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Proceedings of the Second International Symposium on Networking and Network Security (ISNNS ’10)
Jinggangshan, P. R. China, 2-4, April. 2010, pp. 206-209
The Annihilator and its Structure in Lattice
Implication Algebras
Hua Zhu 1 , Weifeng Du 2 *, and Jianbin Zhao 1
1.Department of mathematics, Zhengzhou University, Zhengzhou , China;
Email: zhuhua@zzu.edu.cn, zhaojianbin@zzu.edu.cn
2.School of Mathematics and Information Engineering, Jiaxing University, Jiaxing, China
Email: woodmud@tom.com
Abstract—The notion of annihilator of lattice implication
algebras is proposed. An annihilator is proved to be an ideal
and a sl ideal. Then the special characteristics of an
annihilator are obtained. The relationships between an
annihilator and an ideal, between the lattice implication
homomorphism image of annihilator and the annihilator of
lattice implication homomorphism image are discussed,
respectively.
Index Terms— lattice implication algebra; annihilator;
ideal
Ⅰ. INTRODUCTION
To establish an alternative logic for reasoning and
knowledge representation, in 1993, Xu [7] proposed the
concept of lattice implication algebra by combining
lattice and implication algebra. Lattice implication
algebra can respectively describe the comparable and the
incompletely comparable properties of truth value, which
can more efficiently reflect the people’s thinking, judging
and reasoning. Since then, many researchers[2], [4], [6],
[8],[9], [12], [3], [11], [13], [14] have investigated this
important logic algebra. For example, Jun [1] defined the
notions of LI-ideals in lattice implication algebras and
investigated its some properties. In 2003, Liu et al.[4]
proposed the notions of ILI-ideals and maximal LI-ideal
of lattice implication algebras, respectively, investigated
their properties and obtained the extension theorem of
ILI-ideals. In 2006, Zhu et al.[12] introduced the notions
of prime ideal and primary ideal of lattice implication
algebra respectively, researched their properties and
discussed their relations. In 2008, Pan [11] introduced the
notions of lattice implication n-ordered semigroup and
lattice implication p-ordered semigroup, then discussed
the properties of sl ideals in lattice implication n-ordered
semigroups and lattice implication p-ordered semigroups.
In this paper, as an extension of aforementioned work, In
section 2, we list some basic concepts of lattice
implication algebra which are needed for this topic. In
section 3, we introduce the concept of annihilator and
obtain some special characteristics of annihilator in lattice
implication algebras. Then we prove that an annihilator is
an ideal and a sl ideal , and discuss the relationships
* Corresponding author:Du Weifeng , School of Mathematics &
Information Engineering, Jiaxing University, Jiaxing, Zhejiang, China,
Email:woodmud@tom.com
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, respectively.
Ⅱ. PRELIMINARIES
By a lattice implication algebra we mean a bounded
lattice ( L, ∨∧ , , O, I)
with order-reversing involution ' ,
I and O the greatest and the smallest element of L
respectively, and a binary operation → satisfying the
follow axioms:
(I 1 ) x → ( y → z)
= y → ( x → z)
;
(I 2 ) x → x = I ;
(I 3 ) x → y = y′
→ x′
;
(I 4 )if x→ y = y→ x= 1, then x = y ;
(I 5 ) ( x → y)
→ y = ( y → x)
→ x ;
(L 1 ) ( x ∨ y)
→ z = ( x → z)
∧ ( y → z)
;
(L 2 )( x ∧ y)
→ z = ( x → z)
∨ ( y → z)
for all x, yz , ∈ L.
If a lattice implication algebra L satisfies
∀x, yz , ∈ L , x ∨ y∨(( x∧ y) → z)
= I , then we
call it a lattice H implication algebra.
Definition 2.1([10]). Let L be a lattice implication
algebra. An ideal A is a non-empty subset of L such
that for any x,
y∈ L,
(1)O∈ A;
(2) ( x → y)'
∈ A and y∈ A imply x ∈ A .
Lemma 2.2([10]). Let A be an ideal of a lattice
implication algebra.
If ∀x,
y∈ L, x ≤ y and y ∈ A , then x ∈ A .
Theorem 2.3([10]). Suppose Α is a non-empty
family of ideals of a lattice implication algebra L . Then
I Α is also an ideal of L .
Let A be a subset of a lattice implication algebra L .
The least ideal containing A is called the ideal generated
by A , denoted by p A f.
Specially, if A= {} a , we write p A f=
p a f.
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