# Vague Filters of Residuated Lattices

Authors: Shokoofeh Ghorbani
Journal: Journal of Discrete Mathematics
 (2014)

---

## Abstract

Notions of vague filters, subpositive implicative vague filters, and Boolean vague filters of a residuated lattice are introduced and some related properties are investigated. The characterizations of (subpositive implicative, Boolean) vague filters is obtained. We prove that the set of all vague filters of a residuated lattice forms a complete lattice and we find its distributive sublattices. The relation among subpositive implicative vague filters and Boolean vague filters are obtained and it is proved that subpositive implicative vague filters are equivalent to Boolean vague filters.


---

## Body

## 1. Introduction

In the classical set, there are only two possibilities for any elements: in or not in the set. Hence the values of elements in a set are only one of0 and 1. Therefore, this theory cannot handle the data with ambiguity and uncertainty.Zadeh introduced fuzzy set theory in 1965 [1] to handle such ambiguity and uncertainty by generalizing the notion of membership in a set. In a fuzzy set A each element x is associated with a point-value μ
A
(
x
) selected from the unit interval [
0,1
], which is termed the grade of membership in the set. This membership degree contains the evidences for both supporting and opposing x.A number of generalizations of Zadeh’s fuzzy set theory are intuitionistic fuzzy theory, L-fuzzy theory, and vague theory. Gau and Buehrer proposed the concept of vague set in 1993 [2], by replacing the value of an element in a set with a subinterval of [
0,1
]. Namely, a true membership function t
v
(
x
) and a false-membership function f
v
(
x
) are used to describe the boundaries of membership degree. These two boundaries form a subinterval [
t
v
(
x
)
,
1
-
f
v
(
x
)
] of [
0,1
]. The vague set theory improves description of the objective real world, becoming a promising tool to deal with inexact, uncertain, or vague knowledge. Many researchers have applied this theory to many situations, such as fuzzy control, decision-making, knowledge discovery, and fault diagnosis. Recently in [3], Jun and Park introduced the notion of vague ideal in pseudo MV-algebras and Broumand Saeid [4] introduced the notion of vague BCK/BCI-algebras.The concept of residuated lattices was introduced by Ward and Dilworth [5] as a generalization of the structure of the set of ideals of a ring. These algebras are a common structure among algebras associated with logical systems (see [6–9]). The residuated lattices have interesting algebraic and logical properties. The main example of residuated lattices related to logic is and BL-algebras. A basic logic algebra (BL-algebra for short) is an important class of logical algebras introduced by Hajek [10] in order to provide an algebraic proof of the completeness of “Basic Logic” (BL for short). Continuous t-norm based fuzzy logics have been widely applied to fuzzy mathematics, artificial intelligence, and other areas. The filter theory plays an important role in studying these logical algebras and many authors discussed the notion of filters of these logical algebras (see [11, 12]). From a logical point of view, a filter corresponds to a set of provable formulas.In this paper, the concept of vague sets is applied to residuated lattices. In Section2, some basic definitions and results are explained. In Section 3, we introduce the notion of vague filters of a residuated lattice and investigate some related properties. Also, we obtain the characterizations of vague filters. In Section 4, the smallest vague filters containing a given vague set are established and we obtain the distributive sublattice of the set of all vague filters of a residuated lattice. In Section 5, notions of vague filters, subpositive implicative vague filters, and Boolean vague filters of a residuated lattice are introduced and some related properties are obtained.


## 2. Preliminaries

We recall some definitions and theorems which will be needed in this paper.Definition 1 (see [6]).
Aresiduated lattice is an algebraic structure (
L
,
∧
,
∨
,
*
,
→
,
⇝
,
0,1
) such that(1)
(
L
,
∧
,
∨
,
0,1
) is a bounded lattice with the least element 0 and the greatest element 1,
(2)
(
L
,
*
,
1
) is a monoid; that is, * is associative and x
*
1
=
1
*
x
=
x,
(3)
x
*
y
≤
z if and only if x
≤
y
→
z if and only if y
≤
x
⇝
z, for all x
,
y
,
z
∈
L.In the rest of this paper, we denote the residuated lattice(
L
,
∧
,
∨
,
*
,
→
,
⇝
,
0,1
) by L, ¬
x
=
x
→
0, and ~
x
=
x
⇝
0.Proposition 2 (see [6, 9]).
LetL be a residuated lattice. Then we have the following properties: for all x
,
y
,
z
∈
L,(1)
x
→
x
=
x
⇝
x
=
1,
(2)
1
→
x
=
1
⇝
x
=
x,
(3)
x
→
(
y
⇝
z
)
=
y
⇝
(
x
→
z
),
(4)
x
≤
y if and only if x
→
y
=
1 if and only if x
⇝
y
=
1,
(5)
ifx
≤
y, then x
*
z
≤
y
*
z, z
*
x
≤
z
*
y,
(6)
ifx
≤
y, then z
→
x
≤
z
→
y, z
⇝
x
≤
z
⇝
y,
(7)
ifx
≤
y, then y
→
z
≤
x
→
z, y
⇝
z
≤
x
⇝
z,
(8)
x
≤
(
x
⇝
y
)
→
y, x
≤
(
x
→
y
)
⇝
y,
(9)
y
→
z
≤
(
x
→
y
)
⇝
(
x
→
z
), y
⇝
z
≤
(
x
⇝
y
)
⇝
(
x
⇝
z
),
(10)
x
→
(
y
⇝
(
x
*
y
)
)
=
1, y
⇝
(
x
→
(
x
*
y
)
)
=
1.Definition 3 (see [6, 9]).
LetF be a nonempty subset of a residuated lattice L. F is called afilter if(F1)
x
,
y
∈
F imply x
*
y
∈
F,
(F2)
x
≤
y and x
∈
F imply y
∈
F.Theorem 4 (see [6, 9]).
A nonempty subsetF of a residuated lattice L is a filter if and only if(F3)
1
∈
F and x
,
x
→
y
∈
F imply y
∈
F, or
(F3′)
1
∈
F and x
,
x
⇝
y
∈
F imply y
∈
F.Definition 5 (see [11]).
LetF be a nonempty subset of a residuated lattice L. F is called asubpositive implicative filter of L if(SPIF1)
1
∈
F,
(SPIF2)
(
(
x
→
y
)
*
z
)
⇝
(
(
y
⇝
x
)
→
x
)
∈
F and z
∈
F imply (
x
→
y
)
⇝
y
∈
F for any x
,
y
,
z
∈
L,
(SPIF3)
(
z
*
(
x
⇝
y
)
)
→
(
(
y
→
x
)
⇝
x
)
∈
F and z
∈
F imply (
x
⇝
y
)
→
y
∈
F for any x
,
y
,
z
∈
L.Proposition 6 (see [11]).
LetF be a filter of L. F is a subpositive implicative filter if and only if(SPIF4)
(
x
→
y
)
⇝
(
(
y
⇝
x
)
→
x
)
∈
F imply (
x
→
y
)
⇝
y
∈
F for any x
,
y
∈
L,
(SPIF5)
(
x
⇝
y
)
→
(
(
y
→
x
)
⇝
x
)
∈
F imply (
x
⇝
y
)
→
y
∈
F for any x
,
y
∈
L.Definition 7 (see [1]).
LetX be a nonempty set. Afuzzy set in X is a mapping μ
:
X
→
[
0,1
].Proposition 8 (see [12]).
LetL be a residuated lattice. A fuzzy set μ is called afuzzy filter in L if it satisfies (fF1)
x
≤
y imply μ
(
x
)
≤
μ
(
y
),
(fF2)
min
⁡
{
μ
(
x
)
,
μ
(
y
)
}
≤
μ
(
x
*
y
),for all x
,
y
∈
L.Definition 9 (see [13]).
Avague set
A in the universe of discourse X is characterized by two membership functions given by(i)
a true membership functiont
A
:
X
→
[
0,1
],
(ii)
a false membership functionf
A
:
X
→
[
0,1
],where t
A
(
x
) is a lower bound on the grade of membership of X derived from the “evidence for x,” f
A
(
x
) is a lower bound on the negation of x derived from the “evidence against x,” and t
A
(
x
)
+
f
A
(
x
)
≤
1.
Thus the grade of membership ofx in the vague set A is bounded by a subinterval [
t
A
(
x
)
,
1
-
f
A
(
x
)
] of [
0,1
]. This indicates that if the actual grade of membership of x is μ
(
x
), then t
A
(
x
)
≤
μ
(
x
)
≤
1
-
f
A
(
x
). The vague set A is written as A
=
{
〈
x
,
[
t
A
(
x
)
,
1
-
f
A
(
x
)
]
〉
:
x
∈
X
}, where the interval [
t
A
(
x
)
,
1
-
f
A
(
x
)
] is called thevague value of x in A, denoted by V
A
(
x
).Definition 10 (see [13]).
A vague setA of a set X is called(1)
thezero vague set of X if t
A
(
x
)
=
0 and f
A
(
x
)
=
1 for all x
∈
X,
(2)
theunit vague set of X if t
A
(
x
)
=
1 and f
A
(
x
)
=
0 for all x
∈
X,
(3)
theα
-vague set of X if t
A
(
x
)
=
α and f
A
(
x
)
=
1
-
α for all x
∈
X, where α
∈
(
0,1
).LetI
[
0,1
] denote the family of all closed subintervals of [
0,1
]. Now we define the refined minimum (briefly, imin) and an order ≤ on elements I
1
=
[
a
1
,
b
1
] and I
2
=
[
a
2
,
b
2
] of I
[
0,1
] as
(1)
imin
(
I
1
,
I
2
)
=
[
min
⁡
{
a
1
,
a
2
}
,
min
⁡
{
b
1
,
b
2
}
]
,
I
1
≤
I
2
iff
a
1
≤
a
2
,
b
1
≤
b
2
.Similarly we can define≥, =, and imax. Then the concept of imin and imax could be extended to define iinf and isup of infinite number of elements of I
[
0,1
]. It is a known fact that L
=
{
I
[
0,1
]
,
iinf
,
isup
,
≤
} is a lattice with universal bounds zero vague set and unit vague set. For α
,
β
∈
[
0,1
] we now define (
α
,
β
)-cut and α-cut of a vague set.Definition 11 (see [13]).
LetA be a vague set of a universe X with the true-membership function t
A and false-membership function f
A. The (
α
,
β
)-cut of the vague set A is a crisp subset A
(
α
,
β
) of the set X given by
(2)
A
(
α
,
β
)
=
{
x
∈
X
:
V
A
(
x
)
≥
[
α
,
β
]
}
,
where
α
≤
β
.
Clearly A
(
0,0
)
=
X. The (
α
,
β
)-cuts are also called vague-cuts of the vague set A.Definition 12 (see [13]).
Theα-cut of the vague set A is a crisp subset A
α of the set X given by A
α
=
A
(
α
,
α
). Equivalently, we can define the α-cut as A
α
=
{
x
∈
X
:
t
A
(
x
)
≥
α
}.


## 3. Vague Filters of Residuated Lattices

In this section, we define the notion of vague filters of residuated lattices and obtain some related results.Definition 13.
A vague setA of a residuated lattice L is called avague filter of L if the following conditions hold:(VF1)
ifx
≤
y, then V
A
(
x
)
≤
V
A
(
y
), for all x
,
y
∈
L,
(VF2)
imin
{
V
A
(
x
)
,
V
A
(
y
)
}
≤
V
A
(
x
*
y
), for all x
,
y
∈
L.That is,(1)
x
≤
y implies t
A
(
x
)
≤
t
A
(
y
) and 1
-
f
A
(
x
)
≤
1
-
f
A
(
y
), for all x
,
y
∈
L,
(2)
min
⁡
{
t
A
(
x
)
,
t
A
(
y
)
}
≤
t
A
(
x
*
y
) and min
⁡
{
1
-
f
A
(
x
)
,
1
-
f
A
(
y
)
}
≤
1
-
f
A
(
x
*
y
), for all x
,
y
∈
L.In the following proposition, we will show that the vague filters of a residuated lattice exist.Proposition 14.
Unit vague set andα-vague set of a residuated lattice L are vague filters of L.Proof.
LetA be a α-vague set of L. It is clear that A satisfies (VF1). For x
,
y
∈
L we have
(3)
t
A
(
x
*
y
)
=
α
=
min
⁡
{
α
,
α
}
=
min
⁡
{
t
A
(
x
)
,
t
A
(
y
)
}
,
1
-
f
A
(
x
*
y
)
=
α
=
min
⁡
{
α
,
α
}
=
min
⁡
{
1
-
f
A
(
x
)
,
1
-
f
A
(
y
)
}
.
Thus imin
{
V
A
(
x
)
,
V
A
(
y
)
}
≤
V
A
(
x
*
y
), for all x
,
y
∈
L. Hence it is a vague filter of L. The proof of the other cases is similar.Lemma 15.
LetA be a vague set of a residuated lattice L such that it satisfies (VF1). Then the following are equivalent:(VF4)
imin
{
V
A
(
x
)
,
V
A
(
x
→
y
)
}
≤
V
A
(
y
),
(VF5)
imin
{
V
A
(
x
)
,
V
A
(
x
⇝
y
)
}
≤
V
A
(
y
),
for allx
,
y
∈
L.Proof.
Suppose thatA satisfies (VF4). We have x
≤
(
x
⇝
y
)
→
y. By (VF1), V
A
(
x
)
≤
V
A
(
(
x
⇝
y
)
→
y
). We get that
(4)
imin
{
V
A
(
x
)
,
V
A
(
x
⇝
y
)
}
≤
imin
{
V
A
(
(
x
⇝
y
)
⟶
y
)
,
V
A
(
x
⇝
y
)
}
≤
V
A
(
y
)
.
Conversely, if A satisfies (VF5), then the inequality x
≤
(
x
→
y
)
⇝
y analogously entails (VF4).Theorem 16.
LetA be a vague set of a residuated lattice L. Then A is a vague filter if and only if it satisfies (VF4) and (VF6)
V
A
(
x
)
≤
V
A
(
1
), for all x
∈
L.Proof.
LetA be a vague filter of L. Since x
≤
1, then V
A
(
x
)
≤
V
A
(
1
), for all x
∈
L by (VF1). We have x
*
(
x
→
y
)
≤
y. By (VF1) and (VF2) we get
(5)
imin
{
V
A
(
x
)
,
V
A
(
x
⟶
y
)
}
≤
V
A
(
x
*
(
x
⟶
y
)
)
≤
V
A
(
y
)
.
Hence (VF4) hold.
Conversely, letA satisfy (VF4) and (VF6). Since x
≤
y, then x
→
y
=
1;
(6)
V
A
(
y
)
≥
imin
{
V
A
(
x
⟶
y
)
,
V
A
(
x
)
}
=
imin
{
V
A
(
1
)
,
V
A
(
x
)
}
=
V
A
(
x
)
.
Hence V
A is order-preserving and (VF1) holds. By Proposition 2 part (10), x
→
(
y
⇝
(
x
*
y
)
)
=
1. By (VF4), we have
(7)
V
A
(
y
⇝
(
x
*
y
)
)
≥
imin
{
V
A
(
x
)
,
V
A
(
x
⟶
(
y
⇝
(
x
*
y
)
)
)
}
=
V
A
(
x
)
.
By Lemma 15,
(8)
V
A
(
x
*
y
)
≥
imin
{
V
A
(
y
⇝
(
x
*
y
)
)
,
V
A
(
y
)
}
≥
imin
{
V
A
(
x
)
,
V
A
(
y
)
}
.
Hence (VF2) holds.Theorem 17.
LetA be a vague set of a residuated lattice L. Then A is a vague filter if and only if it satisfies (VF5) and (VF6).Definition 18.
A vague setA of a residuated lattice L is called avague lattice filter of L if it satisfies (VF1) and(VLF2)
imin
{
V
A
(
x
)
,
V
A
(
y
)
}
≤
V
A
(
x
∧
y
), for all x
,
y
∈
L.Proposition 19.
LetA be a vague filter of a residuated lattice L; then A is a vague lattice filter.In the following example, we will show that the converse of Proposition19 may not be true in general.Example 20.
Let{
0
,
a
,
b
,
c
,
1
} such that 0
<
a
<
b, c
<
1, where b and c are incomparable. Define the operations *, →, and ⇝ as follows:
(9)
*
0
a
b
c
1
0
0
0
0
0
0
a
0
0
0
a
a
b
0
a
b
a
b
c
0
0
0
c
c
1
0
a
b
c
1
⟶
0
a
b
c
1
0
1
1
1
1
1
a
c
1
1
1
1
b
c
c
1
c
1
c
0
b
b
1
1
1
0
a
b
c
1
⇝
0
a
b
c
1
0
1
1
1
1
1
a
b
1
1
1
1
b
0
c
1
c
1
c
b
b
b
1
1
1
0
a
b
c
1
.
Then L is a residuated lattice. Let A be the vague set defined as follows:
(10)
A
=
{
〈
0
,
[
0.1,0.2
]
〉
,
〈
a
,
[
0.3,0.4
]
〉
,
〈
b
,
[
0.3,0.4
]
〉
,
h
h
h
〈
c
,
[
0.3,0.8
]
〉
,
〈
0
,
[
0.5,0.9
]
〉
}
.
It is routine to verify that A is a vague lattice filter, but it is not a vague filter of L.
LetB be the vague set defined as follows:
(11)
B
=
{
〈
0
,
[
0.2,0.5
]
〉
,
〈
a
,
[
0.2,0.5
]
〉
,
〈
b
,
[
0.2,0.5
]
〉
,
h
h
h
〈
c
,
[
0.4,0.7
]
〉
,
〈
0
,
[
0.7,0.7
]
〉
}
.
Then B is a vague filter of L.In the following theorems, we will study the relationship between filters and vague filters of a residuated lattice.Theorem 21.
LetA be a vague set of a residuated lattice L. Then A is a vague filter of L if and only if, for all α
,
β
∈
[
0,1
], the set A
(
α
,
β
) is either empty or a filter of L, where α
≤
β.Proof.
Suppose thatA is a vague filter of L and α
,
β
∈
[
0,1
]. Let A
(
α
,
β
)
≠
∅. We will show that A
(
α
,
β
) is a filter of L.(1)
Suppose thatx
≤
y and x
∈
A
(
α
,
β
). By (VF1), V
A
(
y
)
≥
V
A
(
x
)
≥
[
α
,
β
]. Therefor y
∈
A
(
α
,
β
).
(2)
Suppose thatx
,
y
∈
A
(
α
,
β
). Then V
A
(
x
)
,
V
A
(
y
)
≥
[
α
,
β
]. By (VF2), we have V
A
(
x
*
y
)
≥
imin
{
V
A
(
x
)
,
V
A
(
y
)
}
≥
[
α
,
β
]. Hence x
*
y
∈
A
(
α
,
β
) and then A
(
α
,
β
) is a filter of L.
Conversely, suppose that, for allα
,
β
∈
[
0,1
], the sets A
(
α
,
β
) is either empty or a filter of L. Let t
A
(
x
)
=
α
1, t
A
(
y
)
=
α
2, 1
-
f
A
(
x
)
=
β
1, and 1
-
f
A
(
y
)
=
β
2. Put α
=
min
⁡
{
α
1
,
α
2
} and β
=
min
⁡
{
1
-
β
1
,
1
-
β
2
}. Then t
A
(
x
)
,
t
A
(
y
)
≥
α and 1
-
f
A
(
x
)
,
1
-
f
A
(
y
)
≥
β. Hence V
A
(
y
)
,
V
A
(
x
)
≥
[
α
,
β
]; that is, x
,
y
∈
A
(
α
,
β
). So A
(
α
,
β
)
≠
∅ and, by assumption, A
(
α
,
β
) is a filter of L. We obtain that x
*
y
∈
A
(
α
,
β
); that is,
(12)
V
A
(
x
*
y
)
≥
[
α
,
β
]
=
[
min
⁡
⁡
{
α
1
,
α
2
}
,
min
⁡
⁡
{
1
-
β
1
,
1
-
β
2
}
]
=
imin
{
[
t
A
(
x
)
,
1
-
f
A
(
x
)
]
[
t
A
(
y
)
,
1
-
f
A
(
y
)
]
}
=
imin
{
V
A
(
x
)
,
V
A
(
y
)
}
.
Hence (VF2) holds.
Suppose thatx
≤
y, t
A
(
x
)
=
α, and 1
-
f
A
(
x
)
=
β. Hence x
∈
A
(
α
,
β
). By assumption A
(
α
,
β
) is a filter. We get that y
∈
A
(
α
,
β
); that is, V
A
(
y
)
≥
[
α
,
β
]
=
[
t
A
(
x
)
,
1
-
f
A
(
x
)
]
=
V
A
(
x
). Hence A is a vague filter of L.The filters likeA
(
α
,
β
) are also calledvague-cuts filters of L.Corollary 22.
LetA be a vague filter of a residuated lattice L. Then for all α
∈
[
0,1
], the set A
α is either empty or a filter of L.Corollary 23.
Any filterF of a residuated lattice L is a vague-cut filter of some vague filter of L.Proof.
For allx
∈
L, define
(13)
1
-
f
A
(
x
)
=
t
A
(
x
)
=
{
1
if
x
∈
F
α
otherwise
.
Hence we have V
A
(
x
)
=
[
1,1
], if x
∈
F and V
A
(
x
)
=
[
α
,
α
], otherwise, where α
∈
(
0,1
). It is clear that F
=
A
(
1,1
). Let x
,
y
∈
L. If x
,
y
∈
F, then x
*
y
∈
F. We have V
A
(
x
*
y
)
=
[
1,1
]
=
imin
{
V
A
(
x
)
,
V
A
(
y
)
}. If one of x or y does not belong to F, then one of V
A
(
x
) and V
A
(
y
) is equal to [
α
,
α
]. Therefore V
A
(
x
*
y
)
≥
[
α
,
α
]
=
imin
{
V
A
(
x
)
,
V
A
(
y
)
}. Suppose that x
≤
y. If x
∈
F, then y
∈
F. Hence V
A
(
x
)
=
V
A
(
y
). If x
∉
F, then V
A
(
x
)
=
[
α
,
α
]. We have V
A
(
x
)
≤
V
A
(
y
). Hence A is a vague filter of L.In the following proposition, we will show that vague filters of a residuated lattice are a generalization of fuzzy filters.Proposition 24.
LetA be a vague set of a residuated lattice L. Then A is a vague filter of L if and only if the fuzzy sets t
A and 1
-
f
A are fuzzy filters in L.


## 4. Lattice of Vague Filters

LetA and B be vague sets of a residuated lattice L. If V
A
(
x
)
≤
V
B
(
x
), for all x
∈
L, then we say B containing A and denote it by A
⪯
B.The unite vague set of a residuated latticeL is a vague filter of L containing any vague filter of L. For a vague set, we can define the intersection of two vague sets A and B of a residuated lattice L by V
A
∩
B
(
x
)
=
imin
{
V
A
(
x
)
,
V
B
(
x
)
}, for all x
∈
L. It is easy to prove that the intersection of any family of vague filters of L is a vague filter of L. Hence we can define a vague filter generated by a vague set as follows.Definition 25.
LetA be a vague set of a residuated lattice L. A vague filter B is called avague filter generated by A, if it satisfies(i)
A
⪯
B,
(ii)
A
⪯
C that implies B
⪯
C, for all vague filters C of L.
The vague filter generated byA is denoted by 〈
A
〉.Example 26.
Consider the vague lattice filterA in Example 20 which is not a vague filter of L. Then the vague filter generated by A is
(14)
〈
A
〉
=
{
〈
0
,
[
0.3,0.4
]
〉
,
〈
a
,
[
0.3,0.4
]
〉
,
h
h
h
〈
b
,
[
0.3,0.4
]
〉
,
〈
c
,
[
0.3,0.8
]
〉
,
〈
0
,
[
0.5,0.9
]
〉
}
.Theorem 27.
LetA be a vague set of a residuated lattice L. Then the vague value of x in 〈
A
〉 is
(15)
V
〈
A
〉
(
x
)
=
isup
{
imin
{
V
A
(
a
1
)
,
V
A
(
a
2
)
,
…
,
V
A
(
a
n
)
}
:
h
h
h
h
h
h
x
≥
a
1
*
⋯
*
a
n
}
,
for all x
∈
L.Proof.
First, we will prove that〈
A
〉 is a vague set of L:
(16)
t
〈
A
〉
(
x
)
=
sup
⁡
⁡
{
min
⁡
⁡
{
t
A
(
a
1
)
,
t
A
(
a
2
)
,
…
,
t
A
(
a
n
)
}
:
s
s
s
s
s
s
s
s
s
s
x
≥
a
1
*
⋯
*
a
n
}
≤
sup
⁡
⁡
{
min
⁡
⁡
{
1
-
f
A
(
a
1
)
,
1
-
f
A
(
a
2
)
,
…
,
1
-
f
A
(
a
n
)
}
:
s
s
s
s
s
s
s
s
s
s
x
≥
a
1
*
⋯
*
a
n
}
=
sup
⁡
⁡
{
1
-
max
⁡
⁡
{
f
A
(
a
1
)
,
f
A
(
a
2
)
,
…
,
f
A
(
a
n
)
}
:
s
s
s
s
s
s
s
s
s
s
x
≥
a
1
*
⋯
*
a
n
}
=
1
-
⁡
inf
⁡
{
max
⁡
⁡
{
f
A
(
a
1
)
,
f
A
(
a
2
)
,
…
,
f
A
(
a
n
)
}
:
s
s
s
s
s
s
s
s
s
s
s
s
s
s
x
≥
a
1
*
⋯
*
a
n
}
,
=
1
-
f
〈
A
〉
(
x
)
.
Now, we will show that 〈
A
〉 is a vague filter of L.(VF1)
Suppose thatx
,
y
∈
L is arbitrary. For every ε
>
0, we can take a
1
,
…
,
a
n
,
b
1
,
…
,
b
m
∈
L such that x
≥
a
1
*
⋯
*
a
n, y
≥
b
1
*
⋯
*
b
m, and
(17)
t
〈
A
〉
(
x
)
-
ε
<
min
⁡
{
t
A
(
a
1
)
,
t
A
(
a
2
)
,
…
,
t
A
(
a
n
)
}
,
t
〈
A
〉
(
y
)
-
ε
<
min
⁡
{
t
A
(
b
1
)
,
t
A
(
b
2
)
,
…
,
t
A
(
b
m
)
}
.By Proposition 2 part (4), we have x
*
y
≥
a
1
*
⋯
*
a
n
*
b
1
*
⋯
*
b
m and then
(18)
t
〈
A
〉
(
x
*
y
)
≥
min
⁡
{
t
A
(
a
1
)
,
…
,
t
A
(
a
n
)
,
t
A
(
b
1
)
,
…
,
t
A
(
b
m
)
}
=
min
⁡
{
min
⁡
{
t
A
(
a
1
)
,
…
,
t
A
(
a
n
)
}
,
s
s
s
s
s
s
s
s
s
s
s
s
s
s
min
⁡
{
t
A
(
b
1
)
,
…
,
t
A
(
b
m
)
}
}
>
min
⁡
{
(
t
〈
A
〉
(
x
)
-
ε
)
,
(
t
〈
A
〉
(
y
)
-
ε
)
}
=
min
⁡
{
t
〈
A
〉
(
x
)
,
t
〈
A
〉
(
y
)
}
-
ε
.
Since ε
>
0 is arbitrary, we obtain that t
〈
A
〉
(
x
*
y
)
≥
min
⁡
{
t
〈
A
〉
(
x
)
,
t
〈
A
〉
(
y
)
}.
Similarly, we can show that1
-
f
〈
A
〉
(
x
*
y
)
≥
min
⁡
{
1
-
f
〈
A
〉
(
x
)
,
1
-
f
〈
A
〉
(
y
)
}. Hence V
A
(
x
*
y
)
≥
imin
{
V
A
(
x
)
,
V
A
(
y
)
}, for all x
,
y
∈
L.(VF2)
Suppose thatx
≤
y where x
,
y
∈
L. Let a
1
,
…
,
a
n
∈
L such that a
1
*
⋯
*
a
n
≤
x. Then a
1
*
⋯
*
a
n
≤
y. And we get that
(19)
V
A
(
y
)
≥
imin
{
V
A
(
a
1
)
,
V
A
(
a
2
)
,
…
,
V
A
(
a
n
)
}
.Since a
1
,
…
,
a
n
∈
L, where x
≥
a
1
*
⋯
*
a
n is arbitrary, then
(20)
V
〈
A
〉
(
y
)
≥
isup
{
imin
{
V
A
(
a
1
)
,
V
A
(
a
2
)
,
…
,
V
A
(
a
n
)
}
:
h
h
h
h
h
h
h
x
≥
a
1
*
⋯
*
a
n
}
=
V
〈
A
〉
(
x
)
.
Therefore 〈
A
〉 is a vague filter of L by Theorem 16. It is clear that A
⪯
〈
A
〉. Now, let C be a vague filter of L such that A
⪯
C. Then we have
(21)
V
〈
A
〉
(
x
)
=
isup
{
imin
{
V
A
(
a
1
)
,
…
,
V
A
(
a
n
)
}
:
x
≥
a
1
*
⋯
*
a
n
}
≤
isup
{
imin
{
V
C
(
a
1
)
,
…
,
V
C
(
a
n
)
}
:
x
≥
a
1
*
⋯
*
a
n
}
=
V
C
(
x
)
,
for all x
∈
L. Hence 〈
A
〉
⪯
C and then 〈
A
〉 is a vague filter generated by A.We denote the set of all vague filters of a residuated latticeL by VF
(
L
).Corollary 28.
(
VF
(
L
)
,
∩
,
∨
) is a complete lattice.Proof.
Clearly, if{
A
i
}
i
∈
Γ is a family of vague filters of a residuated lattice L, then the infimum of this family is ⋂
i
∈
Γ
A
i and the supremum is ⋁
i
∈
Γ
A
i
=
〈
⋃
i
∈
Γ
A
i
〉. It is easy to prove that (
IFF
(
L
)
,
∩
,
∨
) is a complete lattice.LetA and B be vague sets of a residuated lattice L. Then A
*
B is defined as follows:
(22)
V
(
A
*
B
)
(
z
)
=
isup
{
imin
{
V
A
(
x
)
,
V
B
(
y
)
}
:
z
≥
x
*
y
}
,
for all z
∈
L.Theorem 29.
LetA and B be vague sets of a residuated lattice L. Then A
*
B is a vague set of L.Proof.
SinceA and B are vague sets of L, we have t
A
(
x
)
≤
1
-
f
A
(
x
) and t
B
(
y
)
≤
1
-
f
B
(
y
) for all x
,
y
∈
L. Thus
(23)
t
A
*
B
(
z
)
=
sup
⁡
{
min
⁡
{
t
A
(
x
)
,
t
A
(
y
)
}
:
z
≥
x
*
y
}
≤
sup
⁡
{
min
⁡
{
1
-
f
A
(
x
)
,
1
-
f
A
(
y
)
}
:
z
≥
x
*
y
}
≤
1
-
f
A
*
B
(
z
)
.
Hence t
A
*
B
(
z
)
+
f
A
*
B
(
z
)
≤
1 and then A
*
B is a vague set of L.Theorem 30.
LetA and B be vague filters of a residuated lattice L. Then A
*
B is a vague filter of L.Proof.
By Theorem29, A
*
B is a vague set of L. We will prove that A
*
B is a vague filter of L.(VF2)
Letz
1
≤
z
2 and x
*
y
≤
z
1. We get that imin
{
V
A
(
x
)
,
V
B
(
y
)
}
≤
V
A
*
B
(
z
2
). Since x
*
y
≤
z
1 is arbitrary, we have
(24)
V
A
*
B
(
z
1
)
=
isup
{
imin
{
V
A
(
x
)
,
V
B
(
y
)
}
:
z
1
≥
x
*
y
}
≤
V
A
*
B
(
z
2
)
.
(VF1)
Letx
,
y
∈
L. Then
(25)
imin
{
V
A
*
B
(
x
)
,
V
A
*
B
(
y
)
}
=
imin
{
isup
{
imin
{
V
A
(
a
)
,
V
B
(
b
)
}
:
x
≥
a
*
b
}
,
hhhhhhhh
isup
{
imin
{
V
A
(
c
)
,
V
B
(
d
)
}
:
y
≥
c
*
d
}
}
=
isup
{
isup
{
imin
{
V
A
(
a
*
c
)
,
V
B
(
c
*
d
)
}
:
h
h
h
h
h
h
h
h
h
h
h
h
x
≥
a
*
b
V
A
}
:
y
≥
c
*
d
}
≤
isup
{
imin
{
V
A
(
a
*
c
)
,
V
B
(
c
*
d
)
}
:
h
h
h
h
h
h
h
h
h
x
*
y
≥
(
a
*
b
)
*
(
c
*
d
)
}
≤
isup
{
imin
{
V
A
(
e
)
,
V
B
(
f
)
}
:
x
*
y
≥
e
*
f
}
=
V
A
*
B
(
x
*
y
)
.By Definition 13, A
*
B is a vague filter of L.Theorem 31.
LetA and B be vague filters of a residuated lattice L such that V
A
(
1
)
=
V
B
(
1
). Then A
*
B is the least upper bound of A and B.Proof.
By Definition13, we have V
A
(
1
)
=
V
B
(
1
)
≥
V
A
(
y
) for all y
∈
L. Let z
∈
L. Since z
=
1
*
z, we get that
(26)
V
A
*
B
(
z
)
=
isup
{
imin
{
V
A
(
x
)
,
V
B
(
y
)
}
:
z
≥
x
*
y
}
≥
imin
{
V
A
(
z
)
,
V
B
(
1
)
}
≥
V
A
(
z
)
.
Hence A
⪯
A
*
B. Similarly, we can show that B
⪯
A
*
B. Suppose that C is the vague filter of L containing A and B. Then
(27)
V
A
*
B
(
z
)
=
isup
{
imin
{
V
A
(
x
)
,
V
B
(
y
)
}
:
z
≥
x
*
y
}
≤
isup
{
imin
{
V
C
(
x
)
,
V
C
(
y
)
}
:
z
≥
x
*
y
}
≤
isup
{
imin
{
V
C
(
x
*
y
)
}
:
z
≥
x
*
y
}
=
μ
C
(
z
)
.
Hence A
*
B
⪯
C; that is, A
*
B is the least upper bound of A and B.Lemma 32.
LetC be a vague filter of a residuated lattice L and z
∈
L. Then (1)
V
C
(
z
)
=
isup
{
V
C
(
x
*
y
)
:
z
≥
x
*
y
}, for all x
,
y
,
z
∈
L;
(2)
V
C
(
x
*
y
)
=
imin
{
V
C
(
x
)
,
V
C
(
y
)
}, for x
,
y
∈
L.Proof.
(1) Sincex
*
y
≤
z, then V
C
(
x
*
y
)
≤
V
C
(
z
) by (VF2). Hence V
C
(
z
)
≥
isup
{
V
C
(
x
*
y
)
:
z
≥
x
*
y
}. Also, we have 1
*
z
=
z. We obtain that
(28)
V
C
(
z
)
=
V
C
(
1
*
z
)
≤
isup
{
V
C
(
x
*
y
)
:
z
≥
x
*
y
}
.
(2) It follows from Proposition2 part (2) and Definition 13.Let VFD(L) be the set of all vague filters A and B of L such that V
A
(
1
)
=
V
B
(
1
).Theorem 33.
(
V
F
D
(
L
)
,
∩
,
*
) is a distributive lattice.Proof.
LetA
,
B
,
C
∈
VFD
(
L
) be such that A
⪯
C. We will show that (
A
*
B
)
∩
C
=
(
A
*
C
)
∩
(
B
*
C
). By Lemma 32,
(29)
V
(
A
*
B
)
∩
C
(
z
)
=
imin
{
(
V
A
*
B
)
(
z
)
,
V
C
(
z
)
}
=
imin
{
isup
{
imin
{
V
A
(
x
)
,
V
B
(
y
)
}
:
z
≥
x
*
y
}
,
V
C
(
z
)
}
=
isup
{
imin
{
V
A
(
x
)
,
V
B
(
y
)
,
V
C
(
z
)
}
:
z
≥
x
*
y
}
=
isup
{
imin
{
V
A
(
x
)
,
V
B
(
y
)
,
V
C
(
x
*
y
)
}
:
z
≥
x
*
y
}
=
isup
{
imin
{
V
A
(
x
)
,
V
B
(
y
)
,
V
C
(
x
)
,
V
C
(
y
)
}
:
z
≥
x
*
y
}
=
isup
{
imin
{
V
A
(
x
)
,
V
C
(
x
)
}
,
ssssssssssss
imin
{
V
B
(
y
)
,
V
C
(
y
)
}
:
z
≥
x
*
y
}
=
isup
{
imin
{
V
A
∩
C
(
x
)
,
V
B
∩
C
(
y
)
}
:
z
≥
x
*
y
}
=
V
(
A
∩
C
)
*
(
B
∩
C
)
(
z
)
.


## 5. Subpositive Implicative Vague Filters

Definition 34.
LetA be a vague subset of a residuated lattice L. A is called asubpositive implicative vague filter of L if(VF6)
V
A
(
x
)
≤
V
A
(
1
), for all x
∈
L,
(SVF1)
imin
{
V
A
(
(
(
x
→
y
)
*
z
)
⇝
(
(
y
⇝
x
)
→
x
)
)
,
V
A
(
z
)
}
≤
V
A
(
(
x
→
y
)
⇝
y
) for any x
,
y
,
z
∈
L,
(SVF2)
imin
{
V
A
(
(
z
*
(
x
⇝
y
)
)
→
(
(
y
→
x
)
⇝
x
)
)
,
V
A
(
z
)
}
≤
V
A
(
(
x
⇝
y
)
→
y
) for any x
,
y
,
z
∈
L.Proposition 35.
LetA be a vague filter of a residuated lattice L. Then A is a subpositive implicative vague filter of L if and only if it satisfies(SVF3)
V
A
(
(
x
→
y
)
⇝
(
(
y
⇝
x
)
→
x
)
)
≤
V
A
(
(
x
→
y
)
⇝
y
) for any x
,
y
∈
L,
(SVF4)
V
A
(
(
x
⇝
y
)
→
(
(
y
→
x
)
⇝
x
)
)
≤
V
A
(
(
x
⇝
y
)
→
y
) for any x
,
y
∈
L.Proposition 36.
LetA be a subpositive implicative vague filter of a residuated lattice L. Then A is a vague filter of L.Proof.
We haveimin
{
V
A
(
x
⇝
y
)
,
V
A
(
x
)
}
=
imin
{
V
A
(
(
y
→
y
)
*
x
)
⇝
(
(
y
⇝
y
)
→
y
)
,
V
A
(
x
)
}
≤
V
A
(
(
y
→
y
)
⇝
y
)
=
V
A
(
y
). By (VS1) and Theorem 17, A is a vague filter of a residuated lattice L.The relation between subpositive implicative vague filter and its level subset is as follows.Theorem 37.
LetA be a vague set of a residuated lattice L. Then A is a subpositive implicative vague filter of L if and only if for all α
,
β
∈
[
0,1
], the set A
(
α
,
β
) is either empty or a subpositive implicative filter of L, where α
≤
β.Proof.
Suppose thatA is a subpositive implicative filter vague set of L and α
,
β
∈
[
0,1
]. Let A
(
α
,
β
)
≠
∅. Since A is a vague filter, then A
(
α
,
β
) is a filter of L by Theorem 21. Now, let (
x
→
y
)
⇝
(
(
y
⇝
x
)
→
x
)
∈
A
(
α
,
β
). Then V
A
(
(
x
→
y
)
⇝
(
(
y
⇝
x
)
→
x
)
)
≥
[
α
,
β
]. By (SVF3), V
A
(
(
x
→
y
)
⇝
y
)
≥
[
α
,
β
]; that is, (
x
→
y
)
⇝
y
∈
A
(
α
,
β
). Similarly, we can show that A
(
α
,
β
) satisfies (SVF4). Hence A
(
α
,
β
) is a subpositive implicative filter of L.
Conversely, lett
A
(
(
x
→
y
)
⇝
(
(
y
⇝
x
)
→
x
)
)
=
α and 1
-
f
A
(
(
x
→
y
)
⇝
(
(
y
⇝
x
)
→
x
)
)
=
β. Thus V
A
(
(
x
→
y
)
⇝
(
(
y
⇝
x
)
→
x
)
)
≥
[
α
,
β
]. We get that (
x
→
y
)
⇝
(
(
y
⇝
x
)
→
x
)
∈
A
(
α
,
β
). So A
(
α
,
β
)
≠
∅. Thus (
x
→
y
)
⇝
y
∈
A
(
α
,
β
); that is, V
A
(
(
x
→
y
)
⇝
y
)
≥
[
α
,
β
]
=
V
A
(
(
x
→
y
)
⇝
(
(
y
⇝
x
)
→
x
)
). Similarly, we can prove that A satisfies (SVF4).In the following theorems, we obtain some characterizations of subpositive implicative vague filter.Theorem 38.
LetA be a vague filter of a residuated lattice L. Then A is a subpositive implicative vague filter of L if and only if it satisfies(SVF5)
imin
{
V
A
(
(
y
→
z
)
⇝
(
x
→
y
)
)
,
V
A
(
x
)
}
≤
V
A
(
y
) for any x
,
y
,
z
∈
L,
(SVF6)
imin
{
V
A
(
(
y
⇝
z
)
→
(
x
⇝
y
)
)
,
V
A
(
x
)
}
≤
V
A
(
y
) for any x
,
y
,
z
∈
L.Proof.
LetA be a subpositive implicative vague filter of L. By Proposition 2 part (3), we have imin
{
V
A
(
(
y
→
z
)
⇝
(
x
→
y
)
)
,
V
A
(
x
)
}
=
imin
{
V
A
(
x
→
(
(
y
→
z
)
⇝
y
)
)
,
V
A
(
x
)
}
≤
V
A
(
(
y
→
z
)
⇝
y
). By Proposition 2 part (7), (
y
→
z
)
⇝
y
≤
z
⇝
y
≤
(
y
→
z
)
⇝
(
(
z
⇝
y
)
→
y
). We get that V
A
(
(
y
→
z
)
⇝
y
)
≤
V
A
(
z
⇝
y
)
≤
V
A
(
(
y
→
z
)
⇝
(
(
z
⇝
y
)
→
y
)
)
≤
V
A
(
(
z
⇝
y
)
→
y
). Hence imin
{
V
A
(
(
y
→
z
)
⇝
y
)
,
V
A
(
z
⇝
y
)
}
≤
imin
{
V
A
(
(
z
⇝
y
)
→
y
)
,
V
A
(
z
⇝
y
)
}
≤
V
A
(
y
). Thus it satisfies (SVF5). Similarly, we can prove (SVF6). Conversely, let A satisfy (SVF5) and (SVF6); V
A
(
(
x
→
y
)
⇝
(
(
y
⇝
x
)
→
x
)
)
=
imin
{
V
A
(
[
(
(
x
→
y
)
⇝
y
)
→
1
]
⇝
[
(
x
→
y
)
⇝
(
(
y
⇝
x
)
→
x
)
→
(
(
x
→
y
)
⇝
y
)
]
)
,
V
A
(
1
)
}
≤
V
A
(
(
x
→
y
)
⇝
y
). Similarly, we can prove that (SVF5) holds. Hence A is a vague subpositive implicative filter.Theorem 39.
LetA be a vague filter of a residuated lattice L. Then A is a subpositive implicative vague filter of L if and only if it satisfies(SVF7)
V
A
(
(
x
→
y
)
⇝
(
(
y
→
x
)
⇝
x
)
)
≤
V
A
(
(
y
⇝
x
)
→
x
) for any x
,
y
∈
L;
(SVF8)
V
A
(
(
x
⇝
y
)
→
(
(
y
⇝
x
)
→
x
)
)
≤
V
A
(
(
y
→
x
)
⇝
x
) for any x
,
y
∈
L.Proof.
Let (SVF7) and (SVF8) hold. We will show thatA satisfies (SVF5) and (SVF6). We have imin
{
V
A
(
(
y
→
z
)
⇝
(
x
→
y
)
)
,
V
A
(
x
)
}
=
imin
{
V
A
(
x
→
(
(
y
→
z
)
⇝
y
)
)
,
V
A
(
x
)
}
≤
V
A
(
(
y
→
z
)
⇝
y
)
≤
V
A
(
(
y
→
z
)
⇝
(
(
z
→
y
)
⇝
y
)
)
≤
V
A
(
(
z
⇝
y
)
→
y
)
). Also, we have V
A
(
(
y
→
z
)
⇝
y
)
≤
V
A
(
z
⇝
y
). Since A is a vague filter, V
A
(
(
y
→
z
)
⇝
y
)
≤
imin
{
V
A
(
z
⇝
y
)
,
V
A
(
(
z
⇝
y
)
→
y
)
}
≤
V
A
(
y
). Hence A satisfies (SVF5). Analogously, we can prove that A satisfies (SVF6). Hence A is a subpositive implicative vague filter of L by Theorem 38. Conversely, let A be a subpositive implicative vague filter of L. We have
(30)
[
(
(
y
⇝
x
)
⟶
x
)
⟶
1
]
⇝
[
(
(
x
⟶
y
)
⇝
(
(
y
⟶
x
)
⇝
x
)
)
⟶
(
(
y
⇝
x
)
⟶
x
)
(
(
x
⟶
y
)
⇝
(
(
y
⟶
x
)
⇝
x
)
)
]
=
[
(
x
⟶
y
)
⇝
(
(
y
⟶
x
)
⇝
x
)
]
⟶
[
(
(
y
⟶
x
)
⇝
x
)
⇝
(
(
y
⟶
x
)
⇝
x
)
]
=
(
x
⟶
y
)
⇝
(
(
y
⟶
x
)
⇝
x
)
.
We get that V
A
(
(
x
→
y
)
⇝
(
(
y
→
x
)
⇝
x
)
)
=
imin
{
V
A
(
(
x
→
y
)
⇝
(
(
y
→
x
)
⇝
x
)
),V
A
(
[
(
(
y
⇝
x
)
→
x
)
→
1
]
⇝
[
(
(
x
→
y
)
⇝
(
(
y
→
x
)
⇝
x
)
)
→
(
(
y
⇝
x
)
→
x
)
]
)
}
≤
V
A
(
(
y
⇝
x
)
→
x
) by (SVF5). Similarly, we can prove that (SVF8) holds.Theorem 40.
LetA be a vague filter of a residuated lattice L. Then A is a subpositive implicative vague filter of L if and only if it satisfies(SVF9)
V
A
(
(
x
→
y
)
⇝
x
)
≤
V
A
(
x
) for any x
,
y
∈
L,
(SVF10)
V
A
(
(
x
⇝
y
)
→
x
)
≤
V
A
(
x
) for any x
,
y
∈
L.Proof.
LetA be a subpositive implicative vague filter of L. Then A satisfies (SVF5) and (SVF6):
(31)
V
A
(
(
x
⟶
y
)
⇝
x
)
=
imin
{
V
A
(
1
⟶
(
(
x
⟶
y
)
⇝
x
)
)
,
V
A
(
1
)
}
=
imin
{
V
A
(
(
x
⟶
y
)
⇝
(
1
⟶
x
)
)
,
V
A
(
1
)
}
≤
V
A
(
x
)
.
Similarly, we can prove that (SVF10) holds. Conversely, let A satisfy (SVF9) and (SVF10). We will show that A satisfies (SVF5) and (SVF6); imin
{
V
A
(
(
y
→
z
)
⇝
(
x
→
y
)
)
,
V
A
(
x
)
}
=
imin
{
V
A
(
(
x
→
(
y
→
z
)
⇝
y
)
)
,
V
A
(
x
)
}
≤
V
A
(
(
y
→
z
)
⇝
y
)
≤
V
A
(
y
). Hence A satisfies (SVF5). Similarly, we can prove (SVF7) holds. Therefore A is a subpositive implicative vague filter of L by Theorem 38.Theorem 41.
LetA be a vague filter of a residuated lattice L. Then A is a subpositive implicative vague filter of L if and only if it satisfies(SVF11)
V
A
(
¬
x
⇝
x
)
≤
V
A
(
x
) for any x
∈
L,
(SVF12)
V
A
(
~
x
→
x
)
≤
V
A
(
x
) for any x
∈
L.Proof.
LetA be a subpositive implicative vague filter of L. Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10). Conversely, since 0
≤
y, then x
→
0
≤
x
→
y. Hence (
x
→
y
)
⇝
x
≤
¬
x
⇝
x by Proposition 2 part (6). Therefore V
A
(
(
x
→
y
)
⇝
x
)
≤
V
A
(
¬
x
⇝
x
)
≤
V
A
(
x
). Similarly, we can prove that (SVF10) holds. Hence A is a subpositive implicative vague filter of L by Theorem 40.The extension theorem of subpositive implicative vague filters is obtained from the following theorem.Theorem 42.
LetA and B be two vague filters of a residuated lattice L which satisfies A
≤
B and V
A
(
1
)
=
V
B
(
1
). If A is a subpositive implicative vague filter of L, so B is.Proof.
Suppose thatu
=
(
x
→
y
)
⇝
x. Then
(32)
1
=
u
⟶
u
=
u
⟶
(
(
x
⟶
y
)
⇝
x
)
=
(
x
⟶
y
)
⇝
(
u
⟶
x
)
=
1
.
We have (
x
→
y
)
⇝
(
u
→
x
)
≤
(
(
u
→
x
)
→
y
)
⇝
(
u
→
x
). Therefore V
A
(
(
(
u
→
x
)
→
y
)
⇝
(
u
→
x
)
)
=
V
A
(
1
). By (SVF10), V
A
(
1
)
=
V
A
(
(
(
u
→
x
)
→
y
)
⇝
(
u
→
x
)
)
≤
V
A
(
u
→
x
). Hence V
A
(
u
→
x
)
=
V
A
(
1
)
=
V
B
(
1
). Since A
≤
B, then V
B
(
u
→
x
)
≥
V
A
(
u
→
x
)
=
V
B
(
1
). We get that V
B
(
(
x
→
y
)
⇝
x
)
=
V
B
(
u
)
=
imin
{
V
B
(
u
→
x
)
,
V
B
(
u
)
}
≤
V
B
(
x
). Similarly, we can show that G satisfies (SVF11). Hence G is a vague subpositive implicative filter of L, by Theorem 40.Definition 43.
LetA be a vague filter of a residuated lattice L. A is called avague Boolean filter of L if(BVF1)
V
A
(
x
∨
¬
x
)
=
V
A
(
1
), for all x
∈
L,
(BVF2)
V
A
(
x
∨
~
x
)
=
V
A
(
1
), for all x
∈
L.Similar to Theorem40, we have the following theorem with respect to Boolean vague filters.Theorem 44.
LetA be a vague set of a residuated lattice L. Then A is a Boolean vague filter of L if and only if for all α
,
β
∈
[
0,1
]; the set A
(
α
,
β
) is either empty or a Boolean filter of L, where α
≤
β.Theorem 45.
LetA be a vague filter of a residuated lattice L. Then A is a subpositive implicative vague filter of L if and if only if A is a Boolean vague filter.Proof.
LetA be a subpositive implicative vague filter. We have
(33)
V
A
(
1
)
=
V
A
(
¬
(
x
∨
¬
x
)
⇝
(
x
∨
¬
x
)
)
=
V
A
(
(
¬
x
∧
¬
¬
x
)
⇝
(
x
∨
¬
x
)
)
≤
V
A
(
x
∨
¬
x
)
.
Hence V
A
(
x
∨
¬
x
)
=
V
A
(
1
). Similarly, we can prove that A satisfies (BVF2). Hence A is a Boolean vague filter.
Conversely, letV
A
(
x
∨
¬
x
)
=
V
A
(
1
). We have V
A
(
x
∨
¬
x
)
=
V
A
(
(
x
⇝
x
)
∨
(
¬
x
⇝
x
)
)
V
A
(
(
x
∨
¬
x
)
⇝
x
)
=
imin
{
V
A
(
(
x
∨
¬
x
)
⇝
x
)
,
V
A
(
1
)
}
=
imin
{
V
A
(
(
x
∨
¬
x
)
⇝
x
)
,
V
A
(
x
∨
¬
x
)
}
≤
V
A
(
x
). Similarly, we can show that A satisfies (SVF12). Hence A is a subpositive implicative vague filter by Theorem 41.


## 6. Conclusion

In this paper, we introduced the notions of vague filters, subpositive implicative vague filters, and Boolean vague filters of a residuated lattice and investigated some related properties. We studied the relationship between filters and vague filters of a residuated lattice and showed that vague filters of a residuated lattice is a generalization of fuzzy filters. We proved that the set of all vague filters of a residuated lattice forms a complete lattice and obtained some characterizations of subpositive implicative vague filter. The extension theorem of subpositive implicative vague filters was obtained. Finally, it was proved that subpositive implicative vague filters are equivalent to Boolean vague filters.



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*Source: 120342-2014-09-10.xml*