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= ( b' ∨ y') → x'
= ( b' → x') ∧( y' → x')
= ( b' → x') ∧( x→
y)
= ( b' → x') ∧(( x→ y)' → b)
= ( b' → x') ∧( b' →( x→
y))
= ( b' → x') ∧( b' →( y' → x'))
= ( b' → x')
= x → b . Thus x ∈ B ∗ by Theorem
3.3 . Hence B ∗ is an ideal of L .
Corollary 3.6. Let B be a non-empty subset of L ,
if B ∗ is an annihilator of B, then B ∗ is a sl ideal of L .
Proof. it is trivial by Theorem 3.5 and Theorem 4.2 of
[11].
Next, the special properties of an annihilator are
obtained in lattice implication algebras.
Theorem3.7. Let B and C be non-empty subsets of
L . Then the following hold
∗ ∗
(1) if B ⊆ C , then C ⊆ B ;
(2) B ⊆ B ∗∗ ;
∗ ∗∗∗
(3) B = B ;
∗ ∗ ∗
(4) ( B ∪ C)
= B ∩ C .
Proof. (1). Suppose that B ⊆ C . Then ∀x ∈ C ∗ ,
c∈ C , x ∧ c= O, and so ∀b∈ B, x ∧ b= O . Thus
x ∈ B ∗ .
(2). Because ∀b∈ B, x ∈ B ∗ , x ∧ b= O , then we
have b∈ B ∗∗ . Hence B ⊆ B ∗∗ .
∗ ∗∗∗
(3). By (2), we have B ⊆ B and B ⊆ B ∗∗ , and so
∗∗∗ ∗
∗ ∗∗∗
B ⊆ B by (1). Hence B = B .
(4). Because B ⊆ B∪ C and C ⊆ B∪ C , then
∗ ∗
∗ ∗
( B ∪C)
⊆ B and ( B ∪C)
⊆ C by (1).
∗ ∗ ∗
Thus ( B ∪C)
⊆ B ∩ C .
∗ ∗
On the other hand, ∀x ∈B ∩ C , we have x ∈ B ∗
and x ∈ C ∗ . It follows that ∀b∈B,
c∈ C ,
x ∧ b= O and x ∧ c= O. Then we have
x ∧( b∨ c)
= O , and so x ∈( B∪ C)
∗ . Hence
B ∗ ∩C ∗ ⊆( B∪ C)
∗ .
Corollary 3.8. If A is a non-empty subset of L , then
∗
∗
A =∩ ( a)
.
A
a∈A
Proof. By Theorem 3.7 (4) and A=∪ {} a , we have
∗
∗
= ( ∪ { a})
= ∩ ( ) .
a∈A
a ∗
a∈A a∈A
Corollary 3.9. Let A and B be non-empty subsets of
∗ ∗ ∗
L . Then A ∩B ⊆( A∩ B)
.
Proof. By Theorem 3.7 (4), we have
∗ ∗ ∗
( A∪ B)
= A ∩ B . Because A∩B⊆ A∪ B ,
then ( A B) ∗
∗
∪ ⊆( A∩ B)
by Theorem 3.7 (1). It
∗ ∗ ∗
follows that A ∩B ⊆( A∩ B)
.
Theorem 3.10. Let B be a non-empty subset of L .
If < B >=< B > ∗∗ , then < B >= B ∗∗ .
Proof. Because B ⊆< B > , then we have
∗ ∗
∗∗
∗∗
< B > ⊆ B , and so B ⊆< B> by Theorem 3.7
(1). Then by < B >=< B > ∗∗ , we have B ∗∗ ⊆< B > .
On the other hand, by Theorem 3.7 (2), we have
B ⊆ B ∗∗ , and by Theorem 3.5, we know that B ∗∗ is an
ideal of L . Thus < B >⊆ B ∗∗ by the meaning of the
generated ideal. Hence < B >= B ∗∗ .
Theorem 3.11. Let A and B be non-empty subsets
∗ ∗ ∗
of L , then < A ∪ B >⊆( A∩ B)
.
Proof. Because A∩ B⊆ A and A∩B⊆ B , then
we have A ∗
∗
⊆( A∩ B)
and B ∗
∗
⊆( A∩ B)
by
∗ ∗ ∗
Theorem 3.7 (1). Thus A ∪B ⊆( A∩ B)
. By
Theorem 3.5, we know that ( A∩ B)
∗ is an ideal of L .
∗ ∗ ∗
Hence < A ∪ B >⊆( A∩ B)
by the meaning of the
generated ideal.
The relation between an annihilator and an ideal is
given in the following.
Theorem 3.12. If B is an ideal of L , then
∗
B ∩ B = { O}
.
Proof. That O∈B∩ B ∗ is trivial.
On the other hand, ∀x ∈B∩ B ∗ , we have x ∈ B
and x ∈ B ∗ . Then x = x∧ x= O by the Definition 3.1.
Theorem 3.13. Let B and C be ideals of L . Then
B ∩ C = { O}
if and only if B ⊆ C ∗ .
Proof. Suppose that B ∩ C = { O}
, then
∀x
∈ B , c∈ C , x ∧ c= O . Or x ∧c≠O∈B∩
C
contradicts that B ∩ C = { O}
. Thus x ∈ C ∗ by the
Definition 3.1. Hence B ⊆ C ∗ .
Conversely, if B ⊆ C ∗
∗
, then B ∩C ⊆C ∩ C .
By Theorem 3.12, we have C∩ C ∗ = { O}
. Hence
B ∩ C = { O}
.
Corollary 3.14. Let B and C be ideals of L . If
C = C ∗∗ , then B ⊆ C if and only if B ∩ C ∗ = { O}
.
C ∗
Proof. Suppose that B ⊆ C = C ∗∗ . We know that
is an ideal of L by Theorem 3.5. Hence
∗
B ∩ C = { O}
by Theorem 3.13.
Conversely, if B ∩ C ∗ = { O}
, then we have
∗∗
B ⊆ C = C by Theorem 3.13.
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