set-valued mapping T : Q₁ → P(Q₂) is called a set-valued homomorphism if

• (1)

  T(a)  ∘₂  T(b)⊆T(a  ∘₁  b) for all a, b ∈ Q₁,

• (2)

  $$ for all a_i ∈ Q₁(i ∈ I).

T is called a strong set-valued homomorphism if the equalities in (1), (2) hold.

Example 3.3. (1) Let θ be a congruence on Q₂. Then the set-valued mapping T : Q₁ → P(Q₂) defined by T(x) = [x] _θ is a set-valued homomorphism but not necessarily a strong set-valued homomorphism. If θ is complete, then T is a strong set-valued homomorphism.

(2) Let f be a quantale homomorphism from Q₁ to Q₂. Then the set-valued mapping T : Q₁ → P(Q₂) defined by T(a) = {f(a)} is a strong set-valued homomorphism.

Theorem 3.4. Let T : Q₁ → P(Q₂) be a set-valued homomorphism and ∅ ≠ A, B⊆Q₂. Then

• (1)

  T⁻(A)∨T⁻(B)⊆T⁻(A∨B),

• (2)

  T⁻(A)  ∘₁  T⁻(B)⊆T⁻(A  ∘₂  B),

• (3)

  T⁻(A)∩T⁻(B)⊆T⁻(A∨B),

• (4)

  T⁻(A)∩T⁻(B)⊆T⁻(A  ∘₂  B), if Q₁ is an idempotent quantale.

Proof. (1) Suppose that c ∈ T⁻(A)∨T⁻(B), there exist a ∈ T⁻(A), b ∈ T⁻(B) such that c = a∨b. So there exist x ∈ A∩T(a) and y ∈ B∩T(b). Hence x∨y ∈ A∨B and x∨y ∈ T(a)∨T(b). Since T is a set-valued homomorphism, we have x∨y ∈ T(a∨b). Therefore, T(a∨b)∩(A∨B) ≠ ∅ which implies that c = a∨b ∈ T⁻(A∨B).

(2) The proof is similar to (1).

(3) Suppose that x ∈ T⁻(A)  ∩  T⁻(B), there exist a ∈ A∩T(x) and b ∈ B∩T(x). Since T is a set-valued homomorphism, we have a∨b ∈ T(x)∨T(x)⊆T(x). So a∨b ∈ T(x)∩(A∨B) which implies that x ∈ T⁻(A∨B).

(4) Suppose that x ∈ T⁻(A)  ∩  T⁻(B), there exist a ∈ A  ∩  T(x) and b ∈ B  ∩  T(x). Since T is a set-valued homomorphism and Q₁ is idempotent, we have a∨b ∈ T(x)  ∘₂  T(x)⊆T(x  ∘₁  x) = T(x). So a∨b ∈ T(x)∩(A∨B) which implies that x ∈ T⁻(A  ∘₂  B).

Theorem 3.5. Let T : Q₁ → P(Q₂) be a strong set-valued homomorphism and ∅ ≠ A, B⊆Q₂. Then

• (1)

  T₋(A)∨T₋(B)⊆T₋(A∨B),

• (2)

  T₋(A)  ∘₁  T₋(B)⊆T₋(A  ∘₂  B).

Proof. (1) Suppose that c ∈ T₋(A)∨T₋(B), there exist a ∈ T₋(A), b ∈ T₋(B) such that c = a∨b. Hence T(a)⊆A and T(b)⊆B. Since T is a strong set-valued homomorphism, we have T(a∨b) = T(a)∨T(b)⊆A∨B which implies that c = a∨b ∈ T₋(A∨B).

(2) The proof is similar to (1).

Theorem 4.1. Let T : Q₁ → P(Q₂) be a set-valued mapping. If I and J are, respectively, a right and a left ideal of Q₂, then

• (1)

  T⁻(I  ∘₂  J)⊆T⁻(I)∩T⁻(J),

• (2)

  T₋(I  ∘₂  J)⊆T₋(I)∩T₋(J),

• (3)

  T⁻(I∧J)⊆T⁻(I)∩T⁻(J),

• (4)

  T₋(I)∧T₋(J) = T₋(I∧J),

• (5)

  T⁻(I) ∪ T⁻(J)⊆T⁻(I∨J),

• (6)

  T₋(I) ∪ T₋(J)⊆T₋(I∨J).

If Q₁ is an idempotent quantale and T is a set-valued homomorphism, then the equalities in (1)–(3) hold.

Proof. Since I and J are, respectively, a right and a left ideal of Q₂, we have I  ∘₂  J⊆I∩J, I∧J = I∩J and o ∈ I∩J. By Theorem 2.9, we get the conclusion (1)–(4). By Theorem 3.1, we get (5) and (6).

If Q₁ is idempotent, we first show that T⁻(I)∩T⁻(J)⊆T⁻(I  ∘₂  J). Suppose that x ∈ T⁻(I)∩T⁻(J), there exist y ∈ I, z ∈ J such that y, z ∈ T(x). So, y  ∘₂  z ∈ I  ∘₂  J and y  ∘₂  z ∈ T(x)  ∘₂  T(x). Since T is a set-valued homomorphism and Q₁ is idempotent, we have y  ∘₂  z ∈ T(x  ∘₁  x) = T(x). Therefore, x ∈ T⁻(I  ∘₂  J). So the equality in (1) holds. Since Q₁ is idempotent, we have I∩J⊆I  ∘₂  J. So I∩J = I  ∘₂  J = I∩J. By Theorem 2.9, we get T₋(I  ∘₂  J) = T₋(I∩J) = T₋(I)∩T₋(J). Since the equality in (1) holds, we have T⁻(I∧J) = T⁻(I  ∘₂  J) = T⁻(I)∩T⁻(J).

Lemma 4.2. Let T : Q₁ → P(Q₂) be a set-valued homomorphism. If A is a lower set of Q₂, then T₋(A) is a lower set of Q₁.

Proof. Suppose x ≤ y ∈ T₋(A), then T(y)⊆A. Let z ∈ T(x), a ∈ T(y), we have z∨a ∈ T(x)∨T(y)⊆T(x∨y) = T(y)⊆A. Since A is a lower set, we have z ∈ A. Therefore, T(x)⊆A which implies that x ∈ T₋(A).

Lemma 4.3. Let T : Q₁ → P(Q₂) be a strong set-valued homomorphism. If A is a lower set of Q₂, then T⁻(A) is a lower set of Q₁.

Proof. Suppose that x ≤ y ∈ T⁻(A), there exists z ∈ T(y)∩A. Since T is a strong set-valued homomorphism, we have T(x)∨T(y) = T(x∨y) = T(y). So there exist a ∈ T(x), b ∈ T(y) such that z = a∨b. Since A is a lower set, we have a ∈ A. Thus T(x)∩A ≠ ∅ which follows that x ∈ T⁻(A).

Lemma 4.4. Let T : Q₁ → P(Q₂) be a set-valued homomorphism and A a nonempty subset of Q₂. If A is closed under arbitrary (resp., finite) sups, then T⁻(A) is closed under arbitrary (resp., finite) sups.

Proof. Let B⊆T⁻(A). For each b ∈ B, we have b ∈ T⁻(A), then there exist x_b ∈ T(b)∩A. Since T is a set-valued homomorphism, we have $$. And we have $$ for A is closed under arbitrary sups. So T(⋁ B)∩A ≠ ∅ which implies that ⋁ B ∈ T⁻(A).

Lemma 4.5. Let T : Q₁ → P(Q₂) be a strong set-valued homomorphism and A a nonempty subset of Q₂. If A is closed under arbitrary (resp., finite) sups, then T₋(A) is closed under arbitrary (resp., finite) sups.

Proof. Let B⊆T₋(A). For each b ∈ B, we have T(b)⊆A. Since T is a strong set-valued homomorphism, we have T(⋁ B) = ⋁ T(B). Suppose z ∈ T(⋁ B) = ⋁ T(B), there exist x_b ∈ T(b)⊆A (b ∈ B) such that $$. Since A is closed under arbitrary sups, we have $$. So T(⋁ B)⊆A which implies that ⋁ B ∈ T₋(A).

Theorem 4.6. Let T : Q₁ → P(Q₂) be a set-valued homomorphism. If A is a subquantale of Q₂, then T⁻(A) is a subquantale of Q₁.

Proof. Since T is a set-valued homomorphism and A is a subquantale, by Theorems 3.4 and 2.9, we have T⁻(A)  ∘₁  T⁻(A)⊆T⁻(A  ∘₂  A)⊆T⁻(A). So, T⁻(A) is closed under ∘₁.

Since A is closed under arbitrary sups, by Lemma 4.4, we get T⁻(A) is closed under arbitrary sups.

Theorem 4.7. Let T : Q₁ → P(Q₂) be a strong set-valued homomorphism. If A is a subquantale of Q₂, then T₋(A) is a subquantale of Q₁.

Proof. The proof is similar to Theorem 4.6.

Theorem 4.8. Let T : Q₁ → P^*(Q₂) be a strong set-valued homomorphism and I a right (left) ideal of Q₂. Then T⁻(I) is, if it is nonempty, a right (left) ideal of Q₁.

Proof. Suppose that a, b ∈ T⁻(I). Since I is closed under finite sups, by Lemma 4.4, we have T⁻(I) is closed under finite sups. So a∨b ∈ T⁻(I).

Since I is a lower set, by Lemma 4.3, we have T⁻(I) is a lower set.

Suppose a ∈ Q₁, x ∈ T⁻(I), there exists y ∈ I∩T(x). Since I is a right ideal of Q₂, we have y  ∘₂  b ∈ I for each b ∈ T(a)⊆Q₂. So y  ∘₂  b ∈ T(x)  ∘₂  T(a)⊆T(x  ∘₁  a). Therefore, T(x  ∘₁  a)∩I ≠ ∅ which implies that x  ∘₁  a ∈ T⁻(I).

Theorem 4.9. Let T : Q₁ → P(Q₂) be a strong set-valued homomorphism and I a right (left) ideal of Q₂. Then T₋(I) is, if it is nonempty, a right (left) ideal of Q₁.

Proof. Suppose a, b ∈ T₋(I). Since I is closed under finite sups, by Lemma 4.5, we have T₋(I) is closed under finite sups. So, a∨b ∈ T₋(I).

Since I is a lower set, by Lemma 4.2, we have T₋(I) is a lower set.

Suppose a ∈ Q₁, x ∈ T₋(I), we have T(x)⊆I. Let y ∈ T(x  ∘₁  a). Since T is a strong set-valued homomorphism, we have y ∈ T(x)  ∘₂  T(a), then there exist y₁ ∈ T(x)⊆I, y₂ ∈ T(a) such that y = y₁  ∘₂  y₂. Since I is a right ideal, we have y = y₁  ∘₂  y₂ ∈ I. Therefore, T(x  ∘₁  a)⊆I which implies that x∘₁a ∈ T₋(I).

Definition 4.10. A subset A⊆Q₂ is called a generalized rough ideal (subquantale) of Q₁ if T₋(A) and T⁻(A) are ideals (subquantales) of Q₁.

The following corollary follows from Theorems 4.6–4.9.

Corollary 4.11. Let T : Q₁ → P(Q₂) be a strong set-valued homomorphism and I an ideal (a subquantale) of Q₂. If T₋(I) and T⁻(I) are nonempty, then I is a generalized rough ideal (subquantale) of Q₁.

Example 4.12. Let Q₁ = {0, a, 1} and Q₂ = {0^′, b, c, 1^′} be quantales shown in Figures 1 and 2 and Tables 1 and 2.

Let T : Q₁ → P(Q₂) be a strong set-valued homomorphism as defined by T(0) = {0′}, T(a) = {b, c}, T(1) = {1′}. Let A = {0′, b, c}⊆Q₂, B = {b, c}⊆Q₂, then T₋(A) = {0, a} = T⁻(A) and T₋(B) = {a} = T⁻(B). It is obvious that A is a generalized rough ideal of Q₁ but A is not an ideal of Q₂ and B is a generalized rough subquantale of Q₁ but B is not a subquantale of Q₂.

Theorem 4.13. Let T : Q₁ → P(Q₂) be a strong set-valued homomorphism and I a prime ideal of Q₂. Then T⁻(I) is, if it is nonempty, a prime ideal of Q₁.

Proof. By Theorem 4.8, we get T⁻(I) is an ideal of Q₁.

Let a  ∘₁  b ∈ T⁻(I), there exists x ∈ I∩T(a  ∘₁  b). Since T is a strong set-valued homomorphism, we have x ∈ T(a)  ∘₂  T(b), then there exist y ∈ T(a), z ∈ T(b) such that x = y  ∘₂  z ∈ I. Since I is a prime ideal of Q₂, we have y ∈ I or z ∈ I. So T(a)∩I ≠ ∅ or T(b)∩I ≠ ∅ which implies that a ∈ T⁻(I) or b ∈ T⁻(I).

Theorem 4.14. Let T : Q₁ → P(Q₂) be a strong set-valued homomorphism and I a prime ideal of Q₂. Then T₋(I) is, if it is nonempty, a prime ideal of Q₁.

Proof. By Theorem 4.9, we get T₋(I) is an ideal of Q₁.

Let a  ∘₁  b ∈ T₋(I), we have T(a∘₁b)⊆I. Since T is a strong set-valued homomorphism, we have T(a)  ∘₂  T(b)⊆I. We assume that a ∉ T₋(I), then T(a)⊈I, there exists x ∈ T(a) but x ∉ I. If y ∈ T(b), then x  ∘₂  y ∈ T(a)  ∘₂  T(b)⊆I. Since I is a prime ideal of Q₂, we have y ∈ I. Therefore, T(b)⊆I which implies that b ∈ T₋(I).

Theorem 4.15. Let T : Q₁ → P(Q₂) be a strong set-valued homomorphism and I a semiprime ideal of Q₂. Then T₋(I) is, if it is nonempty, a semiprime ideal of Q₁.

Proof. By Theorem 4.9, we get T₋(I) is an ideal of Q₁.

Suppose that a  ∘₁  a ∈ T₋(I), we have T(a  ∘₁  a)⊆I. Let x ∈ T(a), we have x  ∘₂  x ∈ T(a)  ∘₂  T(a)⊆T(a  ∘₁  a)⊆I. Since I is a semi-prime ideal, we have x ∈ I. So T(a)⊆I which implies that a ∈ T₋(I).

Theorem 4.16. Let T : Q₁ → P(Q₂) be a strong set-valued homomorphism and I a primary ideal of Q₂. Then T⁻(I) is, if T⁻(I) ≠ ∅ and T⁻(I) ≠ Q₁, a primary ideal of Q₁.

Proof. By Theorem 4.8, we get T⁻(I) is an ideal of Q₁.

Suppose that a, b ∈ Q, a  ∘₁  b ∈ T⁻(I) and a ∉ T⁻(I), there exists x ∈ I∩T(a  ∘₁  b). Since T is a strong set-valued homomorphism, we have x ∈ T(a)  ∘₂  T(b), there exist y ∈ T(a), z ∈ T(b) such that x = y  ∘₂  z ∈ I. Since a ∉ T⁻(I), we get y ∉ I. Since I is a primary ideal, we have zⁿ ∈ I for some n > 0 Since T is a strong set-valued homomorphism, we have zⁿ ∈ T(bⁿ). So T(bⁿ)∩I ≠ ∅ which implies that bⁿ ∈ T⁻(I).

Theorem 4.17. Let T : Q₁ → P(Q₂) be a strong set-valued homomorphism and I a primary ideal of Q₂. Let Q₂ be a commutative quantale and T(x) a finite set for each x ∈ Q₁. Then T₋(I) is, if T₋(I) ≠ ∅ and T₋(I) ≠ Q₁, a primary ideal of Q₁.

Proof. By Theorem 4.9, we get T₋(I) is an ideal of Q₁.

Suppose a, b ∈ Q, a  ∘₁  b ∈ T₋(I) and a ∉ T₋(I), then T(a  ∘₁  b)⊆I and T(a)⊈I. So there exists x ∈ T(a) with x ∉ I. We assume that T(b) = {y₁, y₂, …, y_m}, then x  ∘₂  y_i ∈ T(a)  ∘₂  T(b)⊆T(a  ∘₁  b)⊆I(i = 1,2, …, m). Since I is a primary ideal, there exists $$ for some n_i > 0  (i = 1,2, …, m). Let n = max {n_i∣i = 1,2, …, m}. Since I is an ideal, we have $$. Let z ∈ T(b^mbⁿ) = T(b) ^mn. Since Q₂ is commutative, there exists {i₁, i₂, …, i_s}⊆{1,2, …, m} such that $$ with t₁ + t₂ + ⋯+t_s = mn, where t_j > 0  (j = 1,2, …, s). Assume that t_j < n, (j = 1,2, …, s), then t₁ + t₂ + ⋯+t_s < sn ≤ mn. It contradicts with t₁ + t₂ + ⋯+t_s = mn. Therefore, there is 1 ≤ k ≤ s such that t_k ≥ n, (j = 1,2, …, s), we have $$. Since I is an ideal, we get z ∈ I. Hence T(b^mn)⊆I which implies that b^mn ∈ T₋(I).

Theorem 4.18. Let T : Q₁ → P(Q₂) be a set-valued homomorphism. If S is a multiplicative set of Q₂, then T⁻(S) is, if it is nonempty, a multiplicative of Q₁. If T is a strong set-valued homomorphism, then T₋(S) is, if it is nonempty, a multiplicative of Q₁.

Proof. Suppose that a, b ∈ T⁻(S), there exist x ∈ T(a)∩S, y ∈ T(b)∩S. Hence x∘y ∈ T(a)∘₂T(b)⊆T(a∘₁b). Since S is a multiplicative set, we have x∘₂y ∈ S. Therefore, x∘₂y ∈ T(a∘₁b)∩S which implies that a∘₁b ∈ T⁻(S).

Suppose a, b ∈ T₋(S), then T(a)⊆S, T(b)⊆S. Let x ∈ T(a  ∘₁  b). Since T is a strong set-valued homomorphism, we have x ∈ T(a)  ∘₂  T(b). There exist y ∈ T(a)⊆S, z ∈ T(b)⊆S such that x = y  ∘₂  z. Since S is a multiplicative set, we have x = y  ∘₂  z ∈ S. So T(a  ∘₁  b)⊆S which implies that a  ∘₁  b ∈ T₋(S).

Theorem 4.19. Let T : Q₁ → P^*(Q₂) be a set-valued homomorphism. If Q₁ is commutative and A⊆Q₂ with 1 ∉ T⁻(A), then the following statements are equivalent:

• (1)

  A is a generalized rough prime ideal of Q₁,

• (2)

  A is a generalized rough ideal and A^c is a generalized rough multiplicative set of Q₁,

• (3)

  A is a generalized rough ideal and A^c is a generalized rough m-system of Q₁.

Proof. (1)⇒(2): Since A is a generalized rough prime ideal of Q₁, we get A is a generalized rough ideal of Q₁ and T₋(A), T⁻(A) are prime ideals of Q₁. Now we show that T⁻(A^c) is a multiplicative set. Let a, b ∈ T⁻(A^c). By Theorem 2.9(1), we have a, b ∈ (T₋(A)) ^c. Since T₋(A) is a prime ideal, we have a  ∘₁  b ∉ T₋(A) which implies that a  ∘₁  b ∈ (T₋(A)) ^c = T⁻(A^c). So T⁻(A^c) is a multiplicative set. Similarly, T₋(A^c) is a multiplicative set.

(2)⇒(3): Let a, b ∈ T⁻(A^c). Since 1 ∉ T⁻(A) and T(1) ≠ ∅, we have 1 ∈ T⁻(A^c). Since T⁻(A^c) is a multiplicative set, we have a  ∘₁  1  ∘₁  b ∈ T⁻(A^c). So ↓(a  ∘₁  1  ∘₁  b)∩T⁻(A^c) ≠ ∅. Therefore, T⁻(A^c) is an m-system. Similarly, we have T₋(A^c) is a m-system.

(3)⇒(1): Let a  ∘₁  b ∈ T⁻(A). Suppose a ∉ T⁻(A) and b ∉ T⁻(A), then a, b ∈ (T⁻(A)) ^c = T₋(A^c). Since T₋(A^c) is an m-system, we have ↓(a  ∘₁  1  ∘₁  b)∩T₋(A^c) ≠ ∅. Thus ↓(a  ∘₁  1  ∘₁  b)∩(T⁻(A)) ^c ≠ ∅. Since T⁻(A) is an ideal and Q₁ is commutative, we have a  ∘₁  1  ∘₁  b ∈ T⁻(A) and ↓(a  ∘₁  1  ∘₁  b)⊆T⁻(A). It contradicts with ↓(a  ∘₁  1  ∘₁  b)∩(T⁻(A)) ^c ≠ ∅. So, a ∈ T⁻(A) or b ∈ T⁻(A). Therefore, T⁻(A) is a prime ideal. Similarly, we have T₋(A) as a prime ideal.