ili-ideals and prime li-ideals in lattice implication algebras

Information Sciences 155 (2003) 157–175

www.elsevier.com/locate/ins

ILI-ideals and prime LI-idealsin lattice implication algebras

Yong Lin Liu a,b,*, San Yang Liu a, Yang Xu c,Ke Yun Qin c

a Department of Applied Mathematics, Xidian University, Xi’an 710071, Shaanxi, PR Chinab Department of Mathematics, Nanping Teachers College, Nanping 353000, Fujian, PR China

c Department of Applied Mathematics, Southwest Jiaotong University, Chengdu 610031,

Sichuan, PR China

Received 27 December 2002; received in revised form 19 May 2003; accepted 26 May 2003

Abstract

The notions of ILI-ideals and maximal LI-ideals of lattice implication algebras are

introduced, respectively. The properties of ILI-ideals, prime LI-ideals and maximal LI-ideals are investigated. Several characterizations of ILI-ideals and prime LI-ideals are

given. The extension theorem of ILI-ideals is obtained. The relations between ILI-ideals

and LI-ideals, between ILI-ideals and implication filter, between prime LI-ideals and

maximal LI-ideals, between prime LI-ideals and ILI-ideals, between maximal LI-ideals

and ILI-ideals, between maximal LI-ideals and LI-ideals, and between prime LI-ideals

and LI-ideals are investigated, respectively. Some classes of lattice implication algebras

are characterized by their ILI-ideals (respectively prime LI-ideals, maximal LI-ideals).

� 2003 Elsevier Inc. All rights reserved.

Keywords: Lattice implication algebra; ILI-ideal; Prime LI-ideal; Maximal LI-ideal

* Corresponding author. Address: Department of Mathematics, Nanping Teachers College,

Nanping 353000, Fujian, PR China.

E-mail addresses: [email protected], [email protected] (Y.L. Liu), [email protected] (S.Y. Liu),

[email protected] (Y. Xu).

0020-0255/$ - see front matter � 2003 Elsevier Inc. All rights reserved.

doi:10.1016/S0020-0255(03)00159-2

mail to: [email protected],

158 Y.L. Liu et al. / Information Sciences 155 (2003) 157–175

1. Introduction

Non-classical logic has become a considerable formal tool for computer

science and artificial intelligence to deal with fuzzy information and uncertaininformation. Many-valued logic, a great extension and development of classical

logic [1], has always been a crucial direction in non-classical logic. In order to

research the many-valued logical system whose propositional value is given in a

lattice, in 1990 Xu [15,9] proposed the concept of lattice implication algebras

and discussed its some properties. Since then this logical algebra has been ex-

tensively investigated by several researchers (see e.g. [3,7,8,10,13,14]). In [11] Xu

and Qin introduced the notion of lattice H implication algebras, which is an

important class of lattice implication algebras. In [12], Xu and Qin introducedthe notions of filters and implicative filters in lattice implication algebras, and

investigated their some properties. In [6], Jun et al. introduced the notions of

positive implicative filters and associative filters in lattice implication algebras,

and investigated their some properties. In [5], Jun et al. defined the notion of LI-ideals in lattice implication algebras and investigated its some properties. In [4],

Jun defined the notion of prime LI-ideals in lattice implication algebras and

investigated its some properties. In this paper, as an extension of above-mention

work we introduce the notions of ILI-ideals and maximal LI-ideals in latticeimplication algebras, and investigate the properties of ILI-ideals, prime LI-ideals and maximal LI-ideals, respectively. In Section 2, we list some basic in-

formation on the lattice implication algebras which is needed for development

of this topic. In Section 3, we introduce the notion of ILI-ideals in lattice im-

plication algebras. We give seven characterizations of ILI-ideals. The extension

theorem of ILI-ideals is established. The relations between ILI-ideals and LI-ideals, between ILI-ideals and implication filters are investigated, respectively.

We give four LI-ideal characterizations of lattice H implication algebras by ILI-ideals or LI-ideals. In Section 4, the notion of maximal LI-ideals is introduced.

We give seven characterizations of prime LI-ideals in lattice implication alge-

bras and eight characterizations of prime LI-ideals in lattice H implication al-

gebras. The relations between prime LI-ideals and maximal LI-ideals, between

prime LI-ideals and ILI-ideals, between maximal LI-ideals and ILI-ideals, be-

tween maximal LI-ideals and LI-ideals, and between prime LI-ideals and LI-ideals are investigated, respectively. We characterize some classes of quotient

lattice implication algebras by prime LI-ideals or maximal LI-ideals.

2. Preliminaries

A bounded lattice ðL;_;^; 0; 1Þ with order-reversing involution 0 and a bi-

nary operation ! is called a lattice implication algebra if it satisfies the fol-lowing axioms:

Y.L. Liu et al. / Information Sciences 155 (2003) 157–175 159

ðI1Þ x ! ðy ! zÞ ¼ y ! ðx ! zÞ,ðI2Þ x ! x ¼ 1,

ðI3Þ x ! y ¼ y0 ! x0,ðI4Þ x ! y ¼ y ! x ¼ 1 ) x ¼ y,ðI5Þ ðx ! yÞ ! y ¼ ðy ! xÞ ! x,ðL1Þ ðx _ yÞ ! z ¼ ðx ! zÞ ^ ðy ! zÞ,ðL2Þ ðx ^ yÞ ! z ¼ ðx ! zÞ _ ðy ! zÞ,

for all x; y; z 2 L.

A lattice implication algebra L is called a lattice H implication algebra if

it satisfies x _ y _ ððx ^ yÞ ! zÞ ¼ 1 for all x; y; z 2 L.In a lattice implication

algebra, we can define a partial ordering 6 by x6 y if and only if x ! y ¼ 1.

In a lattice implication algebra L, the following hold:(1) 0 ! x ¼ 1; x ! 0 ¼ x0; 1 ! x ¼ x and x ! 1 ¼ 1,

(2) x6 y implies y ! z6 x ! z and z ! x6 z ! y,

(3) x ! y6 ðy ! zÞ ! ðx ! zÞ,(4) x6 ðx ! yÞ ! y,

(5) ððx ! yÞ ! yÞ ! y ¼ x ! y,

(6) x _ y ¼ ðx ! yÞ ! y and x ^ y ¼ ððx0 ! y0Þ ! y0Þ0.In a lattice implication algebra L, the following are equivalent:

(7) x ! ðx ! yÞ ¼ x ! y,(8) x ! ðy ! zÞ ¼ ðx ! yÞ ! ðx ! zÞ,(9) x ! ðy ! zÞ ¼ ðx ^ yÞ ! z,

(10) ðx ! yÞ ! x ¼ x,(11) L is a lattice H implication algebra.

For more details of lattice implication algebras we refer the readers to [15]

or [9,11].

Definition 2.1 (Xu and Qin [12]). Let L be a lattice implication algebra. A

subset F of L is called a filter of L if it satisfies

ðF1Þ 1 2 F ,

ðF2Þ x 2 F and x ! y 2 F imply y 2 F .

A subset F of L is called an implicative filter of L, if it satisfies ðF1Þ and

ðF3Þ x ! ðy ! zÞ 2 F and x ! y 2 F imply x ! z 2 F .

Definition 2.2 (Jun et al. [5]). Let L be a lattice implication algebra. A non-

empty subset I of L is called an LI-ideal of L if it satisfies

ðI1Þ 0 2 I ,ðI2Þ ðx ! yÞ0 2 I and y 2 I imply x 2 I .

Theorem 2.3 (Xu and Qin [12]). An implicative filter is a filter in a lattice im-plication algebra.


Theorem 2.4 (Jun et al. [5]). Let I be an LI-ideal of a lattice implication algebraL. If x6 y and y 2 I then x 2 I .

Next we give a property of LI-ideals which will be needed in the sequel.

Theorem 2.5. Let I be a non-empty subset of a lattice implication algebra L.Then I is an LI-ideal of L if and only if it satisfies for all x; y 2 I and z 2 L,

ðz ! xÞ0 6 y implies z 2 I :

Proof. Suppose that I is an LI-ideal and x; y 2 I ; z 2 L. If ðz ! xÞ0 6 y, then

ðz ! xÞ0 2 I by Theorem 2.4. Using ðI2Þ we obtain z 2 I . Conversely, suppose

that for all x; y 2 I and z 2 L, ðz ! xÞ0 6 y implies z 2 I . Since I is a non-emptysubset of L, we assume x 2 I . Because ð0 ! xÞ0 6 x, we have 0 2 I , and so ðI1Þholds for I . Let ðx ! yÞ0 2 I and y 2 I . Since ðx ! yÞ0 6 ðx ! yÞ0, we have x 2 I ,and so ðI2Þ holds for I . Hence I is an LI-ideal of L. This completes the

proof. h

3. ILI-ideals of lattice implication algebras

Definition 3.1. A non-empty subset I of a lattice implication algebra L is said to

be an implicative LI-ideals (briefly, ILI-ideals) of L if it satisfies ðI1Þ and

ðI3Þ ðððx ! yÞ0 ! yÞ0 ! zÞ0 2 I and z 2 I imply ðx ! yÞ0 2 I

for all x; y; z 2 L:

The following example shows that the ILI-ideal in a lattice implication

algebra exists.

Example 3.2. Let L ¼ f0; a; b; 1g be a set with Cayley tables as follows:

Define _- and ^-operations on L as follows:

x _ y ¼ ðx ! yÞ ! y;

x ^ y ¼ ððx0 ! y0Þ ! y0Þ0;

for all x; y 2 L. Then ðL;_;^;!;0 Þ is a lattice implication algebra [15]. It is easy

to check that I1 ¼ f0g, I2 ¼ f0; ag and I3 ¼ f0; bg are all ILI-ideals of L.


The relation between ILI-ideals and LI-ideals in lattice implication algebras

is as follows:

Theorem 3.3. In a lattice implication algebra, any ILI-ideal is an LI-ideal.Conversely, in a lattice H implication algebra, any LI-ideal is an ILI-ideal.

Proof. Suppose that I is an ILI-ideal of a lattice implication algebra. If

ðx ! yÞ0 2 I and y 2 I , then ðððx ! 0Þ0 ! 0Þ0 ! yÞ0 2 I and y 2 I . By ðI3Þðx ! 0Þ0 2 I , i.e., x 2 I . Hence ðI2Þ holds. Combining ðI1Þ, I is an LI-ideal.

Conversely, suppose that I is an LI-ideal of a lattice H implication algebra. If

ðððx ! yÞ0 ! yÞ0 ! zÞ0 2 I and z 2 I for all x; y; z 2 I , by ðI2Þ we have

ððx ! yÞ0 ! yÞ0 2 I , i.e., ðy0 ! ðy0 ! x0ÞÞ0 2 I . By (7), we have ðy0 ! x0Þ0 2 I ,i.e., ðx ! yÞ0 2 I . Hence ðI3Þ holds. Combining ðI1Þ I is an ILI-ideal. The proof

is complete. h

Remark 3.4. In a lattice implication algebra, an LI-ideal may not be an ILI-ideal. It is shown by the following example:

Example 3.5. Let L ¼ f0; a; b; 1g be a set with Cayley tables as follows:

Define _- and ^-operations on L as follows:

x _ y ¼ ðx ! yÞ ! y;

x ^ y ¼ ððx0 ! y0Þ ! y0Þ0;

for all x; y 2 L. Then ðL;_;^;!;0 Þ is a lattice implication algebra [15]. It iseasy to check that I ¼ f0g is an LI-ideal of L, but not an ILI-ideals of Lbecause: ðððb ! aÞ0 ! aÞ0 ! 0Þ0 ¼ ðb0 ! aÞ0 ¼ ða ! aÞ0 ¼ 0 2 f0g; 0 2 f0g, but

ðb ! aÞ0 ¼ b0 ¼ a 62 f0g.

The relation between ILI-ideals and implication filters in lattice implication

algebras is as follows:

Lemma 3.6 (Xu et al. [15]). Let L be a lattice implication algebra, J L. Thefollowing statements are equivalent:ii(i) J is an implicative filter;i(ii) J is a filter and for any x; y; z 2 L; x ! ðy ! zÞ 2 J implies ðx ! yÞ !

ðx ! zÞ 2 J ;(iii) J is a filter and for any x; y 2 L; x ! ðx ! yÞ 2 J implies x ! y 2 J .


Theorem 3.7. Let S be a non-empty subset of a lattice implication algebra L. S0 isdenoted by S0 ¼ fx0 : x 2 Sg. Then S is an implicative filter of L if and only if S0 isan ILI-ideal of L.

Proof. Suppose that S is an implicative filter of L. Then 1 2 S, and so

0 ¼ 10 2 S0. Let ðððx ! yÞ0 ! yÞ0 ! zÞ0 2 S0 and z 2 S0. There exist some u; v 2 Ssuch that u0 ¼ ðððx ! yÞ0 ! yÞ0 ! zÞ0 and v0 ¼ z. Then ððx ! yÞ0 ! yÞ0 ! z ¼u 2 S and z0 ¼ v 2 S, i.e., z0 ! ððx ! yÞ0 ! yÞ 2 S and z0 2 S. It follows that

ðx ! yÞ0 ! y 2 S by S is a filter, i.e., y0 ! ðy 0 ! x0Þ 2 S. By Lemma 3.6 (iii),

y0 ! x0 2 S, i.e., x ! y 2 S, and so ðx ! yÞ0 2 S0. Hence S0 is an ILI-ideal

of L.

Conversely, if S0 is a ILI-ideal of L, then 0 2 S0, so 1 ¼ 00 2 ðS0Þ0 ¼ S. Letx ! y 2 S and x 2 S. Then ðx ! yÞ0 2 S0 and x0 2 S0, i.e., ðy0 ! x0Þ0 2 S0 and

x0 2 S0, and so y 0 2 S0 by S0 is an LI-ideal. Thus y 2 S. Hence S is a filter of L.

Let x ! ðx ! yÞ 2 S. Then ðx ! ðx ! yÞÞ0 2 S0. That is ðððy 0 ! x0Þ0 !x0Þ0 ! 0Þ0 2 S0. Combining 0 2 S0 we have ðy 0 ! x0Þ0 2 S0 by S0 is an ILI-ideal,

i.e., ðx ! yÞ0 2 S0, and so x ! y 2 S. By Lemma 3.6 (iii), S is an implicative

filter. This completes the proof. h

Next, we give some characterizations of ILI-ideals in lattice implication al-gebras.

Theorem 3.8. If I is an LI-ideal of a lattice implication algebra L, then thefollowing are equivalent:v(i) I is an ILI-ideal of L;i(ii) ððx ! yÞ0 ! yÞ0 2 I implies ðx ! yÞ0 2 I for all x; y 2 I ;(iii) ððx ! yÞ0 ! zÞ0 2 I implies ððx ! zÞ0 ! ðy ! zÞ0Þ0 2 I for all x; y; z 2 L;(iv) ðððx ! yÞ0 ! zÞ0 ! uÞ0 2 I and u 2 I imply ððx ! zÞ0 ! ðy ! zÞ0Þ0 2 I for

all x; y; z; u 2 L.

Proof. (i)) (ii). Let I be an ILI-ideal of L and ððx ! yÞ0 ! yÞ0 2 I . Then

ðððx ! yÞ0 ! yÞ0 ! 0Þ0 2 I . Since 0 2 I , by ðI3Þ we have ðx ! yÞ0 2 I .(ii)) (i). Trivial.

(i)) (iii). Let I be an ILI-ideal of L and ððx ! yÞ0 ! zÞ0 2 I for all x; y; z 2 L.

Then ðx ! yÞ0 ! z 2 I 0, that is z0 ! ðy0 ! x0Þ 2 I 0. By Lemma 3.7, I 0 is an im-

plicative filter of L. By Lemma 3.6 (ii) we have ðz0 ! y0Þ ! ðz0 ! x0Þ 2 I 0, that isðx ! zÞ0 ! ðy ! zÞ0 2 I 0. Hence ððx ! zÞ0 ! ðy ! zÞ0Þ0 2 I .

(iii)) (iv). Trivial.

(iv)) (i). Let ðððx ! yÞ0 ! yÞ0 ! zÞ0 2 I and z 2 I . By (iv) we have

ððx ! yÞ0 ! ðy ! yÞ0Þ0 2 I , and so ðx ! yÞ0 2 I . Hence I is an ILI-ideal of L.

The proof is complete. h

The extension theorem of ILI-ideals is obtained by the following theorem.


Theorem 3.9. Let I and A be two LI-ideals of a lattice implication algebra L withA � I . If I is an ILI-ideal of L, then so is A.

Proof. Suppose that I is an ILI-ideal of L and ððx ! yÞ0 ! yÞ0 2 A for allx; y 2 L. Putting t ¼ ððx ! yÞ0 ! yÞ0, then

ðððx ! tÞ0 ! yÞ0 ! yÞ0

¼ ðy 0 ! ðy0 ! ðt0 ! x0ÞÞÞ0 ¼ ðt0 ! ðy0 ! ðy0 ! x0ÞÞÞ0

¼ ðt0 ! ððx ! yÞ0 ! yÞÞ0 ¼ ðððx ! yÞ0 ! yÞ0 ! tÞ0 ¼ 0 2 I :

By Theorem 3.8 (ii), ððx ! tÞ0 ! yÞ0 2 I A, that is ððx ! yÞ0 ! tÞ0 2 A.

Thus ðx ! yÞ0 2 A as A is an LI-ideal. Hence A is an ILI-ideal of L. This

completes the proof. h

Next, we continue to investigate the characterizations of ILI-ideals.

Theorem 3.10. Let I be an LI-ideal of a lattice implication algebra L. Then I isan ILI-ideal of L if and only if for any t 2 L, the subset At ¼ fx 2 L :ðx ! tÞ0 2 Ig is an LI-ideal of L.

Proof. Assume that for any t 2 L, At is an LI-ideal of L. Let ððx ! yÞ0 ! yÞ0 2 I .Then ðx ! yÞ0 2 Ay . Since y 2 Ay and Ay is an LI-ideal, we have x 2 Ay , and so

ðx ! yÞ0 2 I . By Theorem 3.8 (ii) I is an ILI-ideal of L.

Conversely, let I is an ILI-ideal of L and ðx ! yÞ0 2 At, y 2 At. Then

ððx ! yÞ0 ! tÞ0 2 I and ðy ! tÞ0 2 I . Since

ððx ! tÞ0 ! tÞ0 ! ððx ! yÞ0 ! tÞ0

¼ ððx ! yÞ0 ! tÞ ! ððx ! tÞ0 ! tÞP ðx ! tÞ0 ! ðx ! yÞ0

¼ ðx ! yÞ ! ðx ! tÞP y ! t;

then ½ððx ! tÞ0 ! tÞ0 ! ððx ! yÞ0 ! tÞ0�0 6 ðy ! tÞ0. By Theorem 2.5 we obtain

ððx ! tÞ0 ! tÞ0 2 I . By Theorem 3.8 (ii) ðx ! tÞ0 2 I , and so x 2 At. Hence At is

an LI-ideal of L, ending the proof. h

Corollary 3.11. Let I be an ILI-ideal of a lattice implication algebra L. For anya 2 L, Aa ¼ fx 2 L : ðx ! aÞ0 2 Ig is the least LI-ideal of L containing I and a.

Proof. By Theorem 3.10 Aa is an LI-ideal of L. Clearly I Aa and a 2 Aa. If Bis an LI-ideal containing I and a, then for any x 2 Aa we have ðx ! aÞ0 2 I B.

It follows that x 2 B as a 2 B. Hence Aa B, ending the proof. h

Theorem 3.12. Let I be an LI-ideal of a lattice implication algebra L. Then thefollowing are equivalent:


ii(i) I is an ILI-ideal of L;i(ii) ððx ! ðy ! xÞ0Þ0 ! zÞ0 2 I and z 2 I imply x 2 I for all x; y; z 2 L;(iii) ðx ! ðy ! xÞ0Þ0 2 I implies x 2 I for all x; y 2 L.

Proof. (i)) (iii). Suppose that I is an ILI-ideal of L and ðx ! ðy ! xÞ0Þ0 2 I .Since

ððy ! ðy ! xÞ0Þ0 ! ðy ! xÞ0Þ0 ! ðx ! ðy ! xÞ0Þ0

¼ ðx ! ðy ! xÞ0Þ ! ððy ! ðy ! xÞ0Þ0 ! ðy ! xÞ0ÞP ðy ! ðy ! xÞ0Þ0 ! x ¼ x0 ! ðy ! ðy ! xÞ0Þ ¼ x0 ! ððy ! xÞ ! y0Þ¼ ðy ! xÞ ! ðx0 ! y0Þ ¼ 1;

we have ððy ! ðy ! xÞ0Þ0 ! ðy ! xÞ0Þ0 6 ðx ! ðy ! xÞ0Þ0, and so ððy ! ðy !xÞ0Þ0 ! ðy ! xÞ0Þ0 2 I as Theorem 2.4. By Theorem 3.8 (ii) ðy ! ðy ! xÞ0Þ0 2 I .Since

ðx ! ðy ! ðy ! xÞ0Þ0Þ0

¼ ðx ! ððx0 ! y0Þ ! y0Þ0Þ0 ¼ ðx ! ððy0 ! x0Þ ! x0Þ0Þ0

¼ ðððy0 ! x0Þ ! x0Þ ! x0Þ0 ¼ ðy0 ! x0Þ0 ¼ ðx ! yÞ0

and

ðx ! yÞ0 ! ðx ! ðy ! xÞ0Þ0

¼ ðx ! ðy ! xÞ0Þ ! ðx ! yÞP ðy ! xÞ0 ! y ¼ y0 ! ðy ! xÞ ¼ 1;

we obtain ðx ! ðy ! ðy ! xÞ0Þ0Þ0 6 ðx ! ðy ! xÞ0Þ0 2 I , and so ðx ! ðy !ðy ! xÞ0Þ0Þ0 2 I . Hence x 2 I as I is an LI-ideal.

(iii)) (ii). Assume that (iii) holds. If ððx ! ðy ! xÞ0Þ0 ! zÞ0 2 I and z 2 I ,then ðx ! ðy ! xÞ0Þ0 2 I as I is an LI-ideal. Hence x 2 I .

(ii)) (i). If (ii) holds, we have ððx ! ðy ! xÞ0Þ0 2 I implies x 2 I as I is an LI-ideal. Let ððx ! yÞ0 ! yÞ0 2 I . Since

ððx ! yÞ0 ! ðx ! ðx ! yÞ0Þ0Þ0

¼ ððx ! ðx ! yÞ0Þ ! ðx ! yÞÞ0 ¼ ðððx ! yÞ ! x0Þ ! ðy 0 ! x0ÞÞ0

¼ ðy 0 ! ðððx ! yÞ ! x0Þ ! x0ÞÞ0 ¼ ðy0 ! ððx0 ! ðx ! yÞÞ ! ðx ! yÞÞÞ0

¼ ðy 0 ! ððx0 ! ðy 0 ! x0ÞÞ ! ðy0 ! x0ÞÞÞ0

¼ ðy 0 ! ðy0 ! x0ÞÞ0 ¼ ððx ! yÞ0 ! yÞ0 2 I ;

we have ðx ! yÞ0 2 I . By Theorem 3.8 (ii) I is an ILI-ideal of L. The proof is

complete. h

Next, we investigate the relations between lattice H implication algebras and

ILI-ideals, and characterize lattice H implication algebras by ILI-ideals.


Theorem 3.13. Let L be a lattice implication algebra. The following are equiv-alent:

(i) L is a lattice H implication algebra;(ii) Every LI-ideal of L is an ILI-ideal;

(iii) The zero LI-ideal f0g of L is an ILI-ideal;(iv) For all t 2 L, AðtÞ ¼ fx 2 L : ðx ! tÞ0 ¼ 0g is an LI-ideal;(v) For all LI-ideal I of L and all t 2 L, At ¼ fx 2 L : ðx ! tÞ0 2 Ig is an LI-ideal.

Proof. (i)) (ii). It is immediate from Theorem 3.3.

(ii)) (iii). Trivial.

(iii)) (iv). For any t 2 L and ðx ! yÞ0 2 AðtÞ, y 2 AðtÞ, we have

ððx ! yÞ0 ! tÞ0 ¼ 0 2 f0g and ðy ! tÞ0 ¼ 0 2 f0g. Similar to the proof ofTheorem 3.10, we can obtain ððx ! tÞ0 ! tÞ0 2 f0g. By (iii) f0g is an ILI-ideal,

and hence ðx ! tÞ0 2 f0g as Theorem 3.8 (ii). It means that x 2 AðtÞ. Hence AðtÞis an LI-ideal.

(iv)) (i). Let x ! ðx ! yÞ ¼ 1. Then ðx ! yÞ0 ! x0 ¼ 1, and so

ððx ! yÞ0 ! x0Þ0 ¼ 0. Hence ðx ! yÞ0 2 Aðx0Þ, that is ðy0 ! x0Þ 2 Aðx0Þ. Since

Aðx0Þ is an LI-ideal and x0 2 Aðx0Þ, we have y0 2 Aðx0Þ, that is ðy0 ! x0Þ0 ¼ 0, and

so x ! y ¼ 1. Therefore we have show that if x ! ðx ! yÞ ¼ 1 then x ! y ¼ 1

for all x; y 2 L. Now we show that x ! ðx ! yÞ ¼ x ! y for all x; y 2 L. In fact,putting u ¼ x ! ðx ! yÞ, then x ! ðx ! ðu ! yÞÞ ¼ u ! ðx ! ðx ! yÞÞ ¼ 1.

Hence ðx ! ðu ! yÞÞ ¼ 1, and so ðx ! ðx ! yÞÞ ! ðx ! yÞ ¼ 1. On the other

hand, ðx ! yÞ ! ðx ! ðx ! yÞÞ ¼ x ! ððx ! yÞ ! ðx ! yÞÞ ¼ 1. Hence x !ðx ! yÞ ¼ x ! y and L is a lattice H implication algebra.

(i)) (v). For any LI-ideal I of L and t 2 L, by L is a lattice H implication

algebra and Theorem 3.3 we have I is an ILI-ideal of L. Since I At, by

Theorem 3.9 At is an LI-ideal of L.

(v)) (iii). Let ððx ! yÞ0 ! yÞ0 2 f0g. Denote B ¼ fu 2 L : ðu ! yÞ0 2 f0gg.By the hypothesis B is an LI-ideal. Since ðx ! yÞ0 2 B and y 2 B we have x 2 B,

i.e., ðx ! yÞ0 2 f0g. Hence f0g is an ILI-ideal as Theorem 3.8 (ii). The proof is

complete. h

Theorem 3.14. Let I be an LI-ideal of a lattice implication algebra L. Then I isan ILI-ideal if and only if quotient lattice implication algebra ðL=I ;_;^;C0;C1Þ isa lattice H implication algebra.

Proof. Suppose that I is an ILI-ideal of L. In order to prove that L=I is a lattice

H implication algebra, from Theorem 3.13 (iii) it suffices to show that zero LI-ideal f0g of L=I is an ILI-ideal. If ððCx ! CyÞ0 ! CyÞ0 2 fC0g, i.e.,

Cððx!yÞ0!yÞ0 ¼ C0. Hence ðððx ! yÞ0 ! yÞ0 ! 0Þ0 2 I , i.e., ððx ! yÞ0 ! yÞ0 2 I . By

Theorem 3.8 (ii) ðx ! yÞ0 2 I . It follows that ððx ! yÞ0 ! 0Þ0 2 I . On the other

hand, ð0 ! ðx ! yÞ0Þ0 ¼ 0 2 I . Hence Cðx!yÞ0 ¼ C0, i.e., ðCx ! CyÞ0 2 fC0g.

Thus fC0g is an ILI-ideal by Theorem 3.8 (ii) again.


Conversely, if L=I is a lattice H implication algebra, by Theorem 3.13 (iii)

fC0g is an ILI-ideal. Let ððx ! yÞ0 ! yÞ0 2 I , i.e., ðððx ! yÞ0 ! yÞ0 ! 0Þ0 2 I .Since ð0 ! ððx ! yÞ0 ! yÞ0Þ0 ¼ 0 2 I , we have Cððx!yÞ0!yÞ0 ¼ C0 2 fC0g, i.e.,

ððCx ! CyÞ0 ! CyÞ0 2 fC0g. Hence ðCx ! CyÞ0 2 fC0g, i.e., Cðx!yÞ0 ¼ C0, whichmeans that ðx ! yÞ0 2 I . Therefore I is an ILI-ideal of L. The proof is com-

plete. h

Summarizing Theorems 3.8, 3.10, 3.12 and 3.14, we have the following

corollary.

Corollary 3.15. For a lattice implication algebra L and an LI-ideal I of L, thefollowing are equivalent:iii|(i) I is an ILI-ideal of L;ii|(ii) ððx ! yÞ0 ! yÞ0 2 I implies ðx ! yÞ0 2 I for all x; y 2 I ;i|(iii) ððx ! yÞ0 ! zÞ0 2 I implies ððx ! zÞ0 ! ðy ! zÞ0Þ0 2 I for all x; y;

z 2 L;i|(iv) ðððx ! yÞ0 ! zÞ0 ! uÞ0 2 I and u 2 I imply ððx ! zÞ0 ! ðy ! zÞ0Þ0 2 I for

all x; y; z; u 2 L;iii(v) For all t 2 L, the subset At ¼ fx 2 L : ðx ! tÞ0 2 Ig is an LI-ideal of

L;ii(vi) ððx ! ðy ! xÞ0Þ0 ! zÞ0 2 I and z 2 I imply x 2 I for all x; y; z 2 L;i(vii) ðx ! ðy ! xÞ0Þ0 2 I implies x 2 I for all x; y 2 L;(viii) L=I is a lattice H implication algebra.

Definition 3.16 (Burris and Sankappanavar [2]). Let L be a lattice. An ideal Iof L is a non-empty subset of L such that

i(i) x 2 I , y 2 L and y6 x imply that y 2 I ,(ii) x; y 2 I implies x _ y 2 I .This ideal is called a lattice ideal.

Lemma 3.17 (Jun et al. [5]). Every LI-ideal of a lattice implication algebra is alattice ideal. Conversely, in a lattice H implication algebra L, every lattice ideal isan LI-ideal.

Combining Theorem 3.3 and Lemma 3.17 we have following:

Theorem 3.18. Every ILI-ideal of a lattice implication algebra is a lattice ideal.Conversely, in a lattice H implication algebra L, every lattice ideal is an LI-ideal.

Remark 3.19. In a lattice implication algebra, a lattice ideal may not be an ILI-ideal. In fact, in Example 3.5, it is easy to check that I ¼ f0g is a lattice ideal.

But it is not an ILI-ideal.


4. Prime LI-ideals and maximal LI-ideals of lattice implication algebras

Definition 4.1 (Jun [4]). A proper LI-ideal I of a lattice implication algebra L is

said to be a prime LI-ideal of L if x ^ y 2 I implies x 2 I or y 2 I for anyx; y 2 L.

Remark 4.2. In a lattice implication algebra, an LI-ideal or an ILI-ideal may

not be a prime LI-ideal. For example, the ILI-ideal, and hence LI-ideal I1 ¼ f0gof Example 3.2 is not a prime LI-ideal because: ða ^ bÞ ¼ ðða0 ! b0Þ ! b0Þ0 ¼ððb ! aÞ ! aÞ0 ¼ 0 2 f0g, but a 62 f0g and b 62 f0g. By routine calculation, the

ILI-ideals, and hence LI-ideals I2 ¼ f0; ag and I3 ¼ f0; bg of Example 3.2 are

both prime LI-ideals. This shows that the prime LI-ideals of a lattice impli-cation algebra exists.

Now, we investigate the characterizations of prime LI-ideals in lattice im-

plication algebras.

Lemma 4.3 (Jun [4]). Let I be a proper LI-ideal of a lattice implication algebraL. Then I is prime if and only if ðx ! yÞ0 2 I or ðy ! xÞ0 2 I for all x; y 2 L.

Theorem 4.4. Let I be a proper LI-ideal of a lattice implication algebra L. Thefollowing are equivalent:i(i) I is a prime LI-ideals of L;(ii) x ^ y ¼ 0 implies x 2 I or y 2 I for all x; y 2 L.

Proof. (i)) (ii). Trivial.

(ii)) (i). By Theorem 2.1.10 of [15], for any x; y 2 L we have ðx ! yÞ_ðy ! xÞ ¼ 1. Hence ðx ! yÞ0 ^ ðy ! xÞ0 ¼ 0. By (ii) ðx ! yÞ0 2 I or ðy ! xÞ0 2 I .By Lemma 4.3, I is a prime LI-ideal of L, ending the proof. h

Let A be a subset of a lattice implication algebra L. The least LI-ideal

containing A is called the LI-ideal generated by A, written hAi. If A ¼ fag, hfagiis written hai. For shorter, we shall write ½a1; a2; . . . ; an; x� for a1 ! ða2 !ð� � � ! ðan ! xÞ � � �Þ, and write ½an; x� if a1 ¼ a2 ¼ � � � ¼ an ¼ a. By Theorem 2.9

of [4], hAi ¼ fx 2 L : ½a01; a02; . . . ; a0n; x0� ¼ 1 for some a1; a2; . . . ; an 2 Ag and

hai ¼ fx 2 L : ½ða0Þn; x0� ¼ 1 for some n 2 Ng.

Lemma 4.5 (Jun [4]). Let L be a lattice implication algebra and I an LI-ideal.Then a ^ b 2 I implies hI [ fagi \ hI [ fbgi ¼ I .

Theorem 4.6. If L is a lattice implication algebra and a; b 2 L, then

hai \ hbi ¼ ha ^ bi:


Proof. By Lemma 4.5, we want only to show that hfag [ ha ^ bii ¼ hai and

hfbg [ ha ^ bii ¼ hbi. Clearly hai hfag [ ha ^ bii. If x 2 hfag [ ha ^ bii, then

½ðða ^ bÞ0Þn; ½ða0Þm; x0�� ¼ 1 for some m; n 2 N . Since ða ^ bÞ0 P a0, We have

½ða0Þn; ½ða0Þm; x0��P ½ðða ^ bÞ0Þn; ½ða0Þm; x0�� ¼ 1, and so ½ða0Þmþn; x0� ¼ 1. Hence x 2

hai. This proves that hfag [ ha ^ bii ¼ hai. Similarly we have hfbg [ ha ^ bii ¼hbi. The proof is complete. h

Theorem 4.7. Let L be a lattice implication algebra and I a proper LI-ideal of L.Then the following are equivalent:ii(i) I is a prime LI-ideal;i(ii) I1 \ I2 I implies I1 I or I2 I for any LI-ideals I1 and I2 of L;(iii) hai \ hbi I implies a 2 I or b 2 I for any a; b 2 L;(iv) I1 \ I2 ¼ I implies I1 ¼ I or I2 ¼ I for any LI-ideals I1 and I2 of L.

Proof. (i)) (ii). Suppose that (ii) does not hold, then there exist some LI-ideals

I1 and I2 such that I1 \ I2 I but I1 6 I and I2 6 I . Hence there exist a and bsuch that a 2 I1 n I and b 2 I2 n I . Since a ^ b6 a and a ^ b6 b, by Theorem 2.4

a ^ b 2 I1 and a ^ b 2 I2, and so a ^ b 2 I1 \ I2 I . This is a contradiction

of primeness of I .(ii)) (iii). Trivial.(iii)) (iv). If (iv) does not hold, then there exist LI-ideals I1 and I2 such that

I ¼ I1 \ I2 but I 6¼ I1 and I 6¼ I2. Hence there exist a and b such that a 2 I1 n Iand b 2 I2 n I . By (iii) hai \ hbi 6 I . On the other hand, as a ^ b 2 I1 and

a ^ b 2 I2, then ha ^ bi 2 I1 \ I2. By Theorem 4.6, hai \ hbi ¼ ha ^ bi I1 \ I2 ¼ I . This is a contradiction.

(iv)) (i). Suppose that a ^ b 2 I for some a; b 2 L. By Lemma 4.5 I ¼hI [ fagi \ hI [ fbgi. It follows that hI [ fagi ¼ I or hI [ fbgi ¼ I , and so a 2 Ior b 2 I . Hence I is prime. The proof is complete. h

Remark 4.8. From Theorem 4.7 (iv), it does not need to introduce the notion

of irreducible LI-ideals in lattice implication algebras.

Let L be a lattice implication algebra and S L. For any x 2 L, denote by

x�1S ¼ fy 2 L : x ^ y 2 Sg.

Theorem 4.9. An LI-ideal I of a lattice implication algebra L is prime if and onlyif x�1I ¼ I for all x 2 L n I .

Proof. Let I be a prime LI-ideal of L and x 2 L n I . Clearly I 2 x�1I . Let

y 2 x�1I . Then x ^ y 2 I . Since x 62 I and I is prime, we have y 2 I . Hence

x�1I ¼ I . Conversely, suppose that x�1I ¼ I for all x 2 L n I . If y ^ z 2 I and

z 62 I , then z�1I ¼ I , and hence y 2 z�1I ¼ I . It means that I is a prime LI-ideal.

This completes the proof. h


Next, we investigate the characterizations of prime LI-ideals in lattice Himplication algebras.

Lemma 4.10. Let I be an LI-ideal of a lattice implication algebra. Then x 2 I andy 2 I imply x _ y 2 I .

Proof. Since y P 0, then x ! y P x ! 0 ¼ x0, and so ðx ! yÞ0 6 x 2 I . Hence

ðx ! yÞ0 2 I by Theorem 2.4. Since ððx _ yÞ ! yÞ0 ¼ ððx ! yÞ ^ ðy ! yÞÞ0 ¼ðx ! yÞ0 2 I and y 2 I we have x _ y 2 I . The proof is complete. h

Theorem 4.11. Let L be a lattice H implication algebra and I a proper LI-ideal ofL. Then the following are equivalent:ii(i) I is prime;i(ii) x 2 I if and only if x0 62 I for all x 2 L;(iii) I 0 ¼ L n I , where I 0 ¼ fx0 : x 2 Ig;(iv) x 62 I and y 62 I imply ðx ! yÞ0 2 I and ðy ! xÞ0 2 I ;i(v) I1 ¼ L n I , where I1 is the equivalence class containing 1 in L n I .

Proof. (i)) (ii). If (ii) does not hold, then there exists some x 2 L such that

x 2 I and x0 2 I or x 62 I and x0 62 I . By Theorem 2.2.2 of [15]. x _ x0 ¼ 1, and sox0 ^ x ¼ 0. If x 2 I and x0 2 I , by Lemma 4.10 1 ¼ x _ x0 2 I . Hence I ¼ L by

Theorem 2.4. This is a contradiction. If x 62 I and x0 62 I , but x0 ^ x ¼ 0 2 I ,which contradicts to primeness of I .

(ii)) (iii). Let x 2 I 0. Then x0 2 I . By (ii) x 62 I , and so x 2 L n I . Conversely,

let x 2 L n I then x 62 I . By (ii) x0 2 I , i.e, x 2 I 0. Hence I 0 ¼ L n I .(iii)) (iv). If x 62 I and y 62 I , then x 2 I 0 and y 2 I 0 as (iii), and so x0 2 I and

y0 2 I . By Lemma 4.10, x0 _ y 0 2 I . Since x ^ y6 x ! y and x ^ y6 y ! x, we

have x0 _ y 0 ¼ ðx ^ yÞ0 P ðx ! yÞ0 and x0 _ y 0 ¼ ðx ^ yÞ0 P ðy ! xÞ0. Henceðx ! yÞ0; ðy ! xÞ0 2 I .

(iv)) (v). Let x 2 I1. Then x0 2 I , and hence x 62 I . Otherwise 1 ¼ x _ x0 2 Iimplies I ¼ L, a contradiction. This means x 2 L n I . Conversely, if x 2 L n I ,then x 62 I . Since 1 62 I , by (iv) ðx ! 1Þ0 2 I , and ð1 ! xÞ0 2 I . Hence x � 1, and

so x 2 I1. Therefore I1 ¼ L n I .(v)) (i). If (i) does not hold, then there exist some x 62 I and y 62 I such that

x ^ y 2 I . By (v), x 2 I1 and y 2 I1. It follows that x0 2 I and y0 2 I . Since

ðx0 _ y0Þ ! ðx ! y0Þ ¼ ðx0 ! ðx ! y0ÞÞ ^ ðy0 ! ðx ! y 0ÞÞ¼ ðx0 ! ðy ! x0ÞÞ ^ ðx ! ðy 0 ! y 0ÞÞ ¼ 1;

then x0 _ y 0 6 x ! y0, and so x ^ y ¼ ðx0 _ y0Þ0 P ðx ! y0Þ0. By Theorem 2.4

ðx ! y 0Þ0 2 I , and hence x 2 I . By Lemma 4.10 1 ¼ x _ x0 2 I . It follows that

I ¼ L, which is a contradiction. The proof is complete. h


Let A and B be two subsets of a lattice implication algebra L. A ^ B is de-

noted by the set fa ^ b : a 2 A and b 2 Bg. Recall that for a lattice implication

algebra L and t 2 L, AðtÞ ¼ fx 2 L : ðx ! tÞ0 ¼ 0g ¼ fx 2 L : x6 tg.

Lemma 4.12. If L is a lattice implication algebra and s; t 2 L, then

AðsÞ ^ AðtÞ ¼ Aðs ^ tÞ:

Proof. Let x 2 Aðs ^ tÞ. Then x6 s ^ t6 s; t, and so x 2 AðsÞ and x 2 AðtÞ, which

follows that x 2 AðsÞ ^ AðtÞ. Hence Aðs ^ tÞ AðsÞ ^ AðtÞ. Conversely, if

x 2 AðsÞ ^ AðtÞ, then x ¼ x1 ^ x2 where x1 6 s and x2 6 t. It follows that x6 s ^ tand so x 2 Aðs ^ tÞ. Hence AðsÞ ^ AðtÞ Aðs ^ tÞ. This completes the proof. h

If L is a lattice H implication algebra, by Theorem 3.13 and Corollary 3.11

we obtain that AðtÞ is the least LI-ideal of L containing t, which means that

AðtÞ ¼ hti. By Lemma 4.12, we have following:

Corollary 4.13. If L is a lattice H implication algebra and s; t 2 L, then

hsi ^ hti ¼ hs ^ ti:

Theorem 4.14. If I is a proper LI-ideal of a lattice H implication algebra L, thenI is prime if and only if I1 ^ I2 I implies I1 I or I2 I for any LI-ideals I1 andI2 of L.

Proof. Let I is prime and I1 ^ I2 I for some LI-ideals I1 and I2. If I1 6 I andI2 6 I , then there exist a 2 I1 and b 2 I2 such that a 62 I and b 62 I . But

a ^ b 2 I1 ^ I2 I . It is a contradiction of primeness of I . Conversely, let

a ^ b 2 I where a; b 2 L. By Corollary 4.13 we have hai ^ hbi ¼ ha ^ bi I ,which follows that hai I or hbi I , and so a 2 I or b 2 I . The proof is

complete. h

Definition 4.15. A maximal LI-ideal of a lattice implication algebra L is said to

be a proper LI-ideal of L and not a proper subset of any proper LI-ideal of L.

The relations between prime LI-ideals and maximal LI-ideals of lattice im-

plication algebras are given by the following:

Theorem 4.16. In a lattice implication algebra L, any maximal LI-ideal must beprime.

Proof. Let I be maximal in L and x ^ y 2 I . By Lemma 4.5 hI [ fxg i\hI [ fygi ¼ I . Since I hI [ fxgi, and I is maximal, we have hI [ fxgi ¼ I or


hI [ fxgi ¼ L. If hI [ fxgi ¼ I then x 2 I . If hI [ fxgi ¼ L then hI [ fygi ¼ Iand so y 2 I . Hence I is prime. The proof is complete. h

In a lattice H implication algebra, the converse of Theorem 4.16 holds.

Theorem 4.17. If I is a prime LI-ideal of a lattice H implication algebra L, then Iis a maximal LI-ideal of L.

Proof. If I is not maximal then there exists a proper LI-idealAof L such that I Aand I 6¼ A. Hence there exists some a 2 A such that a 62 I . For any x 2 L, we have

ððx ! aÞ0 ^ aÞ0 ¼ ðx ! aÞ _ a0 ¼ ððx ! aÞ ! a0Þ ! a0

¼ ðða0 ! x0Þ ! a0Þ ! a0 ¼ a0 ! a0 ðby ð10ÞÞ ¼ 1:

Then ðx ! aÞ0 ^ a ¼ 0. Hence ðx ! aÞ0 2 I by I is prime, and so x 2 I . It fol-

lows that L ¼ I . This is a contradiction. The proof is complete. h

The relation between prime LI-ideals and ILI-ideals of lattice implication

algebras is given by the following:

Theorem 4.18. Let L be a lattice implication algebra and I a proper LI-ideal ofL. Then I is both a prime LI-ideal and an ILI-ideal of L if and only if for anyx 2 L, x 2 I or x0 2 I .

Proof. Suppose that for all x 2 L, x 2 I or x0 2 I . We first show that I is prime.

Let x ^ y 2 I . If x 62 I , then x0 2 I . Since x0 ! y 0 P x, then ðx0 ! y 0Þ0 6 x 2 I , and

so ðx0 ! y0Þ0 2 I by Theorem 2.4. Since x ^ y ¼ ððx0 ! y 0Þ ! y0Þ0 ¼ ðy ! ðx0 !y0Þ0Þ0 2 I , we have y 2 I by ðI3Þ. Next we show that I is an ILI-ideal. Let

ððx ! yÞ0 ! yÞ0 2 I . If ðx ! yÞ0 62 I , then x ! y 2 I . Since y6 x ! y then y 2 I ,thus ðx ! yÞ0 2 I . This is a contradiction. Hence ðx ! yÞ0 2 I . By Theorem 3.8

(ii), I is an ILI-ideal.

Conversely, suppose that I is both a prime LI-ideal and an ILI-ideal. By

Theorem 2.1.10 of [15], for any x 2 L we have ðx ! x0Þ _ ðx0 ! xÞ ¼ 1, i.e.,

ðx ! x0Þ0 ^ ðx0 ! xÞ0 ¼ 0 2 I . Hence ðx ! x0Þ0 2 I or ðx0 ! xÞ0 2 I . If ðx ! x0Þ0 2I , i.e., ðð1 ! x0Þ0 ! x0Þ0 2 I , we have ð1 ! x0Þ0 2 I by Theorem 3.8 (ii), i.e.,x 2 I . If ðx0 ! xÞ0 2 I , i.e., ðð1 ! xÞ0 ! xÞ0 2 I , we have ð1 ! xÞ0 2 I , i.e., x0 2 I .The proof is complete. h

The relation between maximal LI-ideals and ILI-ideals of lattice implication

algebras is given by the following:

Theorem 4.19. Let L be a lattice implication algebra and I a proper LI-ideal.Then I is both a maximal LI-ideal and an ILI-ideal if and only if for any x; y 2 L,x 62 I and y 62 I imply ðx ! yÞ0 2 I and ðy ! xÞ0 2 I .


Proof. Suppose that for all x; y 2 L, x 62 I and y 62 I imply ðx ! yÞ0 2 I and

ðy ! xÞ0 2 I . We first prove that I is an ILI-ideal. If I is not an ILI-ideal, by

Theorem 3.12 (iii), there exist some x; y 2 L, x 62 I such that ðx ! ðy ! xÞ0Þ0 2 I .If ðy ! xÞ0 2 I , then x 2 I , which is a contradiction. If ðy ! xÞ0 62 I , since x 62 Iwe have y 2 I by the hypothesis. By ðy ! xÞ0 6 y we obtain ðy ! xÞ0 2 I , which

is a contradiction. Hence I is an ILI-ideal of L.

Next, we show that I is maximal. Note that I is also an ILI-ideal, by Cor-

ollary 3.11, for any a 2 L, Aa ¼ fx 2 L : ðx ! aÞ0 2 Ig is the least LI-ideal

containing I and a. In order to show I is maximal, it suffices to show that for

any a 62 I , Aa ¼ L. Let t 2 L. If t 2 I , then t 2 Aa. If t 62 I , combining a 62 I we

have ðt ! aÞ0 2 I Aa, which follows that t 2 Aa as Aa is an LI-ideal. Hence

Aa ¼ L and so I is maximal.Conversely, let x 62 I and y 62 I . Since I is an ILI-ideal, then Ay ¼ ft 2 L :

ðt ! yÞ0 2 Ig is the least LI-ideal containing I and y. By the maximality of I we

obtain Ay ¼ L, and so x 2 Ay . Hence ðx ! yÞ0 2 I . Similarly, we can prove that

ðy ! xÞ0 2 I . The proof is complete. h

Corollary 4.20. If I is an LI-ideal of a lattice H implication algebra L, then I ismaximal if and only if for any x; y 2 L, x 62 I and y 62 I imply ðx ! yÞ0 2 I andðy ! xÞ0 2 I .

Next, we investigate the relations between prime (respectively maximal)

LI-ideals and quotient lattice implication algebras.

Theorem 4.21. Let I be an LI-ideal of a lattice implication algebra L. Then thefollowing are equivalent:ii(i) I is prime;i(ii) L=I is a chain;(iii) L=I has not any zero divisor.

Proof. (i)) (ii). If I is prime, then for any x; y 2 L, ðx ! yÞ0 2 I or ðy ! xÞ0 2 I .Therefore Cx!y ¼ C1 or Cy!x ¼ C1, i.e., Cx 6Cy or Cy 6Cx. (ii)) (iii). Suppose

that Cx ^ Cy ¼ C0 for any Cx;Cy 2 L=I . Then Cx^y ¼ C0 and so ððx ^ yÞ !0Þ0 2 I , i.e., x ^ y 2 I . Since L=I is a chain, we have Cx 6Cy or Cy 6Cx, i.e.,

ðx ! yÞ0 2 I or ðy ! xÞ0 2 I . If ðx ! yÞ0 2 I , by x ^ y ¼ ððy 0 ! x0Þ ! x0Þ0 ¼ðx ! ðx ! yÞ0Þ0 2 I we have x 2 I , i.e., Cx ¼ C0. If ðy ! xÞ0 2 I , by

x ^ y ¼ ððx0 ! y0Þ ! y0Þ0 ¼ ðy ! ðy ! xÞ0Þ0 2 I we have y 2 I , i.e., Cy ¼ C0.

Thus we have proved that L=I has not any zero divisor. (iii)) (i). Since

ðx ! yÞ _ ðy ! xÞ ¼ 1, then ðx ! yÞ0 ^ ðy ! xÞ0 ¼ 0 2 I . It follows that

Cðx!yÞ0^ðy!xÞ0 ¼ C0, i.e., Cðx!yÞ0 ^ Cðy!xÞ0 ¼ C0. Thus Cðx!yÞ0 ¼ C0 or Cðy!xÞ0 ¼ C0,

i.e., ðx ! yÞ0 2 I or ðy ! xÞ0 2 I . Hence I is a prime LI-ideal. This completes the

proof. h


Theorem 4.22. Let I be an LI-ideal of a lattice implication algebra L. Then I ismaximal if and only if L=I is a simple lattice implication algebra.

Proof. It is easy to verify by routine algebraic calculation and is omitted. h

Summarizing Lemma 4.3, Theorems 4.4, 4.7, 4.9 and 4.21, we have the

following:

Corollary 4.23. Let L be a lattice implication algebra and I a proper LI-ideal ofL. Then the following are equivalent:iii|(i) I is a prime LI-ideals of L;ii|(ii) x ^ y ¼ 0 implies x 2 I or y 2 I for all x; y 2 L;i|(iii) ðx ! yÞ0 2 I or ðy ! xÞ0 2 I for all x; y 2 L;ii(iv) I1 \ I2 I implies I1 I or I2 I for any LI-ideals I1 and I2 of L;iii(v) hai \ hbi I implies a 2 I or b 2 I for any a; b 2 L;ii(vi) I1 \ I2 ¼ I implies I1 ¼ I or I2 ¼ I for any LI-ideals I1 and I2 of L;i(vii) x�1I ¼ I for all x 2 L n I ;(viii) L=I is a chain;ii(ix) L=I has not any zero divisor.

Summarizing Theorems 4.11, 4.14, 4.16–4.18 and 4.22, we have the following:

Corollary 4.24. Let L be a lattice H implication algebra and I a proper LI-idealof L. Then the following are equivalent:iii|(i) I is a prime LI-ideal;ii|(ii) x 2 I if and only if x0 62 I for all x 2 L;i|(iii) I 0 ¼ L n I , where I 0 ¼ fx0 : x 2 Ig;ii(iv) x 62 I and y 62 I imply ðx ! yÞ0 2 I and ðy ! xÞ0 2 I ;iii(v) I1 ¼ L n I , where I1 is the equivalence class containing 1 in L n I ;ii(vi) A ^ B I implies A I or B I for any LI-ideals A and B of L;i(vii) I is a maximal LI-ideal;(viii) x 2 I or x0 2 I for all x 2 L;ii(ix) L=I is a simple lattice implication algebra.

The following two theorems show the relations between maximal LI-ideals

and LI-ideals, and between prime LI-ideals and LI-ideals in a lattice implicationalgebra.

Theorem 4.25. Every proper LI-ideal in a lattice implication algebra L is con-tained in a maximal LI-ideal.

Proof. Assume that I is a proper LI-ideal of L, then 1 62 I . Let IðLÞ be denoted

by the set of all LI-ideals in L and M ¼ fU 2 IðLÞ : I U and 1 62 Ug. By


Zorn�s Lemma, M has a maximal element J . Clearly J is a maximal LI-ideal of

L. The proof is complete. h

Lemma 4.26 (Jun [4]). (Prime LI-ideal theorem) Let L be a lattice implicationalgebra, I an LI-ideal of L, S a non-empty subset of L with ^-closed andI \ S ¼ Ø. Then there exists a prime LI-ideal J such that I J and J \ S ¼ Ø.

Theorem 4.27. Every proper LI-ideal in a lattice implication algebra L is theintersection of prime LI-ideals.

Proof. Suppose that I is a proper LI-ideal of L. By Theorem 4.25, I is contained

in a maximal LI-ideal, and hence I is contained in a prime LI-ideal. Let P ðLÞ bedenoted by the set of all prime LI-ideals in L. Then I \fP 2 P ðLÞ : I Pg. If

x 2 \fP 2 P ðLÞ : I Pg n I , then by the prime LI-ideal theorem there exists

J 2 P ðLÞ such that I J and J \ hxi ¼ Ø. But x 2 \fP 2 P ðLÞ : I Pg J ,

which is a contradiction. This completes the proof. h

5. Conclusion

In order to research the many-valued logical system whose propositional

value is given in a lattice, Xu initiated the concept of lattice implication alge-

bras. Hence for development of this many-valued logical system, it is needed to

make clear the structure of lattice implication algebras. It is well known that to

investigate the structure of an algebraic system, the ideals with special prop-

erties play an important role. In this paper, we proposed the notions of ILI-ideals and maximal LI-ideals in lattice implication algebras, discussed the

properties of ILI-ideals, prime LI-ideals and maximal LI-ideals respectively,established the relations of various LI-ideals and completely characterized

some corresponding classes of lattice implication algebras by these LI-ideals. It

hope above work would serve as a foundation for further study the structure of

lattice implication algebras and develop corresponding many-valued logical

system.

Acknowledgements

Authors would like to express their sincere thanks to the referees for their

valuable suggestions and comments.

This work was supported by the National Natural Science Foundation of

P.R. China (Grant no. 69972036) and the Natural Science Foundation ofFujian.


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