# Fuzzy-Fated Filters ofR

Authors: Jeong Soon Han; Young Bae Jun; Hee Sik Kim
Journal: Discrete Dynamics in Nature and Society
 (2011)

---

## Abstract

Fuzzy set theory of fated filters inR
0-algebras is considered. A characterization
of a fuzzy-fated filter is established, and conditions for a fuzzy filter to
be a fuzzy-fated filter are provided. The notion of an (
∈
,
∈
∨
q
)-fuzzy-fated
filter is introduced. Characterizations of an (
∈
,
∈
∨
q
)-fuzzy-fated filter are provided. Implication-based fuzzy-fated filters are discussed.


---

## Body

## 1. Introduction

One important task of artificial intelligence is to make the computers simulate beings in dealing with certainty and uncertainty in information. Logic appears in a “sacred” (resp., a “profane”) form which is dominant in proof theory (resp., model theory). The role of logic in mathematics and computer science is twofold—as a tool for applications in both areas—and a technique for laying the foundations. Nonclassical logic including many-valued logic and fuzzy logic, takes the advantage of classical logic to handle information with various facets of uncertainty (see [1] for generalized theory of uncertainty), such as fuzziness randomness and Nonclassical logic has become a formal and useful tool for computer science to deal with fuzzy information and uncertain information. Among all kinds of uncertainties, incomparability is an important one which can be encountered in our life. The concept of R
0-algebras was first introduced by Wang in [2] by providing an algebraic proof of the completeness theorem of a formal deductive system [3]. Obviously, R
0-algebras are different from the BL-algebras. Jun and Lianzhen [4] studied (fated) filters of R
0-algebras. Lianzhen and Kaitai [5] discussed the fuzzy set theory of filters in R
0-algebras. The algebraic structure of set theories dealing with uncertainties has been studied by some authors. The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh [6]. Murali [7] proposed a definition of a fuzzy point belonging to fuzzy subset under a natural equivalence on fuzzy subset. The idea of quasicoincidence of a fuzzy point with a fuzzy set, which is mentioned in [8], played a vital role to generate some different types of fuzzy subsets. It is worth pointing out that Bhakat and Das [9, 10] initiated the concepts of (
α
,
β
)-fuzzy subgroups by using the “belongs to” relation (
∈
) and “quasicoincident with” relation (
q
) between a fuzzy point and a fuzzy subgroup and introduced the concept of an (
∈
,
∈
∨
q
)-fuzzy subgroup. In particular, an (
∈
,
∈
∨
q
)-fuzzy subgroup is an important and useful generalization of Rosenfeld's fuzzy subgroup. As a generalization of the notion of fuzzy filters in R
0-algebras, Ma et al. [11] dealt with the notion of (
∈
,
∈
∨
q
)-fuzzy filters in R
0-algebras.In this paper, we deal with the fuzzy set theory of fated filters inR
0-algebras. We provide conditions for a fuzzy filter to be a fuzzy-fated filter. We also introduce the notion of (
∈
,
∈
∨
q
)-fuzzy-fated filters and investigate related properties. We establish a relation between an (
∈
,
∈
∨
q
)-fuzzy filter and an (
∈
,
∈
∨
q
)-fuzzy-fated filter and provide conditions for an (
∈
,
∈
∨
q
)-fuzzy filter to be an (
∈
,
∈
∨
q
)-fuzzy-fated filter. We deal with characterizations of an (
∈
,
∈
∨
q
)-fuzzy-fated filter. Finally, we discuss the implication-based fuzzy-fated filters of an R
0-algebra.


## 2. Preliminaries

LetL be a bounded distributive lattice with order-reversing involution ¬ and a binary operation →, then (
L
,
∧
,
∨
,
¬
,
→
) is called an R
0
-algebra (see [2]) if it satisfies the following axioms:  (R1)
x
→
y
=
¬
y
→
¬
x,
(R2)
1
→
x
=
x,
(R3)
(
y
→
z
)
∧
(
(
x
→
y
)
→
(
x
→
z
)
)
=
y
→
z,
(R4)
x
→
(
y
→
z
)
=
y
→
(
x
→
z
),
(R5)
x
→
(
y
∨
z
)
=
(
x
→
y
)
∨
(
x
→
z
),
(R6)
(
x
→
y
)
∨
(
(
x
→
y
)
→
(
¬
x
∨
y
)
)
=
1.LetL be an R
0-algebra. For any x
,
y
∈
L, we define x
⊙
y
=
¬
(
x
→
¬
y
) and x
⊕
y
=
¬
x
→
y. It is proved that ⊙ and ⊕ are commutative, associative, and x
⊕
y
=
¬
(
¬
x
⊙
¬
y
), and (
L
,
∧
,
∨
,
⊙
,
→
,
0,1
) is a residuated lattice.For any elementsx
,
y, and z of an R
0-algebra L, we have the following properties (see  [12]):  (a1)
x
≤
y  if and only if x
→
y
=
1,
(a2)
x
≤
y
→
x,
(a3)
¬
x
=
x
→
0,
(a4)
(
x
→
y
)
∨
(
y
→
x
)
=
1,
(a5)
x
≤
y implies y
→
z
≤
x
→
z,
(a6)
x
≤
y implies z
→
x
≤
z
→
y,
(a7)
(
(
x
→
y
)
→
y
)
→
y
=
x
→
y,
(a8)
x
∨
y
=
(
(
x
→
y
)
→
y
)
∧
(
(
y
→
x
)
→
x
),
(a9)
x
⊙
¬
x
=
0 and x
⊕
¬
x
=
1,
(a10)
x
⊙
y
≤
x
∧
y and x
⊙
(
x
→
y
)
≤
x
∧
y,
(a11)
(
x
⊙
y
)
→
z
=
x
→
(
y
→
z
),
(a12)
x
≤
y
→
(
x
⊙
y
),
(a13)
x
⊙
y
≤
z  if and only if x
≤
y
→
z,
(a14)
x
≤
y implies x
⊙
z
≤
y
⊙
z,
(a15)
x
→
y
≤
(
y
→
z
)
→
(
x
→
z
),
(a16)
(
x
→
y
)
⊙
(
y
→
z
)
≤
x
→
z.A nonempty subsetA of an R
0-algebra L is called a filter of L if it satisfies the following two conditions:  (b1)
1
∈
A,
(b2)
(
∀
x
∈
A
)
(
∀
y
∈
L
)
(
x
→
y
∈
A
⇒
y
∈
A
).It can be easily verified that a nonempty subsetA of an R
0-algebra L is a filter of L if and only if it satisfies the following conditions:  (b3)
(
∀
x
,
y
∈
A
)
(
x
⊙
y
∈
A
),
(b4)
(
∀
y
∈
L
)
(
∀
x
∈
A
)
(
x
≤
y
⇒
y
∈
A
).Definition 2.1.
A fuzzy subsetμ of an R
0-algebra L is called a fuzzy filter of L if it satisfies:  (c1)
(
∀
x
,
y
∈
L
)
(
μ
(
x
⊙
y
)
≥
min
{
μ
(
x
)
,
μ
(
y
)
}
),
(c2)
μ is order preserving, that is, (
∀
x
,
y
∈
L
)
(
x
≤
y
⇒
μ
(
x
)
≤
μ
(
y
)
).Denote byF
(
L
) the set of all filters of L, and by ℱ
F
(
L
) the set of all fuzzy filters of L.Theorem 2.2.
A fuzzy subsetμ of an R
0-algebra L is a fuzzy filter of L  if and only if it satisfies the following:  (c3)
(
∀
x
∈
L
)
(
μ
(
1
)
≥
μ
(
x
)
),
(c4)
(
∀
x
,
y
∈
L
)
(
μ
(
y
)
≥
min
{
μ
(
x
→
y
)
,
μ
(
x
)
}
).For any fuzzy subsetμ of L and t
∈
(
0,1
], the setU
(
μ
;
t
)
=
{
x
∈
L
∣
μ
(
x
)
≥
t
} is called a level subset of L. It is well known that a fuzzy subset μ of L is a fuzzy filter of L  if and only if the nonempty level subset U
(
μ
;
t
)
,
t
∈
(
0,1
], of μ is a filter of L.A fuzzy subsetμ of a set L of the form (2.1)
μ
(
y
)
:
=
{
t
∈
(
0,1
]
if
y
=
x
,
0
if
y
≠
x
,
is said to be a fuzzy point with support x and value t and is denoted by (
x
,
t
).


## 3. Fuzzy-Fated Filters

In what follows,L is an R
0-algebra unless otherwise specified. In [4], the notion of a fated filter of L is introduced as follows.A nonempty subsetA of L is called a fated filter of L (see [4]) if it satisfies (b1) and(3.1)
(
∀
x
,
y
∈
L
)
(
∀
a
∈
A
)
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
∈
A
⟹
x
∈
A
)
.Denote byF
F
(
L
) the set of all fated filters of L. Note that F
F
(
L
) is a complete lattice under the set inclusion with the largest element L and the least element {
1
}. Now, we consider the fuzzy form of a fated filter of L.Lemma 3.1 (see [4]).
A filterF of L is fated  if and only if the following assertion is valid:
(3.2)
(
∀
x
,
y
,
z
∈
L
)
(
x
⟶
(
y
⟶
z
)
∈
F
,
x
⟶
y
∈
F
⟹
x
⟶
z
∈
F
)
.Lemma 3.2 (see [4]).
A filterF of L is fated  if and only if the following assertion is valid:
(3.3)
(
∀
x
,
y
∈
L
)
(
(
x
⟶
y
)
⟶
x
∈
F
⟹
x
∈
F
)
.Definition 3.3.
A fuzzy subsetμ of L is called a fuzzy-fated filter of L if it satisfies the following assertion:
(3.4)
(
∀
t
∈
[
0,1
]
)
(
U
(
μ
;
t
)
∈
F
F
(
L
)
∪
{
∅
}
)
.Denote byℱ
F
F
(
L
) the set of all  fuzzy-fated filters of L.Example 3.4.
LetL
=
{
0
,
a
,
b
,
c
,
d
,
1
} be a set with Hasse diagram and Cayley tables which are given in Table 1, then (
L
,
∧
,
∨
,
¬
,
→
,
0,1
) is an R
0-algebra (see [5]), where x
∧
y
=
min
{
x
,
y
} and x
∨
y
=
max
{
x
,
y
}. Define a fuzzy subset μ of L by
(3.5)
μ
:
L
⟶
[
0,1
]
,
x
⟼
{
0.7
if
x
∈
{
c
,
d
,
1
}
,
0.2
otherwise
.
Then
(3.6)
U
(
μ
;
t
)
=
{
∅
if
0.7
<
t
≤
1
,
{
c
,
d
,
1
}
if
0.2
<
t
≤
0.7
,
L
if
0
≤
t
≤
0.2
,
which is a fated filter of L. Therefore, μ is a  fuzzy-fated filter of L.Table 1
Hasse diagram and Cayley tables.We provide a characterization of a  fuzzy-fated filter.Theorem 3.5.
For a fuzzy subsetμ of L
,
μ
∈
ℱ
F
F
(
L
)  if and only if it satisfies the following conditions:  (1)
(
∀
x
∈
L
)
(
μ
(
1
)
≥
μ
(
x
)
),
(2)
(
∀
x
,
a
,
y
∈
L
)
(
μ
(
x
)
≥
min
{
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
,
μ
(
a
)
}
).Proof.
Suppose thatμ is a  fuzzy-fated filter of L. For any x
∈
L, let μ
(
x
)
=
t, then x
∈
U
(
μ
;
t
), that is, U
(
μ
;
t
)
≠
∅, and so U
(
μ
;
t
) is a fated filter of L. Thus, 1
∈
U
(
μ
;
t
), and hence μ
(
1
)
≥
t
=
μ
(
x
) for all x
∈
L. For any x
,
a
,
y
∈
L, let
(3.7)
t
a
:
=
min
⁡
{
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
μ
(
a
)
}
.
Then a
∈
U
(
μ
;
t
a
) and a
→
(
(
x
→
y
)
→
x
)
∈
U
(
μ
;
t
a
). Since U
(
μ
;
t
a
) is a fated filter of L, it follows from (3.1) that x
∈
U
(
μ
;
t
a
), so that
(3.8)
μ
(
x
)
≥
t
a
=
min
⁡
{
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
μ
(
a
)
}
,
for all x
,
a
,
y
∈
L.
Conversely, letμ satisfy two conditions (1) and (2), and let t
∈
[
0,1
] be such that U
(
μ
;
t
)
≠
∅, then there exists x
0
∈
U
(
μ
;
t
) such that μ
(
x
0
)
≥
t. Using (1), we have μ
(
1
)
≥
μ
(
x
0
)
≥
t, and so 1
∈
U
(
μ
;
t
). Let x
,
a
,
y
∈
L be such that a
∈
U
(
μ
;
t
) and a
→
(
(
x
→
y
)
→
x
)
∈
U
(
μ
;
t
), then μ
(
a
)
≥
t and μ
(
a
→
(
(
x
→
y
)
→
x
)
)
≥
t. It follows from (2) that
(3.9)
μ
(
x
)
≥
min
⁡
{
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
μ
(
a
)
}
≥
t
,
so that x
∈
U
(
μ
;
t
). Hence, U
(
μ
;
t
) is a fated filter of L, and therefore μ is a  fuzzy-fated filter of L.Theorem 3.6.
For any fuzzy subsetμ of L, one has ℱ
F
F
(
L
)
⊆
ℱ
F
(
L
).Proof.
Letμ be a  fuzzy-fated filter of L. Replacing a and x by x and y, respectively, in Theorem 3.5(2) and using (R2), we have
(3.10)
μ
(
y
)
≥
min
⁡
{
μ
(
x
⟶
(
(
y
⟶
y
)
⟶
y
)
)
,
μ
(
x
)
}
=
min
⁡
{
μ
(
x
⟶
(
1
⟶
y
)
)
,
μ
(
x
)
}
=
min
⁡
{
μ
(
x
⟶
y
)
,
μ
(
x
)
}
,
for all x
,
y
∈
L. Using Theorem 2.2, μ is a fuzzy filter of L.The following example shows that the converse of Theorem3.6 may not be true.Example 3.7.
LetL
=
{
0
,
a
,
b
,
c
,
1
} be a set with Hasse diagram and Cayley tables which are given in Table 2, then (
L
,
∧
,
∨
,
¬
,
→
,
0,1
) is an R
0-algebra (see [5]), where x
∧
y
=
min
{
x
,
y
} and x
∨
y
=
max
{
x
,
y
}. Define a fuzzy subset μ of L by
(3.11)
μ
:
L
⟶
[
0,1
]
,
x
⟼
{
0.6
if
x
∈
{
c
,
1
}
,
0.2
otherwise
.
Then μ is a fuzzy filter of L, but it is not a  fuzzy-fated filter of L since
(3.12)
μ
(
b
)
=
0.2
≱
0.6
=
min
⁡
{
μ
(
c
⟶
(
(
b
⟶
a
)
⟶
b
)
)
,
μ
(
c
)
}
.Table 2
Hasse diagram and Cayley tables.Theorem 3.8.
For any fuzzy filterμ of L,  the following assertions are equivalent:  (1)
μ is a  fuzzy-fated filter of L,
(2)
μ satisfies the following inequality:
(3.13)
(
∀
x
,
y
∈
L
)
(
μ
(
x
)
≥
μ
(
(
x
⟶
y
)
⟶
x
)
)
.Proof.
Assume thatμ is a  fuzzy-fated filter of L. Putting a
=
1 in Theorem 3.5(2) and using (R2) and Theorem 3.5(1), we have
(3.14)
μ
(
x
)
≥
min
⁡
{
μ
(
1
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
μ
(
1
)
}
=
min
⁡
{
μ
(
(
x
⟶
y
)
⟶
x
)
,
μ
(
1
)
}
=
μ
(
(
x
⟶
y
)
⟶
x
)
,
for all x
,
y
∈
L.
Conversely, suppose thatμ satisfies (3.13), it follows from (c4) that
(3.15)
μ
(
x
)
≥
μ
(
(
x
⟶
y
)
⟶
x
)
≥
min
⁡
{
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
μ
(
a
)
}
,
for all x
,
a
,
y
∈
L, so from Theorem 3.5 that μ is a  fuzzy-fated filter of L.Theorem 3.9.
Letμ be a fuzzy filter of L, then μ
∈
ℱ
F
F
(
L
)  if and only if it satisfies
(3.16)
μ
(
x
⟶
z
)
≥
min
⁡
{
μ
(
x
⟶
(
y
⟶
z
)
)
,
μ
(
x
⟶
y
)
}
,
for all x
,
y
,
z
∈
L.Proof.
Assume thatμ
∈
ℱ
F
F
(
L
). If U
(
μ
;
t
)
≠
∅ for all t
∈
[
0,1
], then U
(
μ
;
t
)
∈
F
F
(
L
). Suppose that
(3.17)
μ
(
x
⟶
z
)
<
min
⁡
{
μ
(
x
⟶
(
y
⟶
z
)
)
,
μ
(
x
⟶
y
)
}
,
for some x
,
y
,
z
∈
L, then there exists t
∈
(
0,1
] such that
(3.18)
μ
(
x
⟶
z
)
<
t
≤
min
⁡
{
μ
(
x
⟶
(
y
⟶
z
)
)
,
μ
(
x
⟶
y
)
}
.
It follows that x
→
(
y
→
z
)
∈
U
(
μ
;
t
) and x
→
y
∈
U
(
μ
;
t
), so from Lemma 3.1 that x
→
z
∈
U
(
μ
;
t
), that is, μ
(
x
→
z
)
≥
t. This is a contradiction, and so μ
(
x
→
z
)
≥
min
{
μ
(
x
→
(
y
→
z
)
)
,
μ
(
x
→
y
)
} for all x
,
y
,
z
∈
L.
Conversely, letμ be a fuzzy filter of L that satisfies (3.16). Let t
∈
[
0,1
] be such that U
(
μ
;
t
)
≠
∅, then U
(
μ
;
t
)
∈
F
(
L
) by Theorem 3.5 in [5]. Assume that x
→
(
y
→
z
)
∈
U
(
μ
;
t
) and x
→
y
∈
U
(
μ
;
t
) for all x
,
y
,
z
∈
L, then μ
(
x
→
(
y
→
z
)
)
≥
t and μ
(
x
→
y
)
≥
t. Using (3.16), we have
(3.19)
μ
(
x
⟶
z
)
≥
min
⁡
{
μ
(
x
⟶
(
y
⟶
z
)
)
,
μ
(
x
⟶
y
)
}
≥
t
,
and so x
→
z
∈
U
(
μ
;
t
). Therefore, U
(
μ
;
t
)
∈
F
F
(
L
), and thus μ is a  fuzzy-fated filter of L.Remark 3.10.
Based on Theorem3.9 and [5, Definition 4.1], we know that the notion of a fuzzy-fated filter is equivalent to the notion of a fuzzy implicative filter.


## 4. Fuzzy-Fated Filters Based on Fuzzy Points

For a fuzzy point(
x
,
t
) and a fuzzy subset μ of L, Pu and Liu [8] introduced the symbol (
x
,
t
)
α
μ, where α
∈
{
∈
,
q
,
∈
∨
q
}. We say that  (i)
(
x
,
t
)
belong to
μ, denoted by (
x
,
t
)
∈
μ if μ
(
x
)
≥
t,
(ii)
(
x
,
t
) is quasicoincident with
μ, denoted by (
x
,
t
)
q
μ, if μ
(
x
)
+
t
>
1,
(iii)
(
x
,
t
)
∈
∨
q
μ if (
x
,
t
)
∈
μ or (
x
,
t
)
q
μ,
(iv)
(
x
,
t
)
α
¯
μ if (
x
,
t
)
α
μ does not hold for α
∈
{
∈
,
q
,
∈
∨
q
}.Definition 4.1 (see [11]).
A fuzzy subsetμ of L is said to be an  (
∈
,
∈
∨
q
)
-fuzzy filter of L if it satisfies  (1)
(
x
,
t
)
∈
μ
and
(
y
,
r
)
∈
μ
⇒
(
x
⊙
y
,
min
{
t
,
r
}
)
∈
∨
q
μ,
(2)
(
x
,
t
)
∈
μ
and
x
≤
y
⇒
(
y
,
t
)
∈
∨
q
μ, for all x
,
y
∈
L and t
,
r
∈
(
0,1
].Theorem 4.2 (see [11]).
A fuzzy subsetμ of L is an (
∈
,
∈
∨
q
)-fuzzy filter of L  if and only if the following conditions are valid:  (1)
(
∀
x
∈
L
)
(
μ
(
1
)
≥
min
{
μ
(
x
)
,
0.5
}
),
(2)
(
∀
x
,
y
∈
L
)
(
μ
(
y
)
≥
min
{
μ
(
x
)
,
μ
(
x
→
y
)
,
0.5
}
).Definition 4.3.
A fuzzy subsetμ of L is said to be an  (
∈
,
∈
∨
q
)
-fuzzy-fated filter of L if it satisfies  (1)
(
x
,
t
)
∈
μ
⇒
(
1
,
t
)
∈
∨
q
μ,
(2)
(
a
→
(
(
x
→
y
)
→
x
)
,
t
)
∈
μ
and
(
a
,
r
)
∈
μ
⇒
(
x
,
min
{
t
,
r
}
)
∈
∨
q
μ, for all x
,
a
,
y
∈
L and t
,
r
∈
(
0,1
].If a fuzzy subsetμ of L satisfies (c3) and Definition 4.3(2), then we say that μ is a strong
(
∈
,
∈
∨
q
)
-fuzzy-fated filter of L.Example 4.4.
Consider anR
0-algebra L
=
{
0
,
a
,
b
,
c
,
d
,
1
} which appeared in Example 3.4. Define a fuzzy subset μ of L by
(4.1)
μ
=
(
0
a
b
c
d
1
0.3
0.3
0.3
0.7
0.6
0.8
)
.
It is routine to verify that μ is a strong (
∈
,
∈
∨
q
)-fuzzy-fated filter of L. A fuzzy subset ν of L defined by
(4.2)
ν
=
(
0
a
b
c
d
1
0.4
0.4
0.4
0.8
0.8
0.7
)
is an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L, but it is not a strong (
∈
,
∈
∨
q
)-fuzzy-fated filter of L.Obviously, every strong(
∈
,
∈
∨
q
)-fuzzy-fated filter is an (
∈
,
∈
∨
q
)-fuzzy-fated filter, but not converse as seen in Example 4.4.We provided characterizations of an(
∈
,
∈
∨
q
)-fuzzy-fated filter.Theorem 4.5.
A fuzzy subsetμ of L is an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L  if and only if it satisfies the following inequalities:  (1)
(
∀
x
∈
L
)
(
μ
(
1
)
≥
min
{
μ
(
x
)
,
0.5
}
),
(2)
(
∀
x
,
a
,
y
∈
L
)
(
μ
(
x
)
≥
min
{
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
,
μ
(
a
)
,
0.5
}
).Proof.
Letμ be an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L. Assume that there exists a
∈
L such that μ
(
1
)
<
min
{
μ
(
a
)
,
0.5
}, then μ
(
1
)
<
t
≤
min
{
μ
(
a
)
,
0.5
} for some t
∈
(
0,0.5
], and so (
a
,
t
)
∈
μ. It follows from Definition 4.3(1) that (
1
,
t
)
∈
∨
q
μ, that is, (
1
,
t
)
∈
μ or (
1
,
t
)
q
μ, so that μ
(
1
)
≥
t or μ
(
1
)
+
t
>
1. This is a contradiction. Hence, μ
(
1
)
≥
min
{
μ
(
x
)
,
0.5
} for all x
∈
L. Suppose that there exist a
,
b
,
c
∈
L such that
(4.3)
μ
(
b
)
<
min
⁡
{
μ
(
a
⟶
(
(
b
⟶
c
)
⟶
b
)
)
,
μ
(
a
)
,
0.5
}
,
then μ
(
b
)
<
t
b
≤
min
{
μ
(
a
→
(
(
b
→
c
)
→
b
)
)
,
μ
(
a
)
,
0.5
} for some t
b
∈
(
0,0.5
]. Thus (
a
→
(
(
b
→
c
)
→
b
)
,
t
b
)
∈
μ and (
a
,
t
b
)
∈
μ. Using Definition 4.3(2), we have (
b
,
t
b
)
=
(
b
,
min
{
t
b
,
t
b
}
)
∈
∨
q
μ, which implies that μ
(
b
)
≥
t
b or μ
(
b
)
+
t
b
>
1. This is a contradiction, and therefore
(4.4)
μ
(
x
)
≥
min
⁡
{
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
μ
(
a
)
,
0.5
}
,
for all x
,
a
,
y
∈
L. Conversely, let μ be a fuzzy subset of L that satisfies two conditions (1) and (2). Let x
∈
L and t
∈
(
0,1
] be such that (
x
,
t
)
∈
μ, then μ
(
x
)
≥
t, which implies from (1) that μ
(
1
)
≥
min
{
μ
(
x
)
,
0.5
}
≥
min
{
t
,
0.5
}. If t
≤
0.5, then μ
(
1
)
≥
t, that is, (
1
,
t
)
∈
μ. If t
>
0.5, then μ
(
1
)
≥
0.5 and so μ
(
1
)
+
t
>
0.5
+
0.5
=
1, that is, (
1
,
t
)
q
μ. Hence, (
1
,
t
)
∈
∨
q
μ. Let x
,
a
,
y
∈
L and t
,
r
∈
(
0,1
] be such that (
a
→
(
(
x
→
y
)
→
x
)
,
t
)
∈
μ and (
a
,
r
)
∈
μ, then μ
(
a
→
(
(
x
→
y
)
→
x
)
)
≥
t and μ
(
a
)
≥
r. It follows from (2) that
(4.5)
μ
(
x
)
≥
min
⁡
{
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
μ
(
a
)
,
0.5
}
≥
min
⁡
{
t
,
r
,
0.5
}
.
If min
{
t
,
r
}
≤
0.5, then μ
(
x
)
≥
min
{
t
,
r
}, which shows that (
x
,
min
{
t
,
r
}
)
∈
μ. If min
{
t
,
r
}
>
0.5, then μ
(
x
)
≥
0.5, and thus μ
(
x
)
+
min
{
t
,
r
}
>
1, that is, (
x
,
min
{
t
,
r
}
)
q
μ. Hence, (
x
,
min
{
t
,
r
}
)
∈
∨
q
μ. Consequently, μ is an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L.Corollary 4.6.
Every strong(
∈
,
∈
∨
q
)-fuzzy-fated filter μ of L satisfies the following inequalities:  (1)
(
∀
x
∈
L
)
(
μ
(
1
)
≥
min
{
μ
(
x
)
,
0.5
}
),
(2)
(
∀
x
,
a
,
y
∈
L
)
(
μ
(
x
)
≥
min
{
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
,
μ
(
a
)
,
0.5
}
).Theorem 4.7.
A fuzzy subsetμ of L is an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L  if and only if it satisfies the following assertion:
(4.6)
(
∀
t
∈
(
0,0.5
]
)
(
U
(
μ
;
t
)
∈
F
F
(
L
)
∪
{
∅
}
)
.Proof.
Assume thatμ is an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L. Let t
∈
(
0,0.5
] be such that U
(
μ
;
t
)
≠
∅, then there exists x
∈
U
(
μ
;
t
), and so μ
(
x
)
≥
t. Using Theorem 4.5(1), we get
(4.7)
μ
(
1
)
≥
min
⁡
{
μ
(
x
)
,
0.5
}
≥
min
⁡
{
t
,
0.5
}
=
t
,
that is, 1
∈
U
(
μ
;
t
). Assume that a
→
(
(
x
→
y
)
→
x
)
∈
U
(
μ
;
t
) for all x
,
y
∈
L and a
∈
U
(
μ
;
t
), then μ
(
a
→
(
(
x
→
y
)
→
x
)
)
≥
t and μ
(
a
)
≥
t. It follows from Theorem 4.5 (2) that
(4.8)
μ
(
x
)
≥
min
⁡
{
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
μ
(
a
)
,
0.5
}
≥
min
⁡
{
t
,
0.5
}
=
t
,
so that x
∈
U
(
μ
;
t
). Therefore, U
(
μ
;
t
) is a fated filter of L.
Conversely, letμ be a fuzzy subset of L satisfying the assertion (4.6). Assume that μ
(
1
)
<
min
{
μ
(
a
)
,
0.5
} for some a
∈
L. Putting t
a
:
=
min
{
μ
(
a
)
,
0.5
}, we have a
∈
U
(
μ
;
t
a
) and so U
(
μ
;
t
a
)
≠
∅. Hence, U
(
μ
;
t
a
) is a fated filter of L by (4.6), which implies that 1
∈
U
(
μ
;
t
a
). Thus, μ
(
1
)
≥
t
a, which is a contradiction. Therefore, μ
(
1
)
≥
min
{
μ
(
x
)
,
0.5
} for all x
∈
L. Suppose that
(4.9)
μ
(
x
)
<
min
⁡
{
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
μ
(
a
)
,
0.5
}
,
for some x
,
a
,
y
∈
L. Taking t
x
:
=
min
{
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
,
μ
(
a
)
,
0.5
}, we get a
→
(
(
x
→
y
)
→
x
)
∈
U
(
μ
;
t
x
) and a
∈
U
(
μ
;
t
x
). It follows from (3.1) that x
∈
U
(
μ
;
t
x
), that is, μ
(
x
)
≥
t
x. This is a contradiction. Hence,
(4.10)
μ
(
x
)
≥
min
⁡
{
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
μ
(
a
)
,
0.5
}
,
for all x
,
a
,
y
∈
L. Using Theorem 4.5, we conclude that μ is an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L.Proposition 4.8.
Every(
∈
,
∈
∨
q
)-fuzzy-fated filter μ of L satisfies the following inequalities:  (1)
μ
(
x
→
z
)
≥
min
{
μ
(
x
→
(
y
→
z
)
)
,
μ
(
x
→
y
)
,
0.5
},
(2)
μ
(
x
)
≥
min
{
μ
(
(
x
→
y
)
→
x
)
,
0.5
}, for all x
,
y
,
z
∈
L.Proof.
(1) Suppose that there exista
,
b
,
c
∈
L such that
(4.11)
μ
(
a
⟶
c
)
<
min
⁡
{
μ
(
a
⟶
(
b
⟶
c
)
)
,
μ
(
a
⟶
b
)
,
0.5
}
.
Taking t
:
=
min
{
μ
(
a
→
(
b
→
c
)
)
,
μ
(
a
→
b
)
,
0.5
} implies that t
∈
(
0,0.5
],a
→
(
b
→
c
)
∈
U
(
μ
;
t
) and a
→
b
∈
U
(
μ
;
t
). Since U
(
μ
;
t
)
∈
F
F
(
L
) by Theorem 4.7, it follows from Lemma 3.1 that a
→
c
∈
U
(
μ
;
t
), that is, μ
(
a
→
c
)
≥
t. This is a contradiction, and therefore μ satisfies (1).
(2) Ifμ is an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L, then U
(
μ
;
t
)
∈
F
F
(
L
)
∪
{
∅
} for all t
∈
(
0,0.5
] by Theorem 4.7. Hence, U
(
μ
;
t
)
∈
F
(
L
)
∪
{
∅
} for all t
∈
(
0,0.5
]. Suppose that
(4.12)
μ
(
x
)
<
t
≤
min
⁡
{
μ
(
(
x
⟶
y
)
⟶
x
)
,
0.5
}
,
for some x
,
y
∈
L and t
∈
(
0,0.5
] then (
x
→
y
)
→
x
∈
U
(
μ
;
t
), which implies from Lemma 3.2 that x
∈
U
(
μ
;
t
), that is, μ
(
x
)
≥
t. This is a contradiction. Hence, μ
(
x
)
≥
min
{
μ
(
(
x
→
y
)
→
x
)
,
0.5
} for all x
,
y
∈
L.Theorem 4.9.
IfF is a fated filter of L, then a fuzzy subset μ of L defined by
(4.13)
μ
:
L
⟶
[
0,1
]
,
x
⟼
{
t
1
if
x
∈
F
,
t
2
otherwise,
where t
1
∈
[
0.5,1
] and t
2
∈
(
0,0.5
) is an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L.Proof.
Note that(4.14)
U
(
μ
;
r
)
=
{
F
if
r
∈
(
t
2
,
0.5
]
,
L
if
r
∈
(
0
,
t
2
]
,
which is a fated filter of L. It follows from Theorem 4.7 that μ is an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L.Theorem 4.10.
Letμ be an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L. If μ
(
1
)
<
0.5, then μ is a  fuzzy-fated filter of L.Proof.
It is straightforward.
For any fuzzy subsetμ of L and any t
∈
(
0,1
], we consider two subsets:
(4.15)
Q
(
μ
;
t
)
:
=
{
x
∈
L
∣
(
x
,
t
)
q
μ
}
,
[
μ
]
t
:
=
{
x
∈
L
∣
(
x
,
t
)
∈
∨
q
μ
}
.
It is clear that [
μ
]
t
=
U
(
μ
;
t
)
∪
Q
(
μ
;
t
).Theorem 4.11.
Ifμ is an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L, then
(4.16)
(
∀
t
∈
(
0.5,1
]
)
(
Q
(
μ
;
t
)
∈
F
F
(
L
)
∪
{
∅
}
)
.Proof.
Assume thatQ
(
μ
;
t
)
≠
∅ for all t
∈
(
0.5,1
], then there exists x
∈
Q
(
μ
;
t
), and so μ
(
x
)
+
t
>
1. Using Theorem 4.5(1), we have
(4.17)
μ
(
1
)
≥
min
⁡
{
μ
(
x
)
,
0.5
}
=
{
0.5
if
μ
(
x
)
≥
0.5
,
μ
(
x
)
if
μ
(
x
)
<
0.5
>
1
-
t
,
which implies that 1
∈
Q
(
μ
;
t
). Assume that a
→
(
(
x
→
y
)
→
x
)
∈
Q
(
μ
;
t
) and a
∈
Q
(
μ
;
t
) for all x
,
a
,
y
∈
L, then (
a
→
(
(
x
→
y
)
→
x
)
,
t
)
q
μ and (
a
,
t
)
q
μ, that is, μ
(
a
→
(
(
x
→
y
)
→
x
)
)
>
1
-
t and μ
(
a
)
>
1
-
t. Using Theorem 4.5(2), we get
(4.18)
μ
(
x
)
≥
min
⁡
{
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
μ
(
a
)
,
0.5
}
.
Thus, if min
{
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
,
μ
(
a
)
}
<
0.5, then
(4.19)
μ
(
x
)
≥
min
⁡
{
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
μ
(
a
)
}
>
1
-
t
.
If min
{
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
,
μ
(
a
)
}
≥
0.5, then μ
(
x
)
≥
0.5
>
1
-
t. It follows that (
x
,
t
)
q
μ so that x
∈
Q
(
μ
;
t
). Therefore, Q
(
μ
;
t
) is a fated filter of L.Corollary 4.12.
Ifμ is a strong (
∈
,
∈
∨
q
)-fuzzy-fated filter of L, then
(4.20)
(
∀
t
∈
(
0.5,1
]
)
(
Q
(
μ
;
t
)
∈
F
F
(
L
)
∪
{
∅
}
)
.The converse of Corollary4.12 is not true as shown by the following example.Example 4.13.
Consider the(
∈
,
∈
∨
q
)-fuzzy-fated filter ν of L which is given in Example 4.4, then
(4.21)
Q
(
ν
;
t
)
=
{
{
c
,
d
,
1
}
if
t
∈
(
0.5,0.6
]
,
L
if
t
∈
(
0.6,1
]
is a fated filter of L. But ν is not a strong (
∈
,
∈
∨
q
)-fuzzy-fated filter of L.Theorem 4.14.
For a fuzzy subsetμ of L, the following assertions are equivalent:  (1)
μ is an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L,
(2)
(
∀
t
∈
(
0,1
]
)
(
[
μ
]
t
∈
F
F
(
L
)
∪
{
∅
}
).Proof.
Assume thatμ is an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L, and let t
∈
(
0,1
] be such that [
μ
]
t
≠
∅, then there exists x
∈
[
μ
]
t
=
U
(
μ
;
t
)
∪
Q
(
μ
;
t
), and so x
∈
U
(
μ
;
t
) or x
∈
Q
(
μ
;
t
). If x
∈
U
(
μ
;
t
), then μ
(
x
)
≥
t. It follows from Theorem 4.5(1) that
(4.22)
μ
(
1
)
≥
min
⁡
{
μ
(
x
)
,
0.5
}
≥
min
⁡
{
t
,
0.5
}
=
{
t
if
t
≤
0.5
,
0.5
>
1
-
t
if
t
>
0.5
,
so that 1
∈
U
(
μ
;
t
)
∪
Q
(
μ
;
t
)
=
[
μ
]
t. If x
∈
Q
(
μ
;
t
), then (
x
,
t
)
q
μ, that is, μ
(
x
)
+
t
>
1. Thus,
(4.23)
μ
(
1
)
≥
min
⁡
{
μ
(
x
)
,
0.5
}
≥
min
⁡
{
1
-
t
,
0.5
}
=
{
1
-
t
if
t
>
0.5
,
0.5
≥
t
if
t
≤
0.5
,
and so 1
∈
Q
(
μ
;
t
)
∪
U
(
μ
;
t
)
=
[
μ
]
t. Let x
,
a
,
y
∈
L be such that a
∈
[
μ
]
t and a
→
(
(
x
→
y
)
→
x
)
∈
[
μ
]
t, then μ
(
a
)
≥
t or μ
(
a
)
+
t
>
1, and μ
(
a
→
(
(
x
→
y
)
→
x
)
)
≥
t or μ
(
a
→
(
(
x
→
y
)
→
x
)
)
+
t
>
1. We can consider four cases:
(4.24)
μ
(
a
)
≥
t
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
≥
t
,
(4.25)
μ
(
a
)
≥
t
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
+
t
>
1
,
(4.26)
μ
(
a
)
+
t
>
1
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
≥
t
,
(4.27)
μ
(
a
)
+
t
>
1
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
+
t
>
1
.
For the first case, Theorem 4.5(2) implies that
(4.28)
μ
(
x
)
≥
min
⁡
{
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
μ
(
a
)
,
0.5
}
≥
min
⁡
{
t
,
0.5
}
=
{
0.5
if
t
>
0.5
,
t
if
t
≤
0.5
,
so that x
∈
U
(
μ
;
t
) or μ
(
x
)
+
t
>
0.5
+
0.5
=
1, that is, x
∈
Q
(
μ
;
t
). Hence, x
∈
[
μ
]
t. Case (4.25) implies that
(4.29)
μ
(
x
)
≥
min
⁡
{
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
μ
(
a
)
,
0.5
}
≥
min
⁡
{
1
-
t
,
t
,
0.5
}
=
{
1
-
t
if
t
>
0.5
,
t
if
t
≤
0.5
.
Thus, x
∈
Q
(
μ
;
t
)
∪
U
(
μ
;
t
)
=
[
μ
]
t. Similarly, x
∈
[
μ
]
t for the case (4.26). The final case implies that
(4.30)
μ
(
x
)
≥
min
⁡
{
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
μ
(
a
)
,
0.5
}
≥
min
⁡
{
1
-
t
,
0.5
}
=
{
1
-
t
if
t
>
0.5
,
0.5
if
t
≤
0.5
,
so that x
∈
Q
(
μ
;
t
)
∪
U
(
μ
;
t
)
=
[
μ
]
t. Consequently, [
μ
]
t is a  fuzzy-fated filter of L.
Conversely, letμ be a fuzzy subset of L such that [
μ
]
t is a fated filter of L whenever it is nonempty for all t
∈
(
0,1
]. If there exists a
∈
L such that μ
(
1
)
<
min
{
μ
(
a
)
,
0.5
}, then μ
(
1
)
<
t
a
≤
min
{
μ
(
a
)
,
0.5
} for some t
a
∈
(
0,0.5
]. It follows that a
∈
U
(
μ
;
t
a
) but 1
∉
U
(
μ
;
t
a
). Also, μ
(
1
)
+
t
a
<
2
t
a
≤
1 and so 1
∉
Q
(
μ
;
t
a
). Hence, 1
∉
U
(
μ
;
t
a
)
∪
Q
(
μ
;
t
a
)
=
[
μ
]
t
a, which is a contradiction. Therefore, μ
(
1
)
≥
min
{
μ
(
x
)
,
0.5
} for all x
∈
L. Suppose that
(4.31)
μ
(
x
)
<
min
⁡
{
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
μ
(
a
)
,
0.5
}
,
for some x
,
a
,
y
∈
L. Taking t
:
=
min
{
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
,
μ
(
a
)
,
0.5
} implies that t
∈
(
0,0.5
]
,
a
∈
U
(
μ
;
t
)
⊆
[
μ
]
t, and a
→
(
(
x
→
y
)
→
x
)
∈
U
(
μ
;
t
)
⊆
[
μ
]
t. Since [
μ
]
t
∈
F
F
(
L
), it follows that x
∈
[
μ
]
t
=
U
(
μ
;
t
)
∪
Q
(
μ
;
t
). But (4.31) induces x
∉
U
(
μ
;
t
) and μ
(
x
)
+
t
<
2
t
≤
1, that is, x
∉
Q
(
μ
;
t
). This is a contradiction, and thus μ
(
x
)
≥
min
{
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
,
μ
(
a
)
,
0.5
} for all x
,
a
,
y
∈
L. Using Theorem 4.5, we conclude that μ is an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L.


## 5. Implication-Based  Fuzzy-Fated Filters

Fuzzy logic is an extension of set theoretic multivalued logic in which the truth values are linguistic variables or terms of the linguistic variable truth. Some operators, for example∧
,
∨
,
¬
,
→ in fuzzy logic, are also defined by using truth tables, and the extension principle can be applied to derive definitions of the operators. In fuzzy logic, the truth value of fuzzy proposition Φ is denoted by [
Φ
]. For a universe U of discourse, we display the fuzzy logical and corresponding set-theoretical notations used in this paper(5.1)
[
x
∈
μ
]
=
μ
(
x
)
,
(5.2)
[
Φ
∧
Ψ
]
=
min
⁡
{
[
Φ
]
,
[
Ψ
]
}
,
(5.3)
[
Φ
⟶
Ψ
]
=
min
⁡
{
1,1
-
[
Φ
]
+
[
Ψ
]
}
,
(5.4)
[
∀
x
Φ
(
x
)
]
=
inf
⁡
x
∈
U
[
Φ
(
x
)
]
,
(5.5)
⊨
Φ
if and only if
[
Φ
]
=
1
for all valuations
.
The truth valuation rules given in (5.3) are those in the Łukasiewicz system of continuous-valued logic. Of course, various implication operators have been defined. We show only a selection of them in the following.  (a)
Gaines-Rescher implication operator (I
GR):
(5.6)
I
GR
(
a
,
b
)
=
{
1
if
a
≤
b
,
0
otherwise
.
(b)
Gödel implication operator (I
G):
(5.7)
I
G
(
a
,
b
)
=
{
1
if
a
≤
b
,
b
otherwise
.
(c)
The contraposition of Gödel implication operator (I
cG):
(5.8)
I
cG
(
a
,
b
)
=
{
1
if
a
≤
b
,
1
-
a
otherwise
.Ying [13] introduced the concept of fuzzifying topology. We can expand his/her idea to R
0-algebras, and we define a fuzzifying fated filter as follows.Definition 5.1.
A fuzzy subsetμ of L is called a fuzzifying fated filter of L if it satisfies the following conditions:  (1)
for allx
∈
L, we have
(5.9)
⊨

[
x
∈
μ
]
⟶
[
1
∈
μ
]
,
(2)
for allx
,
a
,
y
∈
L, we get
(5.10)

⊨
[
a
∈
μ
]
∧
[
a
⟶
(
(
x
→
y
)
⟶
x
)
∈
μ
]
⟶
[
x
∈
μ
]
.Obviously, conditions (5.9) and (5.10) are equivalent to Theorem 3.5(1) and Theorem 3.5(2), respectively. Therefore, a fuzzifying fated filter is an ordinary fuzzy-fated filter.In [14], the concept of t-tautology is introduced, that is,(5.11)

⊨
t
Φ
if and only if
[
Φ
]
≥
t
for all
valuations
.Definition 5.2.
Letμ be a fuzzy subset of L and t
∈
(
0,1
], then μ is called a   t
-implication-based fuzzy-fated filter of L if it satisfies the following conditions: (1)
for allx
∈
L, we have
(5.12)

⊨
t
[
x
∈
μ
]
⟶
[
1
∈
μ
]
,
(2)
for allx
,
a
,
y
∈
L, we get
(5.13)

⊨
t
[
a
∈
μ
]
∧
[
a
→
(
(
x
⟶
y
)
⟶
x
)
∈
μ
]
⟶
[
x
∈
μ
]
.
LetI be an implication operator. Clearly, μ is a t-implication-based fuzzy-fated filter of L  if and only if it satisfies  (1)
(
∀
x
∈
L
)
(
I
(
μ
(
x
)
,
μ
(
1
)
)
≥
t
),
(2)
(
∀
x
,
y
∈
L
)
(
I
(
min
{
μ
(
a
)
,
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
}
,
μ
(
x
)
)
≥
t
).Theorem 5.3.
For any fuzzy subsetμ of L, one has the following:  (1)
ifI
=
I
GR, then μ is a 0.5-implication-based fuzzy-fated filter of L  if and only if μ is a fuzzy-fated filter of L,
(2)
ifI
=
I
G, then μ is a 0.5-implication-based fuzzy-fated filter of L  if and only if μ is an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L,
(3)
ifI
=
I
cG, then μ is a 0.5-implication-based fuzzy-fated filter of L  if and only if μ satisfies the following conditions:
(3.1)
max
{
μ
(
1
)
,
0.5
}
≥
min
{
μ
(
x
)
,
1
},
(3.2)
max
{
μ
(
x
)
,
0.5
}
≥
min
{
μ
(
a
)
,
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
,
1
}
for allx
,
a
,
y
∈
L.Proof.
(1) It is Straightforward.
(2) Assume thatμ is a 0.5-implication-based fuzzy-fated filter of L, then(i)
(
∀
x
∈
L
)
(
I
G
(
μ
(
x
)
,
μ
(
1
)
)
≥
0.5
),
(ii)
(
∀
x
,
y
∈
L
)
(
I
G
(
min
{
μ
(
a
)
,
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
}
,
μ
(
x
)
)
≥
0.5
). From (i), we have μ
(
1
)
≥
μ
(
x
) or μ
(
x
)
≥
μ
(
1
)
≥
0.5, and so μ
(
1
)
≥
min
{
μ
(
x
)
,
0.5
} for all x
∈
L. The second case implies that
(5.14)
μ
(
x
)
≥
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
}
or min
{
μ
(
a
)
,
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
}
>
μ
(
x
)
≥
0.5. It follows that
(5.15)
μ
(
x
)
≥
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
0.5
}
,
for all x
,
a
,
y
∈
L. Using Theorem 4.5, we know that μ is an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L.
Conversely, suppose thatμ is an (
∈
,
∈
∨
q
)-fuzzy-fated filter of L. From Theorem 4.5(1), if min
{
μ
(
x
)
,
0.5
}
=
μ
(
x
), then I
G
(
μ
(
x
)
,
μ
(
1
)
)
=
1
≥
0.5. Otherwise, I
G
(
μ
(
x
)
,
μ
(
1
)
)
≥
0.5. From Theorem 4.5(2), if
(5.16)
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
0.5
}
=
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
}
,
then μ
(
x
)
≥
min
{
μ
(
a
)
,
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
}, and so
(5.17)
I
G
(
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
}
,
μ
(
x
)
)
=
1
≥
0.5
.
If min
{
μ
(
a
)
,
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
,
0.5
}
=
0.5, then μ
(
x
)
≥
0.5, and thus
(5.18)
I
G
(
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
}
,
μ
(
x
)
)
≥
0.5
.
Consequently, μ is a 0.5-implication-based fuzzy-fated filter of L.
(3) Suppose thatμ satisfies (3.1) and (3.2). In (3.1), if μ
(
x
)
=
1, then max
{
μ
(
1
)
,
0.5
}
=
1, and hence I
cG
(
μ
(
x
)
,
μ
(
1
)
)
=
1
≥
0.5. If μ
(
x
)
<
1, then
(5.19)
max
⁡
{
μ
(
1
)
,
0.5
}
≥
μ
(
x
)
.
If max
{
μ
(
1
)
,
0.5
}
=
μ
(
1
) in (5.19), then μ
(
1
)
≥
μ
(
x
). Hence,
(5.20)
I
cG
(
μ
(
x
)
,
μ
(
1
)
)
=
1
≥
0.5
.
If max
{
μ
(
1
)
,
0.5
}
=
0.5 in (5.19), then μ
(
x
)
≤
0.5 which implies that
(5.21)
I
cG
(
μ
(
x
)
,
μ
(
1
)
)
=
{
1
≥
0.5
if
μ
(
1
)
≥
μ
(
x
)
,
1
-
μ
(
x
)
≥
0.5
otherwise
.
In (3.2), if min
{
μ
(
a
)
,
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
,
1
}
=
1, then
(5.22)
max
⁡
{
μ
(
x
)
,
0.5
}
=
1
,
and so μ
(
x
)
=
1
≥
min
{
μ
(
a
)
,
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
}. Therefore,
(5.23)
I
cG
(
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
}
,
μ
(
x
)
)
=
1
≥
0.5
.
If
(5.24)
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
1
}
=
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
}
,
then
(5.25)
max
⁡
{
μ
(
x
)
,
0.5
}
≥
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
→
y
)
⟶
x
)
)
}
.
Thus, if max
{
μ
(
x
)
,
0.5
}
=
0.5 in (5.25), then μ
(
x
)
≤
0.5 and
(5.26)
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
}
≤
0.5
.
Therefore,
(5.27)
I
cG
(
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
}
,
μ
(
x
)
)
=
1
≥
0.5
,
whenever μ
(
x
)
≥
min
{
μ
(
a
)
,
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
},
(5.28)
I
cG
(
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
}
,
μ
(
x
)
)
=
1
-
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
}
≥
0.5
,
whenever μ
(
x
)
<
min
{
μ
(
a
)
,
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
}. Now, if
(5.29)
max
⁡
{
μ
(
x
)
,
0.5
}
=
μ
(
x
)
in (5.25), then μ
(
x
)
≥
min
{
μ
(
a
)
,
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
}, and so
(5.30)
I
cG
(
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
}
,
μ
(
x
)
)
=
1
≥
0.5
.
Consequently,μ is a 0.5-implication-based fuzzy-fated filter of L.
Conversely, assume thatμ is a 0.5-implication-based fuzzy-fated filter of L, then  (iii)
I
cG
(
μ
(
x
)
,
μ
(
1
)
)
≥
0.5,
(iv)
I
cG
(
min
{
μ
(
a
)
,
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
}
,
μ
(
x
)
)
≥
0.5, for all x
,
a
,
y
∈
L. The case (iii) implies that I
cG
(
μ
(
x
)
,
μ
(
1
)
)
=
1, that is, μ
(
x
)
≤
μ
(
1
), or 1
-
μ
(
x
)
≥
0.5 and so μ
(
x
)
≤
0.5. It follows that
(5.31)
max
⁡
{
μ
(
1
)
,
0.5
}
≥
μ
(
x
)
=
min
⁡
{
μ
(
x
)
,
1
}
.
From (iv), we have
(5.32)
I
cG
(
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
}
,
μ
(
x
)
)
=
1
,
that is, min
{
μ
(
a
)
,
μ
(
a
→
(
(
x
→
y
)
→
x
)
)
}
≤
μ
(
x
), or
(5.33)
1
-
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
}
≥
0.5
.
Hence,
(5.34)
max
⁡
{
μ
(
x
)
,
0.5
}
≥
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
}
=
min
⁡
{
μ
(
a
)
,
μ
(
a
⟶
(
(
x
⟶
y
)
⟶
x
)
)
,
1
}
,
for all x
,
y
,
z
∈
L. This completes the proof.


## 6. Conclusion

Using the “belongs to” relation (∈) and quasicoincidence with relation (q) between fuzzy points and fuzzy sets, we introduced the notion of an (
∈
,
∈
∨
q
)-fuzzy-fated filter, this is a generalization of a fuzzy implicative filter. In fuzzy logic, one can see that various implication operators have been defined. We used Gaines-Rescher implication operator (I
GR), Gödel implication operator (I
G), and the contraposition of Gödel implication operator (I
cG) to study t-implication-based fuzzy-fated filters.There are also other situations concerning the relations between this kind of results, another type of structures (e.g.,(
∈
¯
,
∈
∨
q
¯
)-fuzzy-fated filter), and (fuzzy) soft and rough set theory. How to deal with these situations will be one of our future topics. We will also try to study the intuitionistic fuzzy version of several type of filters in R
0-algebras related to the intuitionistic “belongs to” relation (∈) and intuitionistic quasicoincidence with relation (q) between intuitionistic fuzzy points and intuitionistic fuzzy sets.



---

*Source: 980315-2011-09-13.xml*