IMP -filters of R0-algebras

Yong-lin Liu

Department of MathematicsWuyi University

Wuyishan, Fujian 354300, Chinaylliu9@126.com

Mei-ying Ren

Department of MathematicslWuyi University

Wuyishan, Fujian 354300, Chinanpmeiyingr@163.com

Abstract—The notion of implication MP -filters (briefly,IMP -filters) in R0-algebras is introduced. The characteristicproperties and extension property of IMP -filters are obtained.The relations among IMP -filters, PIMP -filters and NMP -filters of R0-algebras are established. Finally, the implicativeR0-algebra is completely described by its IMP -filters.

Keywords-IMP -filter; implication R0-algebra

I. INTRODUCTION

It is well known that certain information processing,

especially inferences based on certain information, is based

on classical two-valued logic. Due to strict and complete

logical foundation (classical logic), making inferences about

certain information can be done with high confidence levels.

Thus, it is natural and necessary to attempt to establish some

rational logic system as the logical foundation for uncertain

information processing. It is evident that this kind of logic

cannot be two-valued logic itself but might form a certain

extension of two-valued logic. Various kinds of non-classical

logic systems have therefore been extensively researched in

order to construct natural and efficient inference systems to

deal with uncertainty.

In recent years, motivated by both theory and application,

the study of t-norm-based logic systems and the correspond-

ing pseudo-logic systems has been become a greater focus

in the field of logic (cf. [2]-[15]). Here, t-norm-based logical

investigations were first to the corresponding algebraic inves-

tigations, and in the case of pseudo-logic systems, algebraic

development was first to the corresponding logical develop-

ment. The notion of NM -algebras was introduced by Esteva

and Godo [3] from the views of the left-continuous t-norms

and their residua. In [15], Wang proposed the notion of R0-

algebras. Pei [14] proved that R0-algebras and NM -algebras

are the same algebraic structures. In [8], Liu et al. introduced

the notion of implication R0-algebras. In this paper, the

notion of implication MP -filters (briefly, PIMP -filters)

in R0-algebras is introduced. The characteristic properties

and extension property of PIMP -filters are obtained. The

relations among IMP -filters, PIMP -filters and NMP -

filters of R0-algebras are established. Finally, the implicative

R0-algebra is completely described by its IMP -filters.

II. PRELIMINARIES

By an R0-algebra is meant a bounded distributive lattice

(M,∨,∧, 0, 1) with order-reversing involution ” ′ ” and a

binary operation ”→” satisfying the following axioms:

(R1) a′ → b′ = b → a,

(R2) 1 → a = a,

(R3) b → c ≤ (a → b) → (a → c),(R4) a → (b → c) = b → (a → c),(R5) a → (b ∨ c) = (a → b) ∨ (a → c),(R6) (a → b) ∨ ((a → b) → (a′ ∨ b)) = 1,

for all a, b, c ∈ M .

In an R0-algebra, the following hold:

(1) 0 → a = 1, a → 0 = a′, a → a = 1 and a → 1 = 1,

(2) a ≤ b implies b → c ≤ a → c and c → a ≤ c → b,

(3) a → b ≤ (b → c) → (a → c),(4) ((a → b) → b) → b = a → b,

(5) a → (b ∧ c) = (a → b) ∧ (a → c),(6) (a ∨ b) → c = (a → c) ∧ (b → c),(7) (a ∧ b) → c = (a → c) ∨ (b → c),(8) a ≤ b if and only if a → b = 1.

A subset F of an R0-algebra M is called an MP -filter

of M if it satisfies

(F1) 1 ∈ F ,

(F2) x ∈ F and x → y ∈ F imply y ∈ F for all

x, y ∈ M .

An R0-algebra M is called an implication R0-algebra [8]

if it satisfies (x → y) → x = x for all x, y ∈ M .

III. IMP -FILTERS

Definition 1 A subset F of an R0-algebra M is said to

be an implication MP -filter (briefly, IMP -filters) of M if

it satisfies

(F1) 1 ∈ F ,

(F3) z → ((x → y) → x) ∈ F and z ∈ F imply x ∈ Ffor all x, y, z ∈ M .

Example 1. Let M be the chain {0, a, b, 1} with Cayley

tables as follows:

2010 International Conference on Computational Intelligence and Security

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DOI 10.1109/CIS.2010.158

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2010 International Conference on Computational Intelligence and Security

978-0-7695-4297-3/10 $26.00 © 2010 IEEE

DOI 10.1109/CIS.2010.158

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2010 International Conference on Computational Intelligence and Security

978-0-7695-4297-3/10 $26.00 © 2010 IEEE

DOI 10.1109/CIS.2010.158

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→ 0 a b 1 x x′

0 1 1 1 1 0 1

a b 1 1 1 a bb a a 1 1 b a1 0 a b 1 1 0

Define ∨- and ∧-operations on M by x∨y = max{x, y} and

x∧ y = min{x, y} for all x, y ∈ M . By routine calculation

we then can obtain that M is an R0-algebra and F = {b, 1}is an IMP -filter of M . This shows that the IMP -filter

in an R0-algebra exists. But, unit filter {1} is not an

IMP -filter of M because: 1 → ((b → a) → b) = 1 ∈ {1},and 1 ∈ {1}, but b �∈ {1}.

Next, we give a characterization of IMP -filters in R0-

algebras.

Theorem 1 A MP-filter F of an R0-algebra M is an IMP -

filter if and only if it satisfies (x → y) → x ∈ F implies

x ∈ F for all x, y ∈ M .

Proof Suppose that F is an IMP -filter. Let z = 1 in (F3).We have (x → y) → x ∈ F implies x ∈ F . Conversely, if

z → ((x → y) → x) ∈ F and z ∈ F , then (x → y) → x ∈F as F is an MP -filter. By the hypothesis x ∈ F . Hence

(F3) holds and F is an IMP -filter. The proof is complete.

The relation between IMP -filters and MP -filters in an

R0-algebra is as follows:

Proposition 1 An IMP -filter is an MP-filter, but the

converse is not true.

Proof Suppose that F is an IMP -filter. Putting x = y in

(F3), we have z → x ∈ F and z ∈ F imply x ∈ F for all

x, z ∈ M . Hence (F2) holds and F is an MP -filter. The

last part is shown by Example 1, ending the proof.

Proposition 2 Let F be an MP-filter of an R0-algebra M.

If x ≥ y and y ∈ F then x ∈ F .

Proof If x ≥ y then y → x = 1 ∈ F . Combining y ∈ Fthen x ∈ F .

The extension property of IMP -filters in an R0-algebra

is given by the following:

Theorem 2 Let F and H be two MP-filters of an R0-

algebra M with F ⊆ H . If F is an IMP -filter of M, then

so is H.

Proof Suppose that F is an IMP -filter of M and (x →y) → x ∈ H for all x, y ∈ M . Putting t = (x → y) → x,

then (x → y) → (t → x) = t → ((x → y) → x) = 1 ∈ F .

Since

((x → y) → (t → x)) → (((t → x) → y) → (t → x))≥ ((t → x) → y) → (x → y)≥ x → (t → x) = 1,

then ((t → x) → y) → (t → x) ≥ (x → y) → (t → x) ∈F . By Proposition 2, ((t → x) → y) → (t → x) ∈ F . By

Theorem 1, t → x ∈ F ⊆ H. And so x ∈ F as t ∈ H .

Hence H is an IMP -filter of M . This completes the proof.

Proposition 3 Let F be a non-empty subset of an R0-

algebra M. Then F is an MP -filter of M if and only if it

satisfies for all x, y ∈ F and z ∈ M ,

x → z ≥ y implies z ∈ F.Proof Suppose that F is an MP -filter and x, y ∈ F, z ∈

M. If x → z ≥ y, then x → z ∈ F by Proposition 2.

Using (F2) we obtain z ∈ F . Conversely, suppose that for

all x, y ∈ F and z ∈ M , x → z ≥ y implies z ∈ F . Since

F is a non-empty subset of M , we assume x ∈ M . Because

x → 1 = 1 ≥ x, we have 1 ∈ F , and so (F1) holds for F .

Let x → y ∈ F and x ∈ F . Since x → y ≥ x → y ∈ F ,

we have y ∈ F , and so (F2) holds for F . Hence F is an

MP -filter of M . This completes the proof.

Theorem 3 Let M be an R0-algebra. The following are

equivalent:

(i) M is an implication R0-algebra,

(ii) every MP-filter of M is an IMP-filter,

(iii) The unit MP-filter {1} of M is an IMP-filter,

(iv) for all t ∈ M , Ut = {x ∈ M : t ≤ x} is an IMP-filter.

Proof(i)⇒(ii). Let F be an MP -filter of M and (x →y) → x ∈ F . Then x = (x → y) → x ∈ F . By Theorem 1,

F is an IMP -filter.

(ii)⇒(iii). Trivial.

(iii)⇒(iv). Let x ∈ F and x → y ∈ Ut. Then t → x = 1 ∈{1} and t → (x → y) = 1 ∈ {1}. Since (t → (x → y)) →(t → (t → y)) = (x → (t → y)) → (t → (t → y)) ≥ t →x. By Proposition 3 we obtain t → (t → y) ∈ {1}. Since

((t → y) → y) → (t → y) = t → (((t → y) → y) → y) =t → (t → y) ∈ {1}. By Theorem 1, t → y ∈ {1} and so

y ∈ Ut. Hence Ut is an MP -filter of M . Since Ut ⊇ {1}and Theorem 3, Ut is an IMP -filter of M , completing the

proof.

(iv)⇒(i). By (iv), for any x, y ∈ M , U(x→y)→x is an

IMP -filter. Since (x → y) → x ∈ U(x→y)→x, by Theorem

2, x ∈ U(x→y)→x. That is (x → y) → x ≤ x. Clearly,

x ≤ (x → y) → x. Hence x = (x → y) → x. That is M is

an implication R0-algebra. The proof is complete.

Theorem 4 Let F be an MP-filter of an R0-algebra M.

Then F is an IMP-filter if and only if M/F is an implication

R0-algebra.

Proof Suppose that F is an IMP -filter of M . Now we

show that unit MP -filter {C1} of M/F is an IMP -filter.

If (Cx → Cy) → Cx ∈ {C1}, i.e., C(x→y)→x = C1. Hence

1 → ((x → y) → x) ∈ F , i.e., (x → y) → x ∈ F . By

Theorem 1, x ∈ F , i.e., 1 → x ∈ F . On the other hand,

x → 1 = 1 ∈ F . Hence Cx = C1, i.e., Cx ∈ {C1}. Thus

M/F is an implication R0-algebra by Theorem 3 (ii).

Conversely, if M/F is an implication R0-algebra, by The-

orem 3 (ii) {C1} is an IMP -filter. Let (x → y) → x ∈ F ,

i.e., 1 → ((x → y) → x) ∈ F . Since ((x → y) →x) → 1 ∈ F , we have C(x→y)→x = C1 ∈ {C1}, i.e.,

(Cx → Cy) → Cx ∈ {C1}. Hence Cx ∈ {C1}. It means

that 1 → x = x ∈ F . Therefore F is an IMP -filter of M .

The proof is complete.

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ACKNOWLEDGMENT

This work described here is partially supported by the

National Natural Science Foundation of China (Grant no.

60875034,60775038) and the Science and Technology Foun-

dation of Fujian Education Department (Grant no. JA09242).

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