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The relation between the lattice implication
homomorphism image of annihilator and the annihilator
of lattice implication homomorphism image is
investigated in the following.
Theorem 3.15. Let ( L, ∧∨→ , , ,', O, I)
and
( L, ∧ , ∨ , → , ¬ , O, I ) be lattice implication
1 1 1 1 1 1 1
algebras, B be a non-empty subset of L and
f : L→ L be a lattice implication homomorphism. If
1
B ∗ ∗
∗
is an annihilator of B , then f ( B) ⊆ f( B)
.
Proof. Suppose that B ∗ is an annihilator of B .
Because y f( B ∗
∗
∀ ∈ ) , there exists x ∈B ⊆ L such
that f ( x)
= y , thus ∀b∈ B, x ∧ b= O , and so
f ( x∧ b) = f( x) ∧
1
f( b)
= O1
. In other words, we
have ∀f () b ∈ f( B)
, f ( x)
∈ L1
, f ( x) ∧
1
f( b)
= O1.
Hence y = f( x) ∈ f( B)
∗ by the Definition 3.1.
Theorem 3.16. Let ( L, ∧∨→ , , ,', O, I)
and
( L1, ∧1, ∨1, →
1, ¬
1, O1, I1)
be lattice implication
algebras, B be a non-empty subset of L and
f : L→ L1
be a lattice implication isomorphism. If B ∗
∗
∗
is an annihilator of B , then f ( B ) = f( B)
.
Ⅳ. CONCLUSION
Lattice implication algebra supplies an alternative
theoretical basis for lattice-valued logic. As we know, in
order to investigate the structure of an algebraic system,
the ideals with special properties play an important role.
In this paper, we introduce the notion of annihilators, and
prove that an annihilator is an ideal and a sl ideal in
lattice implication algebra. Then we give some special
properties of the annihilator and discuss the relationships
between an annihilator and an ideal, between the lattice
implication homomorphism image of annihilator and the
annihilator of lattice implication homomorphism image in
lattice implication algebras. We hope that above work
would supply certain theoretical basis for lattice
implication algebras and develop corresponding latticevalued
logical system.
ACKNOWLEDGEMENTS
This work was supported by the National Natural
Science Foundation of P.R. China (Grant no. 60875034)
and Zhejiang province fatal project (priority subjects) key
industrial project (2008C11011).
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