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Theorem 2.4([10]). Let L
1
and L
2
be lattice
implication algebras, f : L1 → L2
be a mapping from
L
1
to L
2
, if for any xy , ∈ L1
,
f ( x→ y) = f( x) → f( y)
holds, then f is called an
implication homomorphism from L 1
to L
2
. If f is an
implication homomorphism and satisfies
f ( x∨ y) = f( x) ∨ f( y)
,
f ( x∧ y) = f( x) ∧ f( y)
,
f ( x') = ( f( x))'
,
then f is called a lattice implication homomorphism
from L
1
to L
2
.
A one-to-one and onto lattice implication
homomorphism is called a lattice implication
isomorphism.
We can define a partial ordering ≤ on a lattice
implication algebra L by x ≤ y if and only if
x → y = I .
In a lattice implication algebra L , ∀x, yz , ∈ L ,
the following hold(see[10]):
(1) x → y≤( y→ z) →( x→ z)
,
x → y≤( z→ x) →( z→ y)
;
(2)if x ≤ y , then
y → z ≤ x→ z,
z→ x≤ z → y ;
(3) x ∨ y = ( x→ y)
→ y;
(4) x ≤( x→ y)
→ y .
In the follows, if not special noted, L denotes a
lattice implication algebra.
Ⅲ.THE ANNIHILATOR IN LATTICE IMPLICATION ALGEBRA
Firstly, we introduce the notion of the annihilator.
Definition 3.1. Let B be a non-empty subset of L , if
B ∗ = { x ∀ b ∈ B, x ∧ b = O, x ∈ L}
,
then B ∗ is called an annihilator of B .
The following example shows that the annihilator of
lattice implication algebra exists.
Example 3.2. Let L= {0, abcd , , , ,1} ,
0' = 1, a' = c, b' = d, c' = a, d' = b,1' = 0, the Hasse
diagram of L be defined as Fig.1 and its implication
operator be defined as Table 1, then ( L, ∨∧ , ,', → ) is a
lattice implication algebra.
c
b
0
1
Fig.1
d
a
Table 1
→ 0 a b c d 1
0 1 1 1 1 1 1
a c 1 b c b 1
b d a 1 b a 1
c a a 1 1 a 1
d b 1 1 b 1 1
1 0 a b c d 1
Let B = {0, c}, then B
∗ = {0, ad , } is the annihilator
of B by simple computing.
Remark. Obviously, the annihilator of {O} is L.
Now, we give the important characteristics of an
annihilator in lattice implication algebras.
Theorem 3.3. Let B be a non-empty subset of L , if
B ∗ is an annihilator of B , then ∀x ∈ L , b∈ B ,
x → b= x' ⇔ x ∈ B ∗ .
Proof. Suppose that ∀x ∈ L , b∈ B, x → b= x'
,
then ( x →b) → x'
= I . It follows that
( b' → x') → x' = b' ∨ x'
= I . Thus x ∧ b= O .
Hence x ∈ B ∗ by the Definition 3.1.
Conversely, if x ∈ B ∗ , then ∀b∈ B, x ∧ b= O .
It follows that
( b∧ x)' = b' ∨ x'
= ( b' → x') → x'
= I . Thus we
have b' → x' ≤ x'
. x ' ≤ b' → x'
is trivial. Hence
b' → x' = x'
, and so x → b= x'
.
Theorem3.4. Let a∈ L, if ( a)
∗ is an annihilator of
{} a , then ∀x
∈ ( a)
∗ , a≤ x'
.
Proof. Suppose that ( a)
∗ is an annihilator of { a },
then ∀x
∈ ( a)
∗ , x → a = x'
by Theorem 3.3. Since
x ' ∨a→( x→a)
= ( x ' →( x→a)) ∧( a→( x→a))
= ( a' →( x' → x')) ∧( x→( a→a))
= I ,
we have x ' ∨a≤ x→ a. It follows that
x ' ≤ x' ∨a≤ x→ a= x'
. Hence x ' ∨ a = x'
, and so
a≤ x' .
The following theorem prove that an annihilator is
an ideal and a sl ideal in lattice implication algebras.
Theorem3.5. Let B be a non-empty subset of L . If
B ∗ is an annihilator of B ,then B ∗ is an ideal of L .
Proof. If B ∗ is an annihilator of B , then O∈
B ∗
is trivial by the Definition 3.1. Assume that
∗ ∗
∀x,
y∈ L , ( x → y)' ∈B , y∈ B , then we have
∀b∈ B , y → b= y'
and ( x → y)'
→ b= x→
y
by Theorem 3.3. It follows that
x ' = I → x'
= (( y →b) → y') → x'
= (( b' → y') → y') → x'
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