# The Axioms Independence of Pseudo-Weak-R

Authors: Yong Lin Liu
Journal: The Scientific World Journal
 (2014)

---

## Abstract

The most simplified axiom systems of pseudo-weak-R
0 algebras and pseudo-R
0 algebras are obtained, and the mutually independence of axioms is proved. We introduce the notions of filters and normal filters in pseudo-weak-R
0 algebras. The structures and properties of the generated filters and generated normal filters in pseudo-weak-R
0 algebras are obtained. These can be seen as noncommutative generalizations of the corresponding ones in weak-R
0 algebras.


---

## Body

## 1. Introduction

In recent years, the study of logic algebras and their noncommutative generalization—pseudo-logic algebras—has become of greater focus in the field of logic. BCK and BCI algebras were introduced by Imai and Iseki [1] and have been extensively investigated by many researchers. Georgescu and Iorgulescu [2] introduced the notion of a pseudo-BCK algebra as a noncommutative generalization of a BCK-algebra. Liu et al. [3] investigated the theory of pseudo-BCK algebras. MV-algebras were introduced by Chang in [4] as an algebraic tool to study the infinitely valued logic of Lukasiewicz. Georgescu and Iorgulescu [5] introduced pseudo MV-algebras which is a noncommutative generalization of MV-algebras. The notion of BL-algebras was introduced by Hajek [6] as the algebraic structures for his Basic Logic. Georgescu and Iorgulescu [7] introduced the notion of pseudo-BL algebras by dropping commutative axioms in BL-algebras. di Nola et al. [8, 9], Zhang and Fan [10], and Zhan et al. [11] investigated in detail the theory of pseudo-BL algebras. MTL-algebras [12] are the algebraic structures for Esteva-Godo monoidal t-norm based logic, many-valued propositional calculus that formalizes the structure of the real unit interval [
0,1
], induced by a left-continuous t-norm. Flondor et al. [13] presented pseudo-MTL algebras as a noncommutative generalization of MTL-algebras. IMTL-algebras [12] are the algebraic counterpart for involutive monoidal t-norm logic, an extension of MTL-algebras. NM-algebras [12] are the algebraic counterpart for nilpotent minimum logic, an extension of IMTL-algebras. Iorgulescu [14] and Liu and zhang [15] introduced and studied the pseudo-IMTL algebras and pseudo-NM algebras. R
0 algebras were introduced by Wang [16] as the algebraic structure for his formal deductive system L
* of fuzzy propositional calculus. Weak-R
0 algebras [16] are the generalization of R
0 algebras. The research on R
0 algebras has attracted more and more attention [17].In [18], we introduced and studied the pseudo-weak-R
0 algebras and pseudo-R
0 algebras. They are noncommutative generalizations of the weak-R
0 algebras and R
0 algebras, respectively. Some properties, the noncommutative forms of the properties in weak-R
0 algebras and R
0 algebras, were investigated. We showed that pseudo-weak-R
0 algebras are categorically isomorphic to pseudo-IMTL algebras, and pseudo-R
0 algebras are categorically isomorphic to pseudo-NM algebras.Based on these results, in this paper, our study focused on the axioms independence and filter theory in pseudo-weak-R
0 algebras and pseudo-R
0 algebras. The most simplified axiom systems of pseudo-weak-R
0 algebras and pseudo-R
0 algebras are obtained, and the mutually independence of axioms is proved. The notions of filters and normal filters in pseudo-weak-R
0 algebras are introduced. The structures and properties of the generated filters and generated normal filters in pseudo-weak-R
0 algebras are obtained. These can be seen as noncommutative generalizations of the corresponding ones in weak-R
0 algebras.


## 2. Preliminaries

We recall some definitions and results which will be used in the sequel.Definition 1 (see [12]).
An IMTL (involutive MTL) algebra is a structure(
A
,
∨
,
∧
,
⊙
,
→
,
0,1
) of type (
2,2
,
2,2
,
0,0
) such that for all x
,
y
,
z
∈
A:(B1)
(
A
,
∨
,
∧
,
0,1
) is a bounded lattice,
(B2)
(
A
,
⊙
,
1
) is a monoid,
(B3)
x
⊙
y
≤
z if and only if x
≤
y
→
z,
(B4)
(
x
→
y
)
∨
(
y
→
x
)
=
1,
(B5)
x
-
-
=
x,where x
-
=
x
→
0.
An NM (nilpotent minimum) algebra is an IMTL algebra satisfying the following condition:(B6)
(
x
⊙
y
)
-
∨
(
(
x
∧
y
)
→
(
x
⊙
y
)
)
=
1.Definition 2 (see [14, 15]).
A pseudo-IMTL (pseudo-involutive MTL) algebra is a structure(
A
,
∨
,
∧
,
⊙
,
→
,
⇝
,
0,1
) of type (
2,2
,
2,2
,
2,0
,
0
) such that for all x
,
y
,
z
∈
A:(pB1)
(
A
,
∨
,
∧
,
0,1
) is a bounded lattice,
(pB2)
(
A
,
⊙
,
1
) is a monoid,
(pB3)
x
⊙
y
≤
z if and only if x
≤
y
→
z if and only if y
≤
x
⇝
z,
(pB4)
(
x
→
y
)
∨
(
y
→
x
)
=
(
x
⇝
y
)
∨
(
y
⇝
x
)
=
1,
(pB5)
x
~
-
=
x
-
~
=
x,where x
-
=
x
→
0 and x
~
=
x
⇝
0.
A pseudo-NM (pseudo-nilpotent minimum) algebra is a pseudo-IMTL algebra satisfying the following condition:(pB6)
(
x
⊙
y
)
-
∨
(
(
x
∧
y
)
→
(
x
⊙
y
)
)
=
(
x
⊙
y
)
~
∨
(
(
x
∧
y
)
⇝
(
x
⊙
y
)
)
=
1.Definition 3 (see [16, 19]).
LetM be a (
¬
,
∧
,
∨
,
→
)-type algebra, where ¬ is a unary operation and ∧, ∨, and → are binary operations. If there is a partial ordering ≤ on M, such that (
M
,
≤
) is a bounded distributive lattice, ∧ and ∨ are infimum and supremum operations with respect to ≤, ¬ is an order-reversing involution with respect to ≤, and the following conditions hold for any a
,
b
,
c
∈
M
(R1)
¬
a
→
¬
b
=
b
→
a,
(R2)
1
→
a
=
a, a
→
a
=
1,
(R3)
b
→
c
≤
(
a
→
b
)
→
(
a
→
c
),
(R4)
a
→
(
b
→
c
)
=
b
→
(
a
→
c
),
(R5)
a
→
(
b
∨
c
)
=
(
a
→
b
)
∨
(
a
→
c
), a
→
(
b
∧
c
)
=
(
a
→
b
)
∧
(
a
→
c
),where 1 is the largest element of M, and then we call M a weak-R
0 algebra.
AnR
0 algebra M is a weak-R
0 algebra satisfying the additional condition as follows:(R6)
(
a
→
b
)
∨
(
(
a
→
b
)
→
(
¬
a
∨
b
)
)
=
1.Definition 4 (see [18]).
A pseudo-weak-R
0 algebra is a structure(
A
,
∧
,
∨
,
→
,
⇝
,
-
,
~
,
0,1
) such that (
A
,
∧
,
∨
,
0,1
) is a bounded distributive lattice, - and ~ are order-reversing pseudo-involution (i.e., if x
≤
y, then y
-
≤
x
- and y
~
≤
x
~; x
~
-
=
x
-
~
=
x), and the following axioms hold for any x
,
y
,
z
∈
A:(pR1)
x
→
y
=
y
-
⇝
x
-, x
⇝
y
=
y
~
→
x
~,
(pR2*)
1
→
x
=
1
⇝
x
=
x; x
→
x
=
x
⇝
x
=
1,
(pR3)
x
→
y
≤
(
z
→
x
)
→
(
z
→
y
), x
⇝
y
≤
(
z
⇝
x
)
⇝
(
z
⇝
y
),
(pR4)
x
→
(
y
⇝
z
)
=
y
⇝
(
x
→
z
),
(pR5*)
x
→
(
y
∨
z
)
=
(
x
→
y
)
∨
(
x
→
z
)
,
x
⇝
(
y
∨
z
)
=
(
x
⇝
y
)
∨
(
x
⇝
z
); x
→
(
y
∧
z
)
=
(
x
→
y
)
∧
(
x
→
z
)
,
x
⇝
(
y
∧
z
)
=
(
x
⇝
y
)
∧
(
x
⇝
z
).
A pseudo-R
0 algebra A is a pseudo-weak-R
0 algebra satisfying the additional axiom as follows:(pR6)
(
x
→
y
)
∨
(
(
x
→
y
)
⇝
(
x
-
∨
y
)
)
=
(
x
⇝
y
)
∨
(
(
x
⇝
y
)
→
(
x
~
∨
y
)
)
=
1.
In [18], we also have another simplified definition.Definition 5 (see [18]).
A pseudo-weak-R
0 algebra is a structure (
A
,
∧
,
∨
,
→
,
⇝
,
-
,
~
,
0,1
) satisfying(pL1)
(
A
,
∧
,
∨
,
0,1
) is a bounded lattice,
(pL2)
ifx
≤
y, then y
-
≤
x
- and y
~
≤
x
~,
(pL3)
x
~
-
=
x
-
~
=
x,
(pR1)
x
→
y
=
y
-
⇝
x
-, x
⇝
y
=
y
~
→
x
~,
(pR2)
1
→
x
=
1
⇝
x
=
x,
(pR3)
x
→
y
≤
(
z
→
x
)
→
(
z
→
y
), x
⇝
y
≤
(
z
⇝
x
)
⇝
(
z
⇝
y
),
(pR4)
x
→
(
y
⇝
z
)
=
y
⇝
(
x
→
z
),
(pR5)
x
→
(
y
∨
z
)
=
(
x
→
y
)
∨
(
x
→
z
), x
⇝
(
y
∨
z
)
=
(
x
⇝
y
)
∨
(
x
⇝
z
).
A pseudo-R
0 algebra A is a pseudo-weak-R
0 algebra satisfying the additional axiom as follows:(pR6)
(
x
→
y
)
∨
(
(
x
→
y
)
⇝
(
x
-
∨
y
)
)
=
(
x
⇝
y
)
∨
(
(
x
⇝
y
)
→
(
x
~
∨
y
)
)
=
1.Proposition 6 (see [18]).
In a pseudo-weak-
R
0
algebra, the following properties hold:
(1)
0
~
=
0
-
=
1
,
1
~
=
1
-
=
0
,
(2)
x
-
=
x
→
0
,
x
~
=
x
⇝
0
,
(3)
x
⇝
x
=
x
→
x
=
1
,
(4)
x
≤
y
if and only if
x
⇝
y
=
1
if and only if
x
→
y
=
1
,
(5)
(
⋀
i
∈
I
x
i
)
~
=
⋁
i
∈
I
x
i
~
,
(
⋀
i
∈
I
x
i
)
-
=
⋁
i
∈
I
x
i
-
, whenever the arbitrary meets and unions exist,
(6)
(
⋁
i
∈
I
x
i
)
~
=
⋀
i
∈
I
x
i
~
,
(
⋁
i
∈
I
x
i
)
-
=
⋀
i
∈
I
x
i
-
, whenever the arbitrary meets and unions exist,
(7)
if
x
≤
y
, then
z
⇝
x
≤
z
⇝
y
and
z
→
x
≤
z
→
y
,
(8)
if
x
≤
y
, then
y
⇝
z
≤
x
⇝
z
and
y
→
z
≤
x
→
z
,
(9)
(
x
∧
y
)
⇝
z
=
(
x
⇝
z
)
∨
(
y
⇝
z
)
,
(
x
∧
y
)
→
z
=
(
x
→
z
)
∨
(
y
→
z
)
,
(10)
x
⇝
(
y
∧
z
)
=
(
x
⇝
y
)
∧
(
x
⇝
z
)
,
x
→
(
y
∧
z
)
=
(
x
→
y
)
∧
(
x
→
z
)
,
(11)
(
x
∨
y
)
⇝
z
=
(
x
⇝
z
)
∧
(
y
⇝
z
)
,
(
x
∨
y
)
→
z
=
(
x
→
z
)
∧
(
y
→
z
)
,
(12)
x
~
∨
y
≤
x
⇝
y
,
x
-
∨
y
≤
x
→
y
,
(13)
x
⇝
y
≤
(
y
⇝
z
)
→
(
x
⇝
z
)
,
x
→
y
≤
(
y
→
z
)
⇝
(
x
→
z
)
,
(14)
(
A
,
∧
,
∨
,
0,1
)
is a bounded distributive lattice,
(15)
x
⇝
y
≤
x
∨
z
⇝
y
∨
z
,
x
→
y
≤
x
∨
z
→
y
∨
z
,
(16)
x
⇝
y
≤
x
∧
z
⇝
y
∧
z
,
x
→
y
≤
x
∧
z
→
y
∧
z
,
(17)
(
x
⇝
y
)
≤
(
x
⇝
z
)
∨
(
z
⇝
y
)
,
(
x
→
y
)
≤
(
x
→
z
)
∨
(
z
→
y
)
,
(18)
(
x
⇝
y
)
∨
(
y
⇝
x
)
=
(
x
→
y
)
∨
(
y
→
x
)
=
1
,
(19)
x
≤
(
x
→
y
)
⇝
y
,
x
≤
(
x
⇝
y
)
→
y
,
(20)
x
→
y
=
(
(
x
→
y
)
⇝
y
)
→
y
,
x
⇝
y
=
(
(
x
⇝
y
)
→
y
)
⇝
y
,
(21)
x
→
(
y
→
x
)
=
x
⇝
(
y
⇝
x
)
=
x
⇝
(
y
→
x
)
=
x
→
(
y
⇝
x
)
=
1
,
(22)
x
-
→
(
x
→
y
)
=
x
~
⇝
(
x
⇝
y
)
=
x
-
⇝
(
x
→
y
)
=
x
~
→
(
x
⇝
y
)
=
1
,
(23)
y
≤
(
x
⇝
y
)
∧
(
x
→
y
)
,
(24)
x
∨
y
=
(
(
x
⇝
y
)
→
y
)
∧
(
(
y
⇝
x
)
→
x
)
=
(
(
x
→
y
)
⇝
y
)
∧
(
(
y
→
x
)
⇝
x
)
,
(25)
(
x
∨
y
)
→
x
=
y
→
x
,
(
x
∨
y
)
⇝
x
=
y
⇝
x
,
(26)
x
→
(
x
∧
y
)
=
x
→
y
,
x
⇝
(
x
∧
y
)
=
x
⇝
y,
(27)
x
≤
y
-
if and only if
y
≤
x
~
,
(28)
x
→
y
~
=
y
⇝
x
-
,
x
⇝
y
-
=
y
→
x
~,
(29)
(
x
→
y
-
)
~
=
(
y
⇝
x
~
)
-
.
In a pseudo-weak-
R
0
algebra (pseudo-
R
0
algebra)
A
, we define a binary operation
⊙
as follows, for any
x
,
y
∈
A
:
(30)
(1)
x
⊙
y
=
(
x
⟶
y
-
)
~
=
(
y
⇝
x
~
)
-
.Proposition 7 (see [18]).
In a pseudo-weak-
R
0
algebra, the following properties hold:
(31)
x
→
y
=
(
x
⊙
y
~
)
-
,
x
⇝
y
=
(
y
-
⊙
x
)
~
,
(32)
(
x
⊙
y
)
⊙
z
=
x
⊙
(
y
⊙
z
)
,
(33)
1
⊙
x
=
x
⊙
1
=
x
,
(34)
x
⊙
y
≤
z
if and only if
x
≤
y
→
z
if and only if
y
≤
x
⇝
z
,
(35)
x
⊙
(
x
⇝
y
)
≤
y
≤
x
⇝
(
x
⊙
y
)
,
(
x
→
y
)
⊙
x
≤
y
≤
x
→
(
y
⊙
x
)
,
(36)
x
⊙
(
x
⇝
y
)
≤
x
≤
y
⇝
(
y
⊙
x
)
,
(
x
→
y
)
⊙
x
≤
x
≤
y
→
(
x
⊙
y
)
,
(37)
if
x
≤
y
, then
x
⊙
z
≤
y
⊙
z
and
z
⊙
x
≤
z
⊙
y
,
(38)
x
⊙
(
x
⇝
y
)
≤
x
∧
y
,
(
x
→
y
)
⊙
x
≤
x
∧
y
,
(39)
x
⊙
0
=
0
⊙
x
=
0
,
(40)
x
⊙
(
⋁
i
∈
I
x
i
)
=
⋁
i
∈
I
(
x
⊙
x
i
)
,
(
⋁
i
∈
I
x
i
)
⊙
x
=
⋁
i
∈
I
(
x
i
⊙
x
)
, whenever the arbitrary unions exist,
(41)
(
x
⊙
y
)
→
z
=
x
→
(
y
→
z
)
,
(
y
⊙
x
)
⇝
z
=
x
⇝
(
y
⇝
z
)
,
(42)
y
⇝
(
⋀
i
∈
I
x
i
)
=
⋀
i
∈
I
(
y
⇝
x
i
)
,
y
→
(
⋀
i
∈
I
x
i
)
=
⋀
i
∈
I
(
y
→
x
i
)
, whenever the arbitrary meets exist,
(43)
(
⋁
i
∈
I
x
i
)
⇝
y
=
⋀
i
∈
I
(
x
i
⇝
y
)
,
(
⋁
i
∈
I
x
i
)
→
y
=
⋀
i
∈
I
(
x
i
→
y
)
, whenever the arbitrary unions and meets exist,
(44)
x
⊙
x
~
=
x
-
⊙
x
=
0
,
(45)
x
⊙
y
≤
x
∧
y
≤
x
,
y
,
(46)
x
∨
(
y
⊙
z
)
≥
(
x
∨
y
)
⊙
(
x
∨
z
)
,
(47)
x
→
y
≤
(
x
⊙
z
)
→
(
y
⊙
z
)
,
x
⇝
y
≤
(
z
⊙
x
)
⇝
(
z
⊙
y
)
,
(48)
x
⊙
(
y
→
z
)
≤
y
→
(
x
⊙
z
)
,
(
y
⇝
z
)
⊙
x
≤
y
⇝
(
z
⊙
x
)
.


## 3. The Axioms Independence of Pseudo-Weak-R
0 Algebras

We investigate the axioms independence of pseudo-R
0 algebras and pseudo-weak-R
0 algebras. Hence, we obtain most simplified axiom systems of pseudo-weak-R
0 algebras and pseudo-R
0 algebras.Theorem 8.
A structure(
A
,
∨
,
∧
,
→
,
⇝
,
-
,
~
,
0,1
) is a pseudo-weak-R
0 algebra if and only if it satisfies the following conditions:(pL1)
(
A
,
∧
,
∨
,
0,1
) is a bounded lattice,
(pL3′)
1
~
-
=
1
-
~
=
1, 0
~
-
=
0
-
~
=
0,
(pR1)
x
→
y
=
y
-
⇝
x
-, x
⇝
y
=
y
~
→
x
~,
(pR2)
1
→
x
=
1
⇝
x
=
x,
(pR3)
x
→
y
≤
(
z
→
x
)
→
(
z
→
y
), x
⇝
y
≤
(
z
⇝
x
)
⇝
(
z
⇝
y
),
(pR5)
x
→
(
y
∨
z
)
=
(
x
→
y
)
∨
(
x
→
z
), x
⇝
(
y
∨
z
)
=
(
x
⇝
y
)
∨
(
x
⇝
z
).Proof.
Necessity is obvious. For sufficiency, it only needs to show axioms (pL2), (pL3), and (pR4) of Definition5 hold. We first show the following three properties hold:(a)
x
→
y
≤
(
y
→
z
)
⇝
(
x
→
z
), x
⇝
y
≤
(
y
⇝
z
)
→
(
x
⇝
z
),
(b)
x
→
x
=
x
⇝
x
=
1,
(c)
x
≤
y if and only if x
→
y
=
1 if and only if x
⇝
y
=
1.
In fact, by (pR1) and (pR3), we havex
→
y
=
y
-
⇝
x
-
≤
(
z
-
⇝
y
-
)
⇝
(
z
-
⇝
x
-
)
=
(
y
→
z
)
⇝
(
x
→
z
), x
⇝
y
=
y
~
→
x
~
≤
(
z
~
→
y
~
)
→
(
z
~
→
x
~
)
=
(
y
⇝
z
)
→
(
x
⇝
z
).
By (a) and (pR2), we have1
=
1
→
1
≤
(
1
⇝
x
)
→
(
1
⇝
x
)
=
x
→
x, and so x
→
x
=
1. Similarly, x
⇝
x
=
1.
Ifx
≤
y, by (pR5) and (b), we have x
→
y
=
x
→
x
∨
y
=
(
x
→
x
)
∨
(
x
→
y
)
=
1. Conversely, if x
→
y
=
1, by (pR2) and (a), we have x
=
1
⇝
x
≤
(
x
→
y
)
⇝
(
1
→
y
)
=
1
⇝
y
=
y. Similarly, x
≤
y if and only if x
⇝
y
=
1.
(pL2): by (c) and (pR1),x
≤
y if and only if x
→
y
=
1 if and only if y
-
⇝
x
-
=
1 if and only if y
-
≤
x
-. Similarly, x
≤
y if and only if x
⇝
y
=
1 if and only if y
~
→
x
~
=
1 if and only if y
~
≤
x
~.
(pL3): since0
≤
1
-, by (pL2), 1
-
~
≤
0
~. By (pL3′), 1
≤
0
~; thus 1
=
0
~ and 1
-
=
0
~
-
=
0. Similarly, 1
=
0
- and 1
~
=
0.
By (pR2) and (pR1),x
=
1
→
x
=
x
-
⇝
0, and so x
-
~
=
x
-
~
-
⇝
0
=
1
→
x
-
~
=
x
-
⇝
0
=
x. Hence, x
-
~
=
x. Similarly, we have x
~
-
=
x.
(pR4): by (pL2) and (pL3), it is easy to verify that pseudo-Kleene dual law holds:(d)
(
x
∧
y
)
~
=
x
~
∨
y
~, (
x
∨
y
)
~
=
x
~
∧
y
~, (
x
∧
y
)
-
=
x
-
∨
y
-, and (
x
∨
y
)
-
=
x
-
∧
y
-.
By (pR1), (pR5), and (d),x
∧
y
→
z
=
z
-
⇝
(
x
∧
y
)
-
=
z
-
⇝
x
-
∨
y
-
=
(
z
-
⇝
x
-
)
∨
(
z
-
⇝
y
-
)
=
(
x
→
z
)
∨
(
y
→
z
). Similarly, we have x
∧
y
⇝
z
=
(
x
⇝
z
)
∨
(
y
⇝
z
).
Ifx
≤
y, then y
→
z
≤
(
x
→
z
)
∨
(
y
→
z
)
=
x
∧
y
→
z
=
x
→
z and y
⇝
z
≤
(
x
⇝
z
)
∨
(
y
⇝
z
)
=
x
∧
y
⇝
z
=
x
⇝
z.
Now we prove that (pR4) holds. Sincex
=
1
→
x
≤
(
x
→
z
)
⇝
(
1
→
z
)
=
(
x
→
z
)
⇝
z, x
→
(
y
⇝
z
)
≥
(
(
x
→
z
)
⇝
z
)
→
(
y
⇝
z
)
≥
y
⇝
(
x
→
z
). Hence, x
→
(
y
⇝
z
)
=
y
⇝
(
x
→
z
).Corollary 9.
A structure(
A
,
∨
,
∧
,
→
,
⇝
,
-
,
~
,
0,1
) is a pseudo-R
0 algebra if and only if it satisfies (pL1), (pL3′), (pR1), (pR2), (pR3), (pR5), and(pR6)
(
x
→
y
)
∨
(
(
x
→
y
)
⇝
(
x
-
∨
y
)
)
=
(
x
⇝
y
)
∨
(
(
x
⇝
y
)
→
(
x
~
∨
y
)
)
=
1.
According to Theorem8 and Corollary 9, one obtains most simplified definitions of pseudo-weak-R
0 algebras and pseudo-R
0 algebras, as the axiom systems are mutually independence (see Theorem 11).Definition 10.
A pseudo-weak-R
0 algebra is a structure (
A
,
∧
,
∨
,
→
,
⇝
,
-
,
~
,
0,1
) such that (
A
,
∧
,
∨
,
0,1
) is a bounded lattice and (
1
~
)
-
=
(
1
-
)
~
=
1 and (
0
~
)
-
=
(
0
-
)
~
=
0, satisfying the following axioms:(P1)
x
→
y
=
y
-
⇝
x
-, x
⇝
y
=
y
~
→
x
~,
(P2)
1
→
x
=
1
⇝
x
=
x,
(P3)
x
→
y
≤
(
z
→
x
)
→
(
z
→
y
), x
⇝
y
≤
(
z
⇝
x
)
⇝
(
z
⇝
y
),
(P4)
x
→
(
y
∨
z
)
=
(
x
→
y
)
∨
(
x
→
z
), x
⇝
(
y
∨
z
)
=
(
x
⇝
y
)
∨
(
x
⇝
z
).
A pseudo-R
0 algebra A is a pseudo-weak-R
0 algebra satisfying the additional axiom as follows:(P5)
(
x
→
y
)
∨
(
(
x
→
y
)
⇝
(
x
-
∨
y
)
)
=
(
x
⇝
y
)
∨
(
(
x
⇝
y
)
→
(
x
~
∨
y
)
)
=
1.Theorem 11.
The five axioms of Definition10 are mutually independent.Proof.
LetA
=
[
0,1
], x
∨
y
=
max
⁡
{
x
,
y
}, x
∧
y
=
min
⁡
{
x
,
y
}, x
-
=
1
-
x, and x
~
=
1
-
x
2. Then A is a bounded lattice satisfying x
-
~
=
x
~
-
=
x for any x
∈
A.(i)
Define operations→ and ⇝ as pseudo-Godel implication on A as follows:
(2)
x
⟶
y
=
x
⇝
y
=
{
1
,
x
≤
y
,
y
,
otherwise
.
ThenA satisfies (P2)–(P5), but not (P1): 1
→
0.5
=
0.5, but 0
.
5
-
⇝
1
-
=
0.5
⇝
0
=
0.(ii)
Define operations→ and ⇝ on A as follows:
(3)
x
⟶
y
=
x
⇝
y
=
1
.
Clearly,A satisfies (P1) and (P3)–(P5), but not (P2): 1
→
0.5
=
1
⇝
0.5
=
1
≠
0.5.(iii)
Define operations→ and ⇝ on A as follows:
(4)
x
⟶
y
=
{
x
-
∨
y
,
x
=
1
or
y
=
0
,
1
,
otherwise
,
x
⇝
y
=
{
x
~
∨
y
,
x
=
1
or
y
=
0
,
1
,
otherwise
.
ThenA satisfies (P1)-(P2) and (P4)-(P5), but not (P3). In fact, let x
=
0.64, y
=
0.1, and z
=
0, then y
→
z
=
y
-
∨
z
=
y
-
=
1
-
y
=
0.9
≈
0.95
,
(
x
→
y
)
→
(
x
→
z
)
=
1
→
x
-
∨
z
=
1
→
x
-
=
x
-
=
0.36
=
0.6.(iv)
Define operations→ and ⇝ as pseudo-Lukasiewicz implication on A as follows:
(5)
x
⟶
y
=
1
∧
(
x
-
+
y
)
,
x
⇝
y
=
1
∧
(
x
~
+
y
)
.
ThenA satisfies (P1)–(P4), but not (P5): (
0.19
→
0.07
)
∨
[
(
0.19
→
0.07
)
⇝
(
0.1
9
-
∨
0.07
)
]
=
0.97
∨
(
0.97
⇝
0.9
)
≤
0.97
∨
0.96
=
0.97
<
1.(v)
Suppose thatA is a bounded lattice given by Figure 1.
The operations-, ~, →, and ⇝ on A are defined by the following:
(6)
x
0
f
c
d
a
b
e
1
x
-
=
x
~
1
e
b
a
d
c
f
0
x
⟶
y
=
x
⇝
y
=
{
1
,
x
≤
y
,
x
-
∨
y
=
x
~
∨
y
,
otherwise
.
ThenA satisfies (P1)–(P3) and (P5), but not (P4): (
a
→
b
)
∨
(
a
→
c
)
=
(
a
-
∨
b
)
∨
(
a
-
∨
c
)
=
(
d
∨
b
)
∨
(
d
∨
c
)
=
b
∨
a
=
e, but a
→
b
∨
c
=
a
→
e
=
1.Figure 1


## 4. Filters and Normal Filters of Pseudo-Weak-R
0 Algebras

We introduce the notions of filters and normal filters in pseudo-weak-R
0 algebras and investigate the structures and properties of the generated filters and generated normal filters in pseudo-weak-R
0 algebras.Definition 12.
A nonempty subsetF of a pseudo-weak-R
0 algebra A is said to be a filter of A if it satisfies(F1)
x
,
y
∈
F
⇒
x
⊙
y
∈
F,
(F2)
x
∈
F, x
≤
y
⇒
y
∈
F.Proposition 13.
For a subsetF of a pseudo-weak-R
0 algebra A, the following are equivalent:(i)
F
is a filter,
(ii)
1
∈
F
and
x
,
x
⇝
y
∈
F
⇒
y
∈
F
,
(iii)
1
∈
F
and
x
,
x
→
y
∈
F
⇒
y
∈
F
.Proof.
(i)⇒(ii). By (F2), we have 1
∈
F. By (F1), x
,
x
⇝
y
∈
F
⇒
x
⊙
(
x
⇝
y
)
∈
F. By (38) and (F2), x
∧
y
∈
F, and so y
∈
F.
(ii)⇒(iii). If x
,
x
→
y
∈
F, by (19), x
≤
(
x
→
y
)
⇝
y. By (4), x
⇝
(
(
x
→
y
)
⇝
y
)
=
1
∈
F. By (ii), y
∈
F.
(iii)⇒(i). If x
∈
F, x
≤
y, then x
→
y
=
1
∈
F, so y
∈
F; that is, (F2) holds; if x
,
y
∈
F, by (41), x
→
(
y
→
(
x
⊙
y
)
)
=
(
x
⊙
y
)
→
(
x
⊙
y
)
=
1
∈
F, and so x
⊙
y
∈
F, which means (F1) holds.Clearly,{
1
} and A are both filters of a pseudo-weak-R
0 algebra A.Proposition 14.
For a subset
F
of a pseudo-weak-
R
0
algebra
A
, the following are equivalent:
(i)
F
is a filter,
(ii)
x
,
y
∈
F
,
y
≤
x
→
z
⇒
z
∈
F
,
(iii)
x
,
y
∈
F
,
y
≤
x
⇝
z
⇒
z
∈
F
.Proof.
(i)⇔(ii). If x
,
y
∈
F, y
≤
x
→
z, by (F2) and Proposition 13 (iii), z
∈
F. Conversely, if x
∈
F, by x
≤
x
→
1, we have 1
∈
F; suppose that x
,
x
→
y
∈
F, by x
→
y
≤
x
→
y, we have y
∈
F. By Proposition 13 (iii), F is a filter.
(i)⇔(iii). Similarly.Next, we consider filter generated by a set. It is easy to verify that the intersection of filters ofA is also a filter. If S
⊆
A, the least filter containing S; that is, the intersection of all filters of A containing S is called the filter generated by S and denoted by [
S
). If S
=
{
a
}, [
{
a
}
) is written [
a
). Clearly
(7)
[
S
)
=
∩
{
T
∣
S
⊆
T
⊆
A
,
T
is
a
filter
of
A
}
.Theorem 15.
LetA be a pseudo-weak-R
0 algebra and let S be a nonempty subset of A. Then
(8)
[
S
)
=
{
x
∈
A
∣
t
h
e
r
e
a
r
e
n
≥
1
,
a
1
,
a
2
,
…
,
a
n
∈
S
,
k
s
u
c
h
t
h
a
t
a
1
⊙
⋯
⊙
a
n
≤
x
}
=
{
x
∈
A
∣
t
h
e
r
e
a
r
e
n
≥
1
,
a
1
,
a
2
,
…
,
a
n
∈
S
,
k
s
u
c
h
t
h
a
t
a
n
⇝
(
⋯
⇝
(
a
1
⇝
x
)
⋯
)
=
1
}
=
{
x
∈
A
∣
t
h
e
r
e
a
r
e
n
≥
1
,
a
1
,
a
2
,
…
,
a
n
∈
S
,
k
s
u
c
h
t
h
a
t
a
n
⟶
(
⋯
⟶
(
a
1
⟶
x
)
⋯
)
=
1
}
.Proof.
Only prove the first equality. Using (34) to the first equality, we can get the rest of the two equalities. LetB denote the right side of the first equality. If x
,
y
∈
B, then there are a
1
,
a
2
,
…
,
a
n
,
b
1
,
b
2
,
…
,
b
m
∈
S such that a
1
⊙
⋯
⊙
a
n
≤
x and b
1
⊙
⋯
⊙
b
m
≤
y. By (37), a
1
⊙
⋯
⊙
a
n
⊙
b
1
⊙
⋯
⊙
b
m
≤
x
⊙
y, so x
⊙
y
∈
B. If x
∈
B and x
≤
y, we have a
1
⊙
⋯
⊙
a
n
≤
x
≤
y, so y
∈
B. Hence B is a filter. If C is a filter and S
⊆
C, for any x
∈
B, there are a
1
,
a
2
,
…
,
a
n
∈
S such that a
1
⊙
⋯
⊙
a
n
≤
x. By (F2), x
∈
C, hence B
⊆
C.For convenience, we shall writea
n
:
=
a
⊙
⋯
⊙
a
︷
n and a
0
:
=
1; a
⇝
n
x
:
=
a
⇝
(
⋯
⇝
(
a
︷
n
⇝
x
)
⋯
) and a
⇝
0
x
:
=
x; a
→
n
x
:
=
a
→
(
⋯
→
(
a
︷
n
→
x
)
⋯
) and a
→
0
x
:
=
x.Corollary 16.
IfA is a pseudo-weak-R
0 algebra and a
∈
A, then
(9)
[
a
)
=
{
x
∈
A
∣
n
≥
1
,
a
n
≤
x
}
=
{
x
∈
A
∣
n
≥
1
,
a
⇝
n
x
=
1
}
=
{
x
∈
A
∣
n
≥
1
,
a
→
n
x
=
1
}
.Corollary 17.
Let
F
be a filter of a pseudo-weak-
R
0
algebra
A
and
a
∈
A
; then
(10)
[
F
∪
{
a
}
)
=
{
x
∈
A
∣
(
s
1
⊙
a
n
1
)
⊙
⋯
⊙
(
s
m
⊙
a
n
m
)
≤
x
,
w
h
e
r
e
m
≥
1
,
n
1
,
…
,
n
m
≥
0
,
s
1
,
…
,
s
m
∈
F
}
.Theorem 18.
LetF be a filter of a pseudo-weak-R
0 algebra A and a
,
b
∈
A; then
(11)
[
F
∪
{
a
}
)
∩
[
F
∪
{
b
}
)
=
[
F
∪
{
a
∨
b
}
)
.Proof.
Assume thatx
∈
[
F
∪
{
a
}
)
∩
[
F
∪
{
b
}
), by Corollary 17, there are n
1
,
…
,
n
m
,
l
1
,
…
,
l
k
≥
0
,
s
1
,
…
,
s
m
,
t
1
,
…
,
t
k
∈
F such that
(12)
(
s
1
⊙
a
n
1
)
⊙
⋯
⊙
(
s
m
⊙
a
n
m
)
≤
x
,
(
t
1
⊙
b
l
1
)
⊙
⋯
⊙
(
t
k
⊙
b
l
k
)
≤
x
.
Put p
=
s
1
⊙
⋯
⊙
s
m
⊙
t
1
⊙
⋯
⊙
t
k and q
=
max
⁡
{
n
1
,
…
,
n
m
,
l
1
,
…
,
l
k
}, and then
(13)
(
p
⊙
a
q
)
m
≤
x
,
(
p
⊙
b
q
)
k
≤
x
.
Thus, by (46) x
≥
(
p
⊙
a
q
)
m
∨
(
p
⊙
b
q
)
k
≥
(
(
p
⊙
a
q
)
m
∨
(
p
⊙
b
q
)
)
k
≥
(
(
p
⊙
a
q
)
∨
(
p
⊙
b
q
)
)
m
k
=
(
p
⊙
(
a
q
∨
b
q
)
)
m
k
≥
(
p
⊙
(
a
∨
b
)
q
2
)
m
k. x
∈
[
F
∪
{
a
∨
b
}
). Hence [
F
∪
{
a
}
)
∩
[
F
∪
{
b
}
)
⊆
[
F
∪
{
a
∨
b
}
). Inverse contains is obvious.Corollary 19.
LetF be a filter of a pseudo-weak-R
0 algebra A and a
,
b
∈
A. If a
∨
b
∈
F, then
(14)
[
F
∪
{
a
}
)
∩
[
F
∪
{
b
}
)
=
F
.Corollary 20.
LetA be a pseudo-weak-R
0 algebra and a
,
b
∈
A; then [
a
)
∩
[
b
)
=
[
a
∨
b
).Proof.
TakingF
=
{
1
} in Theorem 18.Next we introduce the notion of normal filters in a pseudo-weak-R
0 algebra.Definition 21.
A filterF of a pseudo-weak-R
0 algebra A is called normal if x
,
y
∈
A, x
→
y
∈
F if and only if x
⇝
y
∈
F.Proposition 22.
LetF be a normal filter of a pseudo-weak-R
0 algebra A. Then there is s
∈
F such that b
⊙
s
≤
c if and only if there is t
∈
F such that t
⊙
b
≤
c.Proof.
If there iss
∈
F such that b
⊙
s
≤
c, by (34), s
≤
b
⇝
c. By s
∈
F, we have b
⇝
c
∈
F, and so b
→
c
∈
F. Put b
→
c
=
t
∈
F
,
and then t
⊙
b
≤
c. Converse is similar.Theorem 23.
If
F
is a normal filter of a pseudo-weak-
R
0
algebra
A
and
a
∈
A
, then
(15)
[
F
∪
{
a
}
)
=
{
x
∈
A
∣
t
h
e
r
e
a
r
e
s
∈
F
,
n
≥
0
,
s
u
c
h
t
h
a
t
s
⊙
a
n
≤
x
}
=
{
x
∈
A
∣
t
h
e
r
e
a
r
e
s
∈
F
,
n
≥
0
,
s
u
c
h
t
h
a
t
a
n
⊙
s
≤
x
}
.Proof.
We show the first equality. By Corollary17,
(16)
[
F
∪
{
a
}
)
=
{
x
∈
A
∣
(
s
1
⊙
a
n
1
)
⊙
⋯
⊙
(
s
m
⊙
a
n
m
)
≤
x
,
m
≥
1
,
n
1
,
…
,
n
m
≥
0
s
1
,
…
,
s
m
∈
F
}
.
Since
(17)
(
s
1
⊙
a
n
1
)
⊙
(
s
2
⊙
a
n
2
)
⊙
⋯
⊙
(
s
m
⊙
a
n
m
)
≤
x
,
by (34),
(18)
(
s
1
⊙
a
n
1
)
⊙
s
2
≤
(
a
n
2
⊙
(
s
3
⊙
a
n
3
)
⊙
⋯
⊙
(
s
m
⊙
a
n
m
)
)
⟶
x
,
by Proposition 22, there is t
2
∈
F such that
(19)
t
2
⊙
(
s
1
⊙
a
n
1
)
≤
(
a
n
2
⊙
(
s
3
⊙
a
n
3
)
⊙
⋯
⊙
(
s
m
⊙
a
n
m
)
)
⟶
x
,
and so
(20)
(
t
2
⊙
s
1
)
⊙
a
n
1
+
n
2
⊙
(
s
3
⊙
a
n
3
)
⊙
⋯
⊙
(
s
m
⊙
a
n
m
)
≤
x
.
Repeating the above steps, there are t
2
,
…
,
t
m
∈
F such that
(21)
(
t
m
⊙
⋯
⊙
t
2
⊙
s
1
)
⊙
a
n
1
+
⋯
+
n
m
≤
x
.
Let s
=
t
m
+
⋯
+
t
2
+
s
1
∈
F and n
=
n
1
+
⋯
+
n
m, we have s
⊙
a
n
≤
x. That is that the first equality holds.By the first equality and Proposition22, we can obtain the second equation.Corollary 24.
If
F
is a normal filter of a pseudo-weak-
R
0
algebra
A
and
a
∈
A
, then
(22)
[
F
∪
{
a
}
)
=
{
x
∈
A
∣
t
h
e
r
e
i
s
n
≥
0
,
s
u
c
h
t
h
a
t
a
→
n
x
∈
F
}
=
{
x
∈
A
∣
t
h
e
r
e
i
s
n
≥
0
,
s
u
c
h
t
h
a
t
a
⇝
n
x
∈
F
}
.Proof.
By Theorem23,
(23)
[
F
∪
{
a
}
)
=
{
x
∈
A
∣
there
are
s
∈
F
,
n
≥
0
,
such
that
s
⊙
a
n
≤
x
}
.
Since there iss
∈
F such that s
⊙
a
n
≤
x, if and only if there is s
∈
F such that s
≤
a
→
(
⋯
→
(
a
︷
n
→
x
)
⋯
); that is, there is s
∈
F such that s
≤
a
→
n
x, if and only if a
→
n
x
∈
F. Thus, we prove the first equality.
Similarly, by(24)
[
F
∪
{
a
}
)
=
{
x
∈
A
∣
there
are
s
∈
F
,
n
≥
0
,
such
that
a
n
⊙
s
≤
x
}
.
Since there is s
∈
F such that a
n
⊙
s
≤
x, if and only if there is s
∈
F such that s
≤
a
⇝
(
⋯
⇝
(
a
︷
n
⇝
x
)
⋯
); that is, there is s
∈
F such that s
≤
a
⇝
n
x, if and only if a
⇝
n
x
∈
F, Thus, we have the second equality.Corollary 25.
If
F
is a normal filter of a pseudo-weak-
R
0
algebra
A
and
a
∈
A
, then
(25)
[
F
∪
{
a
}
)
=
{
x
∈
A
∣
t
h
e
r
e
i
s
n
≥
0
,
s
u
c
h
t
h
a
t
a
n
⟶
x
∈
F
}
=
{
x
∈
A
∣
t
h
e
r
e
i
s
n
≥
0
,
s
u
c
h
t
h
a
t
a
n
⇝
x
∈
F
}
.Proof.
There iss
∈
F such that s
⊙
a
n
≤
x, if and only if there is s
∈
F such that s
≤
a
n
→
x, if and only if a
n
→
x
∈
F. There is s
∈
F such that a
n
⊙
s
≤
x, if and only if there is s
∈
F such that s
≤
a
n
⇝
x, if and only if a
n
⇝
x
∈
F. By Theorem 23, Corollary 25 holds.


## 5. Conclusions

We obtained the most simplified axiom systems of pseudo-weak-R
0 algebras and pseudo-R
0 algebras and proved the mutually independence of axioms. We introduced the notions of filters and normal filters in pseudo-weak-R
0 algebras and gave the structures and properties of the generated filters and generated normal filters in pseudo-weak-R
0 algebras. These will be conducive to further study pseudo-weak-R
0 algebras (pseudo-IMTL algebras) and pseudo-R
0 algebras (pseudo-NM algebras). In the future, we will investigate relations between various kinds of filters of pseudo-logic algebras. We may also study fuzzy type of filters of pseudo-weak-R
0 algebras and pseudo-R
0 algebras.



---

*Source: 854168-2014-07-07.xml*