# On Some Fuzzy Filters in Pseudo-BCI Algebras

Authors: Xiaohong Zhang
Journal: The Scientific World Journal
 (2014)

---

## Abstract

Some new properties of fuzzy associative filters (also known as fuzzy associative pseudo-filters), fuzzyp-filter (also known as fuzzy pseudo-p-filters), and fuzzy a-filter (also known as fuzzy pseudo-a-filters) in pseudo-BCI algebras are investigated. By these properties, the following important results are proved: (1) a fuzzy filter (also known as fuzzy pseudo-filters) of a pseudo-BCI algebra is a fuzzy associative filter if and only if it is a fuzzy a-filter; (2) a filter (also known as pseudo-filter) of a pseudo-BCI algebra is associative if and only if it is an a-filter (also call it pseudo-a filter); (3) a fuzzy filter of a pseudo-BCI algebra is fuzzy a-filter if and only if it is both a fuzzy p-filter and a fuzzy q-filter.


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## Body

## 1. Introduction and Preliminaries

In the field of artificial intelligence research, nonclassical logics (fuzzy logic, epistemic logic, nonmonotonic logic, default logic, etc.) are extensively used (see [1]). In this paper we discuss a kind of logic algebra system, that is, pseudo-B
C
I algebra, which originated from B
C
I-logic; it is a kind of nonclassical logic and inspired by the calculus of combinators [2]. Newly, in [3], we show that pseudo-B
C
I algebra plays an important role in weakly integral residuated algebraic structure, which is in close connection with various fuzzy logic formal systems [4, 5].In 1966, Iséki [2] introduced the concept of B
C
I-algebra as an algebraic counterpart of the B
C
I-logic. Since then, the ideal theory of B
C
I-algebras gets in-depth research and development. In 2008, as a generalization of B
C
I-algebra, Dudek and Jun [6] introduced the notion of pseudo-B
C
I algebra which is also generalization of pseudo-B
C
K algebra introduced by Georgescu and Iorgulescu in [7]. We investigated some classes of pseudo-B
C
I algebras in [8]. Recently, the pseudoideal theory of pseudo-B
C
I algebras has been studied: the notion of pseudo-B
C
I ideal (or pseudoideal) of pseudo-B
C
I algebra is introduced in [9]; some special pseudo-B
C
I ideals are discussed in [10], for example, associative pseudoideal and pseudo-a ideal, which are generalization of associative ideal of B
C
I-algebra (it is introduced by X.H. Zhang and R.G. Ling).The notion of fuzzy sets has been applied to many algebraic systems (see [4, 11, 12]); naturally, it has been applied to pseudo-B
C
K
/
B
C
I algebra; for example, fuzzy pseudoideals have been investigated in [13–15]. As continuums of the above works, we further study fuzzy associative pseudoideal and fuzzy pseudo-a ideal in pseudo-B
C
I algebras.Note that the notion of pseudo-B
C
I algebra in this paper is indeed dual form of original definition in [6]; accordingly, the notion of pseudo-filter (or pseudo-B
C
I filter) is the dual form of pseudoideal (or pseudo-B
C
I ideal) in [9]. Moreover, for short, the notion of pseudo-filter (or pseudo-B
C
I filter) is simply called “filter” in this paper.At first, we recall some basic concepts and properties of pseudo-B
C
I algebras.Definition 1 (see [6]).
A pseudo-B
C
I algebra is a structure (
X
;
≤
,
→
,
⇝
,
1
), where “≤” is a binary relation on X, “→” and “⇝” are binary operations on X, and “1” is an element of X, verifying the axioms: for all x
,
y
,
z
∈
X,(1)
y
→
z
≤
(
z
→
x
)
⇝
(
y
→
x
), y
⇝
z
≤
(
z
⇝
x
)
→
(
y
⇝
x
);
(2)
x
≤
(
x
→
y
)
⇝
y, x
≤
(
x
⇝
y
)
→
y;
(3)
x
≤
x;
(4)
x
≤
y, y
≤
x
⇒
x
=
y;
(5)
x
≤
y
⇔
x
→
y
=
1
⇔
x
⇝
y
=
1.If(
X
;
≤
,
→
,
⇝
,
1
) is a pseudo-B
C
I algebra satisfying x
→
y
=
x
⇝
y for all x
,
y
∈
X, then (
X
;
≤
,
→
,
⇝
,
1
) is a B
C
I-algebra.Proposition 2 (see [6, 9, 10]).
Let(
X
;
≤
,
→
,
⇝
,
1
) be a pseudo-B
C
I algebra; then X satisfy the following properties (
∀
x
,
y
,
z
∈
X
):(1)
1
≤
x
⇒
x
=
1
;
(2)
x
≤
y
⇒
y
→
z
≤
x
→
z
,
y
⇝
z
≤
x
⇝
z
;
(3)
x
≤
y
,
y
≤
z
⇒
x
≤
z
;
(4)
x
⇝
(
y
→
z
)
=
y
→
(
x
⇝
z
)
;
(5)
x
≤
y
→
z
⇒
y
≤
x
⇝
z
;
(6)
x
→
y
≤
(
z
→
x
)
→
(
z
→
y
)
,
x
⇝
y
≤
(
z
⇝
x
)
⇝
(
z
⇝
y
)
;
(7)
x
≤
y
⇒
z
→
x
≤
z
→
y
,
z
⇝
x
≤
z
⇝
y
;
(8)
1
→
x
=
x
,
1
⇝
x
=
x
;
(9)
(
(
y
→
x
)
⇝
x
)
→
x
=
y
→
x
,
(
(
y
⇝
x
)
→
x
)
⇝
x
=
y
⇝
x
;
(10)
x
→
y
≤
(
y
→
x
)
⇝
1
,
x
⇝
y
≤
(
y
⇝
x
)
→
1
;
(11)
(
x
→
y
)
→
1
=
(
x
→
1
)
⇝
(
y
⇝
1
)
,
(
x
⇝
y
)
⇝
1
=
(
x
⇝
1
)
→
(
y
→
1
)
;
(12)
x
→
1
=
x
⇝
1
.Definition 3 (see [9]).
A nonempty subsetF of a pseudo-B
C
I algebra X is called a pseudo-B
C
I filter (briefly, filter) of X if it satisfies(F1)
1
∈
F;
(F2)
x
∈
F, x
→
y
∈
F
⇒
y
∈
F;
(F3)
x
∈
F, x
⇝
y
∈
F
⇒
y
∈
F.Definition 4 (see [10]).
A nonempty subsetF of a pseudo-B
C
I algebra X is called an associative pseudo-B
C
I filter (briefly, associative filter) of X if it satisfies(1)
1
∈
F;
(2)
x
→
(
y
⇝
z
)
∈
F and x
→
y
∈
F
⇒
z
∈
F;
(3)
x
⇝
(
y
→
z
)
∈
F and x
⇝
y
∈
F
⇒
z
∈
F.Definition 5 (see [10]).
A nonempty subsetF of a pseudo-B
C
I algebra X is called a pseudo-a-filter (briefly, a-filter) of X if it satisfies(1)
1
∈
F;
(2)
(
x
→
1
)
⇝
(
y
→
z
)
∈
F and y
∈
F
⇒
z
⇝
x
∈
F;
(3)
(
x
⇝
1
)
→
(
y
⇝
z
)
∈
F and y
∈
F
⇒
z
→
x
∈
F.Definition 6 (see [13–15]).
A fuzzy setμ
:
X
→
[
0,1
] is called a fuzzy pseudofilter (briefly, fuzzy filter) of pseudo-B
C
I algebra X if it satisfies(FF1)
μ
(
1
)
≥
μ
(
x
), ∀
x
∈
X;
(FF2)
μ
(
y
)
≥
min
⁡
{
μ
(
x
→
y
)
,
μ
(
x
)
}, ∀
x
,
y
∈
X;
(FF3)
μ
(
y
)
≥
min
⁡
{
μ
(
x
⇝
y
)
,
μ
(
x
)
}, ∀
x
,
y
∈
X.Proposition 7.
Letμ be a fuzzy filter of a pseudo-B
C
I algebra X. If x
≤
y, then μ
(
x
)
≤
μ
(
y
), where x
,
y
∈
X.As a consequence of the so-calledTransfer Principle for Fuzzy Sets in [11], we have the following.Theorem 8 (see [11, 15]).
LetX be a pseudo-B
C
I algebra. Then a fuzzy set μ
:
X
→
[
0,1
] is a fuzzy filter of X if and only if the level set μ
t
=
{
x
∈
X
∣
μ
(
x
)
≥
t
} is filter of X for all t
∈
Im
⁡
(
μ
).Theorem 9 (see [15]).
LetX be a pseudo-B
C
I algebra. Then a fuzzy set μ
:
X
→
[
0,1
] is a fuzzy filter of X if and only if it satisfies(1)
for allx
,
y
,
z
∈
X, x
≤
y
→
z
⇒
μ
(
z
)
≥
min
⁡
{
μ
(
x
)
,
μ
(
y
)
};
(2)
for allx
,
y
,
z
∈
X, x
≤
y
⇝
z
⇒
μ
(
z
)
≥
min
⁡
{
μ
(
x
)
,
μ
(
y
)
}.Definition 10 (see [13, 15]).
A fuzzy setμ
:
X
→
[
0,1
] is called a fuzzy p-filter of a pseudo-B
C
I algebra X if it satisfies (FF1) and(FPF1)
for allx
,
y
,
z
∈
X, μ
(
z
)
≥
min
⁡
{
μ
(
(
x
→
y
)
⇝
(
x
→
z
)
)
,
μ
(
y
)
};
(FPF2)
for allx
,
y
,
z
∈
X, μ
(
z
)
≥
min
⁡
{
μ
(
(
x
⇝
y
)
→
(
x
⇝
z
)
)
,
μ
(
y
)
}.Definition 11 (see [13, 15]).
A fuzzy setμ
:
X
→
[
0,1
] is called a fuzzy a-filter of a pseudo-B
C
I algebra X if it satisfies (FF1) and(FaF1)
for allx
,
y
,
z
∈
X, μ
(
z
→
x
)
≥
min
⁡
{
μ
(
(
x
⇝
1
)
→
(
y
⇝
z
)
)
,
μ
(
y
)
};
(FaF2)
for allx
,
y
,
z
∈
X, μ
(
z
⇝
x
)
≥
min
⁡
{
μ
(
(
x
→
1
)
⇝
(
y
→
z
)
)
,
μ
(
y
)
}.Definition 12 (see [15]).
A fuzzy setμ
:
X
→
[
0,1
] is called a fuzzy associative filter of a pseudo-B
C
I algebra X if it satisfies (FF1) and(FAF1)
for allx
,
y
,
z
∈
X, μ
(
z
)
≥
min
⁡
{
μ
(
x
→
(
y
⇝
z
)
)
,
μ
(
x
→
y
)
};
(FAF2)
for allx
,
y
,
z
∈
X, μ
(
z
)
≥
min
⁡
{
μ
(
x
⇝
(
y
→
z
)
)
,
μ
(
x
⇝
y
)
}.Definition 13 (see [13, 15]).
A fuzzy setμ
:
X
→
[
0,1
] is called a fuzzy q-filter of a pseudo-B
C
I algebra X if it satisfies (FF1) and(FqF1)
for allx
,
y
,
z
∈
X, μ
(
x
→
z
)
≥
min
⁡
{
μ
(
(
x
⇝
y
)
→
z
)
,
μ
(
y
)
};
(FqF2)
for allx
,
y
,
z
∈
X, μ
(
x
⇝
z
)
≥
min
⁡
{
μ
(
(
x
→
y
)
⇝
z
)
,
μ
(
y
)
}.


## 2. New Properties of Fuzzya-Filters and Fuzzy Associative Filters

Lemma 14 (see [15]).
Letμ be a fuzzy a-filter of a pseudo-B
C
I algebra X. Then μ satisfies
(1)
μ
(
x
⟶
y
)
≥
μ
(
(
y
⇝
1
)
⟶
x
)
,
μ
(
x
⇝
y
)
≥
μ
(
(
y
⟶
1
)
⇝
x
)
,
∀
x
,
y
∈
X
.Lemma 15 (see [15]).
Letμ be a fuzzy a-filter of a pseudo-B
C
I algebra X. Then μ satisfies
(2)
μ
(
x
)
=
μ
(
x
⟶
1
)
=
μ
(
x
⇝
1
)
,
∀
x
∈
X
.Theorem 16.
Letμ be a fuzzy a-filter of a pseudo-B
C
I algebra X. Then the following statements hold for all x
,
y
,
z
∈
X:(1)
for allx
,
y
∈
X, μ
(
x
)
≥
min
⁡
{
μ
(
y
)
,
μ
(
x
→
y
)
}, μ
(
x
)
≥
min
⁡
{
μ
(
y
)
,
μ
(
x
⇝
y
)
};
(2)
for allx
,
y
∈
X
,
μ
(
x
→
y
)
=
μ
(
x
⇝
y
);
(3)
for allx
∈
X, μ
(
(
x
→
1
)
→
x
)
=
μ
(
(
x
⇝
1
)
⇝
x
)
=
μ
(
1
);
(4)
for allx
,
y
,
z
∈
X, μ
(
x
→
z
)
≥
min
⁡
{
μ
(
(
x
⇝
y
)
→
z
)
,
μ
(
y
)
};
(5)
for allx
,
y
,
z
∈
X, μ
(
x
⇝
z
)
≥
min
⁡
{
μ
(
(
x
→
y
)
⇝
z
)
,
μ
(
y
)
};
(6)
for allx
∈
X, μ
(
x
→
(
x
⇝
1
)
)
=
μ
(
x
⇝
(
x
→
1
)
)
=
μ
(
1
);
(7)
for allx
,
y
,
z
∈
X, μ
(
(
x
→
y
)
→
y
)
≥
μ
(
x
), μ
(
(
x
⇝
y
)
⇝
y
)
≥
μ
(
x
).Proof.
(1) For anyx
,
y
∈
X, by Proposition 2(6), we have y
→
1
≤
(
x
→
y
)
→
(
x
→
1
). From this, applying Proposition 7, we get
(3)
μ
(
y
⟶
1
)
≤
μ
(
(
x
⟶
y
)
⟶
(
x
⟶
1
)
)
.
Using Definition 6(FF2), we have
(4)
μ
(
x
⟶
1
)
≥
min
⁡
{
μ
(
(
x
⟶
y
)
⟶
(
x
⟶
1
)
)
,
μ
(
x
⟶
y
)
}
.
Thus,
(5)
μ
(
x
⟶
1
)
≥
min
⁡
⁡
{
μ
(
(
x
⟶
y
)
⟶
(
x
⟶
1
)
)
,
μ
(
x
⟶
y
)
}
≥
min
⁡
{
μ
(
y
⟶
1
)
,
μ
(
x
⟶
y
)
}
.
By Lemma 15, μ
(
x
→
1
)
=
μ
(
x
) and μ
(
y
→
1
)
=
μ
(
y
). Therefore, μ
(
x
)
≥
min
⁡
{
μ
(
y
)
,
μ
(
x
→
y
)
}.
Similarly, we haveμ
(
x
)
≥
min
⁡
{
μ
(
y
)
,
μ
(
x
⇝
y
)
}.
(2) For anyx
,
y
∈
X, by Proposition 2(11) and (12), we have (
x
→
y
)
→
1
=
(
x
→
1
)
⇝
(
y
→
1
). By Lemma 14, it follows that
(6)
μ
(
(
x
⟶
y
)
⟶
1
)
=
μ
(
(
x
⟶
1
)
⇝
(
y
⟶
1
)
)
≤
μ
(
(
y
⟶
1
)
⇝
x
)
≤
μ
(
x
⇝
y
)
.
From this, applying Lemma 15, we get μ
(
x
→
y
)
≤
μ
(
x
⇝
y
). Similarly, we can get μ
(
x
⇝
y
)
≤
μ
(
x
→
y
). By Definition 1(4), μ
(
x
→
y
)
=
μ
(
x
⇝
y
).
(3) For anyx
∈
X, applying Proposition 2(12) and Lemma 14, we have
(7)
μ
(
(
x
⟶
1
)
⟶
x
)
=
μ
(
(
x
⇝
1
)
⟶
x
)
≥
μ
(
(
x
⇝
1
)
⟶
(
x
⇝
1
)
)
=
μ
(
1
)
.
By Definition 6(FF1), it follows that μ
(
(
x
→
1
)
→
x
)
=
μ
(
1
).
Similarly, we haveμ
(
(
x
⇝
1
)
⇝
x
)
=
μ
(
1
).
(4) For anyx
,
y
∈
X, by Proposition 2(6) and Definition 1(1), we have
(8)
y
⇝
1
≤
(
x
⇝
y
)
⇝
(
x
⇝
1
)
≤
(
(
x
⇝
1
)
⇝
x
)
⟶
(
(
x
⇝
y
)
⇝
x
)
.
From this, by Theorem 8, we get
(9)
μ
(
(
x
⇝
y
)
⇝
x
)
≥
min
⁡
{
μ
(
y
⇝
1
)
,
μ
(
(
x
⇝
1
)
⇝
x
)
}
.
Applying (3), Lemma 15, and Definition 6(FF1), μ
(
(
x
⇝
y
)
⇝
x
)
≥
μ
(
y
). From this and (2), we get μ
(
(
x
⇝
y
)
→
x
)
≥
μ
(
y
).
On the other hand, by Definition1(1), (
x
⇝
y
)
→
x
≤
(
x
→
z
)
⇝
(
(
x
⇝
y
)
→
z
). Using Proposition 7 and the above result, we have
(10)
μ
(
y
)
≤
μ
(
(
x
⇝
y
)
⟶
x
)
≤
μ
(
(
x
⟶
z
)
⇝
(
(
x
⇝
y
)
⟶
z
)
)
.
Moreover, by (1), we have
(11)
μ
(
x
⟶
z
)
≥
min
⁡
{
(
(
x
⟶
z
)
⇝
(
(
x
⇝
y
)
⟶
z
)
)
μ
(
(
x
⇝
y
)
⟶
z
)
,
μ
(
(
x
⟶
z
)
⇝
(
(
x
⇝
y
)
⟶
z
)
)
}
.
Therefore,
(12)
μ
(
x
⟶
z
)
≥
min
⁡
⁡
{
(
(
x
⟶
z
)
⇝
(
(
x
⇝
y
)
⟶
z
)
)
μ
(
(
x
⇝
y
)
⟶
z
)
,
μ
(
(
x
⟶
z
)
⇝
(
(
x
⇝
y
)
⟶
z
)
)
}
≥
min
⁡
{
μ
(
(
x
⇝
y
)
⟶
z
)
,
μ
(
y
)
}
.
This means that (4) holds.
(5) The proof is similar to (4).
(6) For anyx
∈
X, by (4), we have
(13)
μ
(
x
⟶
(
x
⇝
1
)
)
≥
min
⁡
⁡
{
μ
(
(
x
⇝
1
)
⟶
(
x
⇝
1
)
)
,
μ
(
1
)
}
=
min
⁡
{
μ
(
1
)
,
μ
(
1
)
}
=
μ
(
1
)
.
It follows that μ
(
x
→
(
x
⇝
1
)
)
=
μ
(
1
). Similarly, μ
(
x
⇝
(
x
→
1
)
)
=
μ
(
1
).
(7) For anyx
∈
X, by Definition 1(3), we have x
≤
(
x
→
y
)
⇝
y. From this, (2), and Proposition 7, we get
(14)
μ
(
x
)
≤
μ
(
(
x
⟶
y
)
⇝
y
)
=
μ
(
(
x
⟶
y
)
⟶
y
)
.
Similarly, we haveμ
(
(
x
⇝
y
)
⇝
y
)
≥
μ
(
x
).Lemma 17.
Letμ be a fuzzy associative filter of a pseudo-B
C
I algebra X. Then μ satisfies
(15)
μ
(
y
)
≥
μ
(
x
⟶
(
x
⇝
y
)
)
,
μ
(
y
)
≥
μ
(
x
⇝
(
x
⟶
y
)
)
,
∀
x
,
y
∈
X
.Proof.
It is easily proved by Definition12; the proof is omitted.Theorem 18.
Letμ be a fuzzy associative filter of a pseudo-B
C
I algebra X. Then the following statements hold:(1)
for allx
∈
X, μ
(
x
→
(
x
⇝
1
)
)
=
μ
(
x
⇝
(
x
→
1
)
)
=
μ
(
1
);
(2)
for allx
,
y
∈
X, μ
(
y
)
≥
μ
(
(
x
→
1
)
⇝
(
x
→
y
)
);
(3)
for allx
,
y
∈
X, μ
(
y
)
≥
μ
(
(
x
⇝
1
)
→
(
x
⇝
y
)
);
(4)
for allx
,
y
∈
X, μ
(
x
→
y
)
=
μ
(
x
⇝
y
);
(5)
for allx
,
y
∈
X, μ
(
x
→
(
x
→
y
)
)
≥
μ
(
y
), μ
(
x
⇝
(
x
⇝
y
)
)
≥
μ
(
y
);
(6)
for allx
∈
X, μ
(
x
)
≥
μ
(
(
x
→
1
)
→
1
)
), μ
(
x
)
≥
μ
(
(
x
⇝
1
)
⇝
1
)
);
(7)
for allx
,
y
∈
X, μ
(
x
→
(
y
→
1
)
)
≥
μ
(
x
→
y
), μ
(
x
⇝
(
y
⇝
1
)
)
≥
μ
(
x
⇝
y
);
(8)
for allx
,
y
∈
X, μ
(
y
→
x
)
)
≥
μ
(
x
→
y
), μ
(
y
⇝
x
)
)
≥
μ
(
x
⇝
y
);
(9)
for allx
∈
X, μ
(
(
x
⇝
1
)
→
x
)
=
μ
(
(
x
→
1
)
⇝
x
)
=
μ
(
1
);
(10)
for allx
,
y
∈
X, μ
(
(
x
→
y
)
→
y
)
≥
μ
(
x
), μ
(
(
x
⇝
y
)
⇝
y
)
≥
μ
(
x
);
(11)
for allx
∈
X, μ
(
x
)
=
μ
(
x
→
1
)
=
μ
(
x
⇝
1
);
(12)
for allx
,
y
∈
X, μ
(
x
)
≥
min
⁡
{
μ
(
y
)
,
μ
(
x
→
y
)
}, μ
(
x
)
≥
min
⁡
{
μ
(
y
)
,
μ
(
x
⇝
y
)
}.Proof.
(1) For anyx
∈
X (by Definition 1 and Proposition 2),
(16)
(
x
⇝
1
)
⇝
(
(
x
⇝
1
)
⟶
(
x
⟶
(
x
⇝
1
)
)
)
=
(
x
⇝
1
)
⇝
(
x
⟶
(
(
x
⇝
1
)
⟶
(
x
⇝
1
)
)
)
=
(
x
⇝
1
)
⇝
(
x
⟶
1
)
=
(
x
⟶
1
)
⇝
(
x
⟶
1
)
=
1
.
From this and Lemma 17, we have
(17)
μ
(
x
⟶
(
x
⇝
1
)
)
≥
μ
(
(
x
⇝
1
)
⇝
(
(
x
⇝
1
)
⟶
(
x
⟶
(
x
⇝
1
)
)
)
)
=
μ
(
1
)
.
This means that μ
(
x
→
(
x
⇝
1
)
)
=
μ
(
1
). Similarly, μ
(
x
⇝
(
x
→
1
)
)
=
μ
(
1
).
(2) For anyx
,
y
∈
X, by Definition 1(1), we have
(18)
x
⇝
(
x
⟶
1
)
≤
(
(
x
⟶
1
)
⇝
(
x
⟶
y
)
)
⟶
(
x
⇝
(
x
⟶
y
)
)
.
Applying Theorem 9(1), we get
(19)
μ
(
x
⇝
(
x
⟶
y
)
)
≥
min
⁡
{
μ
(
x
⇝
(
x
⟶
1
)
)
,
μ
(
(
x
⟶
1
)
⇝
(
x
⟶
y
)
)
}
.
By (1) and Lemma 17, μ
(
x
⇝
(
x
→
1
)
)
=
μ
(
1
), μ
(
y
)
≥
μ
(
x
⇝
(
x
→
y
)
). Therefore,
(20)
μ
(
y
)
≥
μ
(
x
⇝
(
x
⟶
y
)
)
≥
min
⁡
{
μ
(
1
)
,
μ
(
(
x
⟶
1
)
⇝
(
x
⟶
y
)
)
}
=
μ
(
(
x
⟶
1
)
⇝
(
x
⟶
y
)
)
.
(3) It is similar to (2).
(4) For anyx
,
y
∈
X, by (2), we have μ
(
x
⇝
y
)
≥
μ
(
(
y
→
1
)
⇝
(
y
→
(
x
⇝
y
)
)
). On the other hand, applying Definition 1 and Proposition 2,
(21)
(
y
⟶
1
)
⇝
(
y
⟶
(
x
⇝
y
)
)
=
(
y
⟶
1
)
⇝
(
x
⇝
(
y
⟶
y
)
)
=
(
y
⟶
1
)
⇝
(
x
⇝
1
)
=
(
y
⟶
1
)
⇝
(
x
⟶
1
)
≥
x
⟶
y
.
From this and Proposition 7, μ
(
(
y
→
1
)
⇝
(
y
→
(
x
⇝
y
)
)
)
≥
μ
(
x
→
y
). Thus, μ
(
x
⇝
y
)
≥
μ
(
x
→
y
). Similarly, we can get μ
(
x
→
y
)
≥
μ
(
x
⇝
y
). Therefore, μ
(
x
→
y
)
=
μ
(
x
⇝
y
).
(5) For anyx
,
y
∈
X, since (by Proposition 2)
(22)
y
⇝
(
x
⟶
(
x
⟶
y
)
)
=
x
⟶
(
y
⇝
(
x
⟶
y
)
)
=
x
⟶
(
x
⟶
(
y
⇝
y
)
)
=
x
⟶
(
x
⟶
1
)
=
x
⟶
(
x
⇝
1
)
,
then μ
(
y
⇝
(
x
→
(
x
→
y
)
)
)
=
μ
(
x
→
(
x
⇝
1
)
). By (1), μ
(
x
→
(
x
⇝
1
)
)
=
μ
(
1
); hence, μ
(
y
⇝
(
x
→
(
x
→
y
)
)
)
=
μ
(
1
). Moreover, by Definition 6(FF3), we have
(23)
μ
(
x
⟶
(
x
⟶
y
)
)
≥
min
⁡
{
μ
(
y
⇝
(
x
⟶
(
x
⟶
y
)
)
)
,
μ
(
y
)
}
.
Thus,
(24)
μ
(
x
⟶
(
x
⟶
y
)
)
≥
min
⁡
⁡
{
μ
(
y
⇝
(
x
⟶
(
x
⟶
y
)
)
)
,
μ
(
y
)
}
=
min
⁡
{
μ
(
1
)
,
μ
(
y
)
}
=
μ
(
y
)
.
Similarly,μ
(
x
⇝
(
x
⇝
y
)
)
≥
μ
(
y
).
(6) By (2) and (3), we can get (6).
(7) For anyx
,
y
∈
X, by Definition 1, we have y
→
(
y
→
1
)
)
≤
(
x
→
y
)
→
(
x
→
(
y
→
1
)
).
Applying (1) and Theorem9, we get
(25)
μ
(
x
⟶
(
y
⟶
1
)
)
≥
min
⁡
{
μ
(
y
⟶
(
y
⟶
1
)
)
,
μ
(
x
⟶
y
)
}
=
min
⁡
{
μ
(
1
)
,
μ
(
x
⟶
y
)
}
=
μ
(
x
⟶
y
)
.
Similarly,μ
(
x
⇝
(
y
⇝
1
)
)
≥
μ
(
x
⇝
y
).
(8) For anyx
,
y
∈
X, by Lemma 17, μ
(
y
→
x
)
≥
μ
(
x
⇝
(
x
→
(
y
→
x
)
)
). And, using Proposition 2(4), we have
(26)
x
⇝
(
x
⟶
(
y
⟶
x
)
)
=
x
⟶
(
x
⇝
(
y
⟶
x
)
)
=
x
⟶
(
y
⟶
(
x
⇝
x
)
)
=
x
⟶
(
y
⟶
1
)
.
Hence, μ
(
y
→
x
)
≥
μ
(
x
→
(
y
→
1
)
). From this and (7), we get
(27)
μ
(
y
⟶
x
)
≥
μ
(
x
⟶
(
y
⟶
1
)
)
≥
μ
(
x
⟶
y
)
.
Similarly,μ
(
y
⇝
x
)
≥
μ
(
x
⇝
y
).
(9) By (1) and (8), we can get (8).
(10) It is similar to the proof of Theorem16(7).
(11) For anyx
∈
X, by Lemma 17, μ
(
x
)
≥
μ
(
x
→
(
x
⇝
x
)
)
=
μ
(
x
→
1
). On the other hand, using (8), μ
(
x
→
1
)
≥
μ
(
1
→
x
)
=
μ
(
x
). Hence, μ
(
x
)
=
μ
(
x
→
1
)
=
μ
(
x
⇝
1
).
(12) It is similar to the proof of Theorem16(1).


## 3. Some Necessary and Sufficient Conditions for Fuzzya-Filters and Fuzzy Associative Filters

Checking the proof of Theorem18 in detail, we know that the proof only applies the properties of fuzzy filters and the conditions in Lemma 17. From this, we can get the following.Lemma 19.
Letμ be a fuzzy filter of a pseudo-B
C
I algebra X. If μ satisfies(C1)
for allx
,
y
∈
X, μ
(
y
)
≥
μ
(
x
→
(
x
⇝
y
)
);
(C2)
for allx
,
y
∈
X, μ
(
y
)
≥
μ
(
x
⇝
(
x
→
y
)
),
then the following statements hold:
(C3)
for allx
∈
X, μ
(
x
→
(
x
⇝
1
)
)
=
μ
(
x
⇝
(
x
→
1
)
)
=
μ
(
1
);
(C4)
for allx
,
y
∈
X, μ
(
x
→
y
)
=
μ
(
x
⇝
y
);
(C5)
for allx
∈
X, μ
(
x
)
=
μ
(
x
→
1
)
=
μ
(
x
⇝
1
);
(C6)
for allx
,
y
∈
X, μ
(
x
)
≥
min
⁡
{
μ
(
y
)
,
μ
(
x
→
y
)
}, μ
(
x
)
≥
min
⁡
{
μ
(
y
)
,
μ
(
x
⇝
y
)
}.Proof.
It is similar to Theorem18 (the conditions (FAF1) and (FAF2) are not applied); the proof is omitted.Theorem 20.
Letμ be a fuzzy filter of a pseudo-B
C
I algebra X. Then μ is a fuzzy associative filter of X if and only if it satisfies(C1)
for allx
,
y
∈
X, μ
(
y
)
≥
μ
(
x
→
(
x
⇝
y
)
);
(C2)
for allx
,
y
∈
X, μ
(
y
)
≥
μ
(
x
⇝
(
x
→
y
)
).Proof.
Assume thatμ is a fuzzy associative filter of X; by Lemma 17, (C1) and (C2) hold.
Conversely, assume thatμ satisfies conditions (C1) and (C2). For any x
,
y
,
z
∈
X, by Proposition 2(6), y
→
(
x
→
z
)
≤
(
x
→
y
)
→
(
x
→
(
x
→
z
)
). Using Theorem 9, we get
(28)
μ
(
x
⟶
(
x
⟶
z
)
)
≥
min
⁡
{
μ
(
y
⟶
(
x
⟶
z
)
)
,
μ
(
x
⟶
y
)
}
.
By Lemma 19(C4) and Proposition 2(4), μ
(
y
→
(
x
→
z
)
)
=
μ
(
y
⇝
(
x
→
z
)
)
=
μ
(
x
→
(
y
⇝
z
)
).
Thus,
(P1)μ
(
x
→
(
x
→
z
)
)
≥
min
⁡
{
μ
(
x
→
(
y
⇝
z
)
)
,
μ
(
x
→
y
)
}.
On the other hand, applying Lemma19(C5), Proposition 2(11), and (12),
(29)
μ
(
x
⟶
(
x
⟶
z
)
)
=
μ
(
(
x
⟶
(
x
⟶
z
)
)
⟶
1
)
=
μ
(
(
x
⟶
1
)
⇝
(
(
x
⟶
1
)
⇝
(
z
⟶
1
)
)
)
=
μ
(
(
x
⟶
1
)
⇝
(
z
⟶
(
(
x
⟶
1
)
⇝
1
)
)
)
=
μ
(
z
⟶
(
(
x
⟶
1
)
⇝
(
(
x
⟶
1
)
⇝
1
)
)
)
.
And, by Lemma 19(C6),
(30)
μ
(
z
)
≥
min
⁡
{
μ
(
(
x
⟶
1
)
⇝
(
(
x
⟶
1
)
⇝
1
)
)
,
μ
(
z
⟶
(
(
x
⟶
1
)
⇝
(
(
x
⟶
1
)
⇝
1
)
)
)
}
.
By Lemma 19(C3) and the above result, we get
(P2)μ
(
z
)
≥
min
⁡
{
μ
(
1
)
,
μ
(
x
→
(
x
→
z
)
)
}
=
μ
(
x
→
(
x
→
z
)
).
Combining (P1) and (P2), we get thatμ
(
z
)
≥
min
⁡
{
μ
(
x
→
(
y
⇝
z
)
)
,
μ
(
x
→
y
)
}. That is, (FAF1) holds. Similarly, condition (FAF2) holds. Therefore, by Definition 12, μ is a fuzzy associative filter of X.Theorem 21 (see [13, 15]).
Letμ be a fuzzy filter of a pseudo-B
C
I algebra X. Then μ is a fuzzy a-filter of X if and only if it satisfies(a1)
for allx
,
y
∈
X, μ
(
x
→
y
)
≥
μ
(
(
y
⇝
1
)
→
x
);
(a2)
for allx
,
y
∈
X, μ
(
x
⇝
y
)
≥
μ
(
(
y
→
1
)
⇝
x
).Theorem 22.
Letμ be a fuzzy filter of a pseudo-B
C
I algebra X. Then the following conditions are equivalent:(i)
μ is a fuzzy a-filter of X;
(ii)
μ is a fuzzy associative filter of X.Proof.
(i)⇒ (ii). Suppose that μ is a fuzzy a-filter of X. For any x
,
y
∈
X, by Definition 1(1),
(31)
x
⟶
(
x
⇝
1
)
≤
(
(
x
⇝
1
)
⟶
(
x
⇝
y
)
)
⇝
(
x
⟶
(
x
⇝
y
)
)
.
Applying Theorem 16(6) and Proposition 7, we get
(32)
μ
(
1
)
=
μ
(
x
⟶
(
x
⇝
1
)
)
≤
μ
(
(
(
x
⇝
1
)
⟶
(
x
⇝
y
)
)
⇝
(
x
⟶
(
x
⇝
y
)
)
)
.
It follows that
(33)
μ
(
1
)
=
μ
(
(
(
x
⇝
1
)
⟶
(
x
⇝
y
)
)
⇝
(
x
⟶
(
x
⇝
y
)
)
)
.
Moreover, by Theorem 16(1),
(34)
μ
(
(
x
⇝
1
)
⟶
(
x
⇝
y
)
)
≥
min
⁡
{
⇝
(
x
⟶
(
x
⇝
y
)
)
)
μ
(
x
⟶
(
x
⇝
y
)
)
,
μ
(
(
(
x
⇝
1
)
⟶
(
x
⇝
y
)
)
⇝
(
x
⟶
(
x
⇝
y
)
)
)
}
.
Then we get
(35)
μ
(
(
x
⇝
1
)
⟶
(
x
⇝
y
)
)
≥
min
⁡
⁡
{
μ
(
x
⟶
(
x
⇝
y
)
)
,
μ
(
1
)
}
=
μ
(
x
⟶
(
x
⇝
y
)
)
.
Using Theorem 16(2), μ
(
(
x
⇝
1
)
⇝
(
x
⇝
y
)
)
=
μ
(
(
x
⇝
1
)
→
(
x
⇝
y
)
). Thus,
(36)
μ
(
(
x
⇝
1
)
⇝
(
x
⇝
y
)
)
≥
μ
(
x
⟶
(
x
⇝
y
)
)
.
On the other hand, applying Proposition 2(6) and (8), y
≤
1
→
y
≤
(
x
⇝
1
)
⇝
(
x
⇝
y
). From this and Theorem 16(1), we get
(37)
μ
(
y
)
≥
min
⁡
{
μ
(
(
x
⇝
1
)
⇝
(
x
⇝
y
)
)
,
μ
(
y
⟶
(
(
x
⇝
1
)
⇝
(
x
⇝
y
)
)
)
}
.
From y
≤
(
x
⇝
1
)
⇝
(
x
⇝
y
), we have y
→
(
(
x
⇝
1
)
⇝
(
x
⇝
y
)
)
=
1; it follows that
(38)
μ
(
y
)
≥
min
⁡
⁡
{
μ
(
(
x
⇝
1
)
⇝
(
x
⇝
y
)
)
,
μ
(
1
)
}
=
μ
(
(
x
⇝
1
)
⇝
(
x
⇝
y
)
)
.
Therefore,
(39)
μ
(
y
)
≥
μ
(
(
x
⇝
1
)
⇝
(
x
⇝
y
)
)
≥
μ
(
x
⟶
(
x
⇝
y
)
)
.
This means that (C1) holds. Similarly, μ
(
y
)
≥
μ
(
x
⇝
(
x
→
y
)
). By Theorem 20, μ is a fuzzy associative filter of X.
(ii)⇒ (i). Suppose that μ is a fuzzy associative filter of X. For any x
,
y
∈
X, by Definition 1(1), (
y
⇝
1
)
→
x
≤
(
x
→
y
)
⇝
(
y
⇝
1
). Applying Proposition 7, we get
(40)
μ
(
(
x
⟶
y
)
⇝
(
y
⇝
1
)
)
≥
μ
(
(
y
⇝
1
)
⟶
x
)
.
On the other hand, using Theorem 18(12),
(41)
μ
(
x
⟶
y
)
≥
min
⁡
{
μ
(
(
y
⇝
1
)
⟶
y
)
,
μ
(
(
x
⟶
y
)
⇝
(
y
⇝
1
)
)
}
.
By Theorem 18(9), μ
(
(
y
⇝
1
)
→
y
)
=
μ
(
1
). It follows that
(42)
μ
(
x
⟶
y
)
≥
min
⁡
{
μ
(
1
)
,
μ
(
(
x
⟶
y
)
⇝
(
y
⇝
1
)
)
}
=
μ
(
(
x
⟶
y
)
⇝
(
y
⇝
1
)
)
.
Therefore,
(43)
μ
(
x
⟶
y
)
≥
μ
(
(
x
⟶
y
)
⇝
(
y
⇝
1
)
)
≥
μ
(
(
y
⇝
1
)
⟶
x
)
.
Similarly, we can getμ
(
x
⇝
y
)
≥
μ
(
(
y
→
1
)
⇝
x
). By Theorem 21, μ is a fuzzy a-filter of X.Now, we discuss the relationship between associative filters anda-filters of pseudo-B
C
I algebras. At first, we give the following results (the proofs are omitted).Proposition 23.
A nonempty subsetF of pseudo-B
C
I algebra X is a filter (associative filter, a-filter) of X if and only if the characteristic function χ
F of F is a fuzzy filter (fuzzy associative filter, fuzzy a-filter) of X.Proposition 24.
LetX be a pseudo-B
C
I algebra. Then a fuzzy set μ
:
X
→
[
0,1
] is a fuzzy associative filter (fuzzy a-filter) of X if and only if the level set μ
t
=
{
x
∈
X
∣
μ
(
x
)
≥
t
} is associative filter (a-filter) of X for all t
∈
Im
⁡
(
μ
).In fact, the above proposition is a consequence of the so-calledTransfer Principle for Fuzzy Sets in [11].Combining Propositions23 and 24, Theorems 8 and 22, we get the following.Theorem 25.
LetF be a filter of a pseudo-B
C
I algebra X. Then the following conditions are equivalent:(i)
F is an a-filter of X;
(ii)
F is an associative filter of X.Remark 26.
In [10], the authors proved Theorem 18, but the proof is wrong (from “Conversely …” to the end of proof). In fact, by Theorem 25, Theorem 4 in [10] is true.Finally, we discuss the relationship among fuzzy associative filters, fuzzya-filters, and fuzzy q-filters in pseudo-B
C
I algebras.Lemma 27.
Letμ be a fuzzy p-filter of a pseudo-B
C
I algebra X. Then μ satisfies(1)
for allx
∈
X, μ
(
x
)
=
μ
(
(
x
→
1
)
⇝
1
);
(2)
for allx
∈
X, μ
(
(
(
x
→
1
)
⇝
1
)
⇝
x
)
=
μ
(
1
).Proof.
(1) For anyx
∈
X, by Definition 10(FPF1), we have
(44)
μ
(
x
)
≥
min
⁡
{
μ
(
(
x
→
1
)
⇝
(
1
→
1
)
)
,
μ
(
1
)
}
=
μ
(
(
x
→
1
)
⇝
1
)
.
And, using Definition 1(2) and Proposition 7, μ
(
(
x
→
1
)
⇝
1
)
≥
μ
(
x
). It follows that μ
(
x
)
=
μ
(
(
x
→
1
)
⇝
1
).
(2) For anyx
∈
X, by Proposition 2(11), (12), and (9), we have
(45)
(
(
(
(
x
⟶
1
)
⇝
1
)
⇝
x
)
⟶
1
)
⇝
1
=
(
(
(
(
x
⟶
1
)
⇝
1
)
⇝
x
)
⟶
1
)
⟶
1
=
(
(
(
(
x
⟶
1
)
⇝
1
)
⇝
x
)
⇝
1
)
⟶
1
=
(
(
(
(
x
⟶
1
)
⇝
1
)
⇝
1
)
⟶
(
x
⇝
1
)
)
⟶
1
=
(
(
(
(
x
⟶
1
)
⇝
1
)
⟶
1
)
⟶
(
x
⇝
1
)
)
⟶
1
=
(
(
x
⟶
1
)
⟶
(
x
⇝
1
)
)
⟶
1
=
(
(
x
⟶
1
)
⟶
(
x
⟶
1
)
)
⟶
1
=
1
⟶
1
=
1
.
From this and (1), we get
(46)
μ
(
(
(
x
⟶
1
)
⇝
1
)
⇝
x
)
=
μ
(
(
(
(
(
x
⟶
1
)
⇝
1
)
⇝
x
)
⟶
1
)
⇝
1
)
=
μ
(
1
)
.
This means that (2) holds.Theorem 28.
Letμ be a fuzzy filter of a pseudo-B
C
I algebra X. Then the following conditions are equivalent:(1)
μ is a fuzzy a-filter of X;
(2)
μ is both a fuzzy p-filter and a fuzzy q-filter of X.Proof.
Assume thatμ is a fuzzy a-filter of X. It is easy to prove that μ is both a fuzzy p-filter and a fuzzy q-filter of X.
Conversely, letμ be both a fuzzy p-filter and fuzzy q-filter of X. For any x
,
y
∈
X, by Definition 13(FqF1), we have
(47)
μ
(
x
⟶
y
)
≥
min
⁡
{
μ
(
(
x
⇝
(
(
y
⇝
1
)
⟶
x
)
)
⟶
y
)
,
μ
(
(
y
⇝
1
)
⟶
x
)
(
(
x
⇝
(
(
y
⇝
1
)
⟶
x
)
)
⟶
y
)
}
.
And, by Proposition 2(4),
(48)
(
x
⇝
(
(
y
⇝
1
)
⟶
x
)
)
⟶
y
=
(
(
y
⇝
1
)
⟶
(
x
⇝
x
)
)
⟶
y
=
(
(
y
⇝
1
)
⟶
1
)
⟶
y
.
From this and Lemma 27(2),
(49)
μ
(
(
x
⇝
(
(
y
⇝
1
)
⟶
x
)
)
⟶
y
)
=
μ
(
(
(
y
⇝
1
)
⟶
1
)
⟶
y
)
=
μ
(
1
)
.
Therefore,
(50)
μ
(
x
⟶
y
)
≥
min
⁡
{
μ
(
1
)
,
μ
(
(
y
⇝
1
)
⟶
x
)
}
=
μ
(
(
y
⇝
1
)
⟶
x
)
.
This means that Theorem 21(a1) holds. Similarly, we can prove (a2). By Theorem 21, we know that μ is a fuzzy a-filter of X.



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*Source: 718972-2014-04-08.xml*