Fuzzy 1-type and 2-type positive implicative filters of pseudo-BCK algebras

Abstract

The notions of fuzzy 1-type (2-type) positive implicative filter and fuzzy normal filter of pseudo-BCK algebras are introduced, some properties and equivalent conditions are given. By using the new notions, the characterizations of 1-type positive implicative pseudo-BCK algebra and 2-type positive implicative pseudo-BCK algebra (positive implicative BCK-algebra) are displayed. Moreover, some conclusions related with positive implicative filters in previous literature are revised. Finally, the concepts of fuzzy commutative filter and fuzzy implicative filter are proposed, and the relationships among fuzzy 1-type (2-type) positive implicative filters, fuzzy commutative filters and fuzzy implicative filters of pseudo-BCK algebras are investigated.

Full text

AUTHOR COPY
Journal of Intelligent & Fuzzy Systems 28 (2015) 2309–2317
DOI:10.3233/IFS-141514
IOS Press
2309
Fuzzy 1-type and 2-type positive implicative
filters of pseudo-BCK algebras1
Xiaohong Zhang∗
Department of Mathematics, College of Arts and Sciences, Shanghai Maritime University, Shanghai, P.R. China
Abstract. The notions of fuzzy 1-type (2-type) positive implicative filter and fuzzy normal filter of pseudo-BCK algebras are
introduced, some properties and equivalent conditions are given. By using the new notions, the characterizations of 1-type positive
implicative pseudo-BCK algebra and 2-type positive implicative pseudo-BCK algebra (positive implicative BCK-algebra) are
displayed. Moreover, some conclusions related with positive implicative filters in previous literature are revised. Finally, the
concepts of fuzzy commutative filter and fuzzy implicative filter are proposed, and the relationships among fuzzy 1-type (2-type)
positive implicative filters, fuzzy commutative filters and fuzzy implicative filters of pseudo-BCK algebras are investigated.
Keywords: Fuzzy logic, Pseudo-BCK algebra, Fuzzy 1-type positive implicative filter, Fuzzy normal filter, Fuzzy implicative
filter
1. Introduction and Preliminaries
The notion of pseudo-BCK algebra were introduced
by Georgescu and Iorgulescu [4] as a generalization of
well-known BCK-algebra. Pseudo-BCK algebra has
close connection with BCC-algebra and various non-
commutative fuzzy logics, it can be regarded as a
uniform algebraic structure (implicational fragment) of
various non-commutative fuzzy logic formal systems
(see [5, 12–14, 18, 21, 23, 24]).
The notions of positive implicative pseudo-BCK
algebra and positive implicative pseudo-filter are intro-
duced by Jun et al. in [8, 9], they are renamed by
2-type positive implicative pseudo-BCK algebra and
1This work is supported by National Natural Science Founda-
tion of China (No. 61175044, 61473239) and Innovation Program of
Shanghai Municipal Education Commission (No. 13ZZ122).
∗Corresponding author. Xiaohong Zhang, Department of
Mathematics, College of Arts and Sciences, Shanghai Maritime
University, Shanghai 201306, P.R. China. Tel.: +8602138282231;
Fax: +8602138282209; E-mails: zxhonghz@263.net; zhangxh@
shmtu.edu.cn.
2-type positive implicative pseudo-filter (shortly, 2-type
positive implicative filter) in [22], respectively. In fact,
the notion of 1-type positive implicative pseudo-BCK
algebra is a proper generalization of well-known posi-
tive implicative BCK-algebra, and the characterization
of 1-type positive implicative pseudo-BCK algebra is
displayed by the notion of 1-type positive implicative
filter (see [22]). In [20], three open problems related
with positive implicative filters are solved.
On the other hand, the notion of fuzzy sets has been
applied to many algebraic systems and rough sets the-
ory (see [1, 2, 7, 17, 25]), naturally, it has been applied
to pseudo-BCK algebras, that is, fuzzy pseudo-filters
(pseudo-ideals) have been investigated in [10]. As con-
tinuums of the above works, we further study fuzzy
positive implicative filters of pseudo-BCK algebras.
Note that, the notion of pseudo-BCK algebra in this
paper is indeed dual form (this is similar to [11]) of
original definition in [4], accordingly, the notion of
pseudo-filter is the dual form of pseudo-ideal in [9].
Moreover, for short, the notion of pseudo-filter is simply
called “filter” in this paper.
1064-1246/15/$35.00 © 2015 – IOS Press and the authors. All rights reserved

AUTHOR COPY
2310 X. Zhang / Fuzzy 1-type and 2-type positive implicative filters of pseudo-BCK algebras
At first, we recall some basic concepts and properties
of pseudo-BCK algebras.
Definition 1. [4] A pseudo-BCK algebra is a structure
(X;≤,→,→,1), where “≤” is a binary relation on
“X”, “→” and “→” are binary operations on Xand
“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≤1;
(5) x≤y, y ≤x=⇒ x=y;
(6) x≤y⇐⇒ x→y=1⇐⇒ x→y=1.
If (X;≤,→,→,1) is a pseudo-BCK algebra satis-
fying x→y=x→yfor all x, y ∈X, then (X;≤,→,
→,1)isaBCK-algebra.
Proposition 1. [4, 5, 6] Let (X;≤,→,→,1) be a
pseudo-BCK algebra, then Xsatisfy the following
properties (∀x, y, z ∈X):
(1) x≤y=⇒ y→z≤x→z, y →z≤x→z;
(2) x≤y, y ≤z=⇒ x≤z;
(3) x→(y→z)=y→(x→z);
(4) x≤y→z=⇒ y≤x→z;
(5) x→y≤(z→x)→(z→y),
x→y≤(z→x)→(z→y);
(6) x≤y→x, x ≤y→x;
(7) 1 →x=x, 1→x=x;
(8) x≤y=⇒ z→x≤z→y, z →x≤z→y;
(9) x≤y=⇒ y→z≤x→z, y →z≤x→z;
(10) ((x→y)→y)→y=x→y,
((x→y)→y)→y=x→y.
Definition 2. [22] A pseudo-BCK algebra Xis called
a 1-type positive implicative pseudo-BCK algebra if it
satisfies:
(1) x→(x→y)=x→y, ∀x, y ∈X;
(2) x→(x→y)=x→y, ∀x, y ∈X.
Definition 3. [8, 9, 22] A pseudo-BCK algebra Xis
called a 2-type positive implicative pseudo-BCK alge-
bra (also known as positive implicative pseudo-BCK
algebra in [9, 10]) if it satisfies:
(1) x→(y→z)=(x→y)→(x→z),∀x, y, z
∈X;
(2) x→(y→z)=(x→y)→(x→z),∀x, y, z
∈X.
Proposition 2. [8] If Xis a 2-type positive implica-
tive pseudo-BCK algebra, then x→y=x→yfor
all x, y ∈X.
Note from Proposition 2 and Example 3.2 in [22] that
every 2-type positive implicative pseudo-BCK algebra
is a BCK-algebra, but there are some 1-type posi-
tive implicative pseudo-BCK algebras which are not
BCK-algebra. That is, the notion of 1-type positive
implicative pseudo-BCK algebra is a good generaliza-
tion of the notion of positive implicative BCK-algebra.
Definition 4. [9, 12] A nonempty subset Fof a pseudo-
BCK algebra Xis called a pseudo-filter (briefly, filter)
of Xif it satisfies:
(F1) 1 ∈F;
(F2) x∈F, x →y∈F=⇒ y∈F;
(F3) x∈F, x →y∈F=⇒ y∈F.
Note from the results in [12] that the condition (F2)
(or the condition (F3)) is not necessary (one of them is
necessary). Here, we keep the original definition.
Definition 5. [12, 20] A filter Fof a pseudo-BCK
algebra Xis normal if it satisfies:
(NF) x→y∈F⇐⇒ x→y∈F.
Definition 6. [22] A filter Fof a pseudo-BCK algebra
Xis called a 1-type positive implicative filter of Xif it
satisfies:
(1) x→(x→y)∈F=⇒ x→y∈F;
(2) x→(x→y)∈F=⇒ x→y∈F.
Definition 7. [9, 22] A filter Fof a pseudo-BCK alge-
bra Xis called a 2-type positive implicative filter (also
known as positive implicative filter in [10]) of Xif it
satisfies:
(1) x→(y→z)∈Fand x→y∈F=⇒ x→
z∈F;
(2) x→(y→z)∈Fand x→y∈F=⇒ x→
z∈F.
Definition 8. [10] A fuzzy set µ:X→[0,1] is called
a fuzzy pseudo-filter (briefly, fuzzy filter) of pseudo-
BCK algebra Xif 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 3. Let µbe a fuzzy filter of a pseudo-BCK
algebra X.Ifx≤y, then µ(x)≤µ(y), where x, y ∈X.

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X. Zhang / Fuzzy 1-type and 2-type positive implicative filters of pseudo-BCK algebras 2311
By Proposition 1 (10) and Proposition 3 (it follows
from (FF2) or (FF3)) we know that the condition (FF2)
(or the condition (FF3)) is not necessary (one of them
is necessary).
As a consequence of the so-called Transfer Principle
for Fuzzy Sets in [7], we have the following.
Theorem 1. [7, 10] Let Xbe a pseudo-BCK algebra.
Then a fuzzy set µ:X→[0,1] is a fuzzy filter of X
if and only if the level set µt={x∈X|µ(x)≥t}is
filter of Xfor all t∈Im(µ).
Theorem 2. [10] Let Xbe a pseudo-BCK algebra. Then
a fuzzy set µ:X→[0,1] is a fuzzy filter of Xif and
only if it satisfies
(1) ∀x, y, z ∈X, x ≤y→z=⇒ µ(z)≥min{µ(x),
µ(y)};
(2) ∀x, y, z ∈X, x ≤y→z=⇒ µ(z)≥min{µ(x),
µ(y)}.
Definition 9. [10] A fuzzy set µ:X→[0,1] is called a
fuzzy positive implicative filter (it is called fuzzy 2-type
positive implicative filter in this paper) of pseudo-BCK
algebra Xif it satisfies (FF1) and
(FPIF21)∀x, y, z ∈X, µ(x→z)≥min{µ(x→
(y→z)),µ(x→y)};
(FPIF22)∀x, y, z ∈X, µ(x→z)≥min{µ(x→
(y→z)),µ(x→y)}.
2. Fuzzy normal filters and new properties
of fuzzy 2-type positive implicative filters
Definition 10. A fuzzy set µ:X→[0,1] is called
a fuzzy normal filter of pseudo-BCK algebra Xif it
satisfies:
(FNF) µ(x→y)=µ(x→y),∀x, y ∈X.
Example 1. Let Xbe the set {a, b, c, d, 1}with the
following operations:
→abc d 1
a1 1 11 1
b a 1 1 1 1
ca b 1d1
d a c c 1 1
1a b c d 1
→a b c d 1
a1 1 111
ba1 1 1 1
c a d 1d1
d a bc1 1
1a b c d 1
Then (X;→,→,≤,1) is a pseudo-BCK algebra,
where ≤is defined by x≤yiff x→y=1. Define
fuzzy set µ:X→[0,1] as following:
µ(1) =µ(c)=µ(b)=µ(d)=1,µ(a)=0.
Then µis a fuzzy normal filter of X.
Theorem 3. Let µbe a fuzzy 2-type positive implicative
filter of a pseudo-BCK algebra X. Then µis a fuzzy
normal filter of X.
Proof. For any x, y ∈X, by Definition 9 we have
µ(x→y)
≥min{µ(x→(x→y),µ(x→x)}
=min{µ(x→(x→y),µ(1)}
=µ(x→(x→y)).
On the other hand, using Proposition 1 (6),
x→y≤x→(x→y).
From this and Proposition 3 we get
µ(x→y)≤µ(x→(x→y)).
Thus,
µ(x→y)≥µ(x→(x→y)≥µ(x→y).
Similarly, we can get
µ(x→y)≥µ(x→y).
Therefore, µ(x→y)=µ(x→y). By Definition 10
we know that µis a fuzzy normal filter of X.
Theorem 4. Let µbe a fuzzy filter of a pseudo-BCK
algebra X. Then the following statements are equiva-
lent:
(1) µis a fuzzy 2-type positive implicative filter of X;
(2) ∀x, y ∈X, µ(x→(x→y)) =µ(x→y)=µ(x
→y);
(3) ∀x, y ∈X, µ(x→(x→y)) =µ(x→y),µ(x
→(x→y)) =µ(x→y);
(4) ∀x, y ∈X, µ(x→(x→y)) ≤µ(x→y),µ(x
→(x→y)) ≤µ(x→y).
Proof. (1) =⇒ (2) : For any x∈X, by Definition 9 and
Theorem 3 we have
µ(x→y)

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2312 X. Zhang / Fuzzy 1-type and 2-type positive implicative filters of pseudo-BCK algebras
≥min{µ(x→(x→y),µ(x→x)}
=µ(x→(x→y)
≥µ(x→y)=µ(x→y).
It follows that
µ(x→(x→y)) =µ(x→y)=µ(x→y).
(2) =⇒ (3) : For any x, y ∈X, by (2) we have
µ(x→(x→y))
=µ(x→(x→y))
=µ(x→(x
→y))
=µ(x→y);
µ(x→(x→y))
=µ(x→(x→y)) =µ(x→y).
It follows that (3) holds.
(3) =⇒ (4) : Obviously.
(4) =⇒ (1) : For any x, y ∈X, by (2) we have
µ(x→(x→y))
≤µ(x→(x→y))
≤µ(x→(x→y)) =µ(x→y).
µ(x→y)
≤µ(x→(x→y))
≤µ(x→(x→y)) =µ(x→y).
It follows that µ(x→y)=µ(x→y). That is, µis a
fuzzy normal filter of X.
Moreover, for any x, y, z ∈X, by Proposition 1 (5),
y→(x→z)≤(x→y)→(x→(x→z)).
Applying Theorem 2 we get
µ(x→(x→z))
≥min{µ(y→(x→z)),µ(x→y)}.
By (4) and the property of normal filter,
µ(x→(x→z))
≤µ(x→z)=µ(x→z);
µ(y→(x→z))
=µ(y→(x→z)) =µ(x→(y→z)).
Then
µ(x→z)
≥µ(x→(x→z))
≥min{µ(x→(y→z)),µ(x→y)}.
This meas that (FPIF21) holds for fuzzy filter µ. Sim-
ilarly, we can get (FPIF22). By Definition 9 we know
that µis a fuzzy 2-type positive implicative filter of X.
By Definition 8, Definition 9 and Theorem 1 we can
prove the following results (the proofs are omitted). In
fact, Theorem 5 (3) is a consequence of the so-called
Transfer Principle for Fuzzy Sets in [7].
Theorem 5. Let Xbe a pseudo-BCK algebra. Then
(1) if Fis filter of X, then the character function χF
of Fis a fuzzy filter of X;
(2) if Fis a 2-type positive implicative filter of X,
then the character function χFof Fis a fuzzy
2-type positive implicative filter of X;
(3) if µis a fuzzy set in X, then µis a fuzzy 2-
type positive implicative filter of Xif and only
if the level set µt={x∈X|µ(x)≥t}is 2-type
positive implicative filter of Xfor all t∈Im(µ).
By Theorem 3 and Theorem 5 we can prove the
following result (the proof is omitted).
Theorem 6. Every 2-type positive implicative filter of
a pseudo-BCK algebra Xis a normal filter of X.
The following example shows that the converse of
Theorem 6 is not true in general.
Example 2. Let Xbe the set {a, b, c, d, 1}with the
following operations:
→a b c d 1
a111 1 1
b b 1d1 1
cd d 1 1 1
d b d d1 1
1a b c d1
→a b c d 1
a111 1 1
bd1d1 1
c c d 1 1 1
dc d d11
1ab c d 1
Then (X;→,→,≤,1) is a pseudo-BCK algebra,
where ≤is defined by x≤yiff x→y=1. Assume
F={b, c, d, 1}, then Fis a normal filter of X. But F
is not a 2-type positive implicative filter of X, since
d→(b→a)=1/=d=(d→b)→(d→a).
By Theorem 6 and Theorem 3.3 in [20] we have
Theorem 7. Let Xbe a pseudo-BCK algebra and F
a filter of X. For every w∈X, denote Fw={x|w→
x∈F, w →x∈F}. Then
(1) if Fis a 2-type positive implicative filter of X,
then Fwis a filter of Xfor all w∈X;
(2) if Fis a normal filter of X, and Fwis a filter
of Xfor all w∈X, then Fis a 2-type positive
implicative filter of X.
By Theorem 6 and Theorem 4.5 in [20] we have
Theorem 8. Let Xbe a pseudo-BCK algebra. Then the
following statements are equivalent:

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X. Zhang / Fuzzy 1-type and 2-type positive implicative filters of pseudo-BCK algebras 2313
(1) (X, →,1) is a positive implicative BCK-algebra;
(2) (X, →,1) is a positive implicative BCK-
algebra;
(3) ∀x, y, z ∈X, x →y=x→y, x →(y→
z)=(x→y)→(x→z);
(4) every filter of Xis a 2-type positive implicative
filter of X;
(5) {1}and ↑w={x∈X|w≤x}(∀w∈X) are
positive implicative filter of X.
3. Fuzzy 1-type positive implicative filters
Definition 11. A fuzzy filter µ:X→[0,1] of pseudo-
BCK algebra Xis called a fuzzy 1-type positive
implicative filter if it satisfies
(FPIF11)µ(x→y)≥µ(x→(x→y)),∀x, y ∈
X;
(FPIF12)µ(x→y)≥µ(x→(x→y)),∀x, y ∈
X.
Example 3. Let Xbe the set {a, b, c, 1}with the fol-
lowing operations:
→a b c 1
a11 1 1
ba1c1
cb b 1 1
1a b c 1
→a b c1
a11 1 1
b c 1c1
c a b 1 1
1ab c 1
Then (X;→,→,≤,1) is a pseudo-BCK algebra,
where ≤is defined by x≤yiff x→y=1. Define
fuzzy set µ:X→[0,1] as following:
µ(1) =µ(b)=1,µ(a)=µ(c)=0.
Then µis a fuzzy 1-type positive implicative filter of
X,butµ(c→a)=0 and
min{µ(c→(c→a)),µ(c→c)}=1.
That is, µis not a fuzzy 2-type positive implicative filter
of X(by Definition 9).
Example 4. Let (X;→,→,≤,1) be the pseudo-BCK
algebra in Example 1. Define fuzzy set µ:X→[0,1]
as following:
µ(1) =µ(c)=µ(b)=µ(d)=0.92,µ(a)=0.25.
Then µis both a fuzzy 1-type positive implicative filter
and fuzzy 2-type positive implicative.
From Theorem 4, by using the notions of fuzzy nor-
mal filter and fuzzy 1-type positive implicative filter, we
can give the characterization of fuzzy 2-type positive
implicative filter.
Theorem 9. Let µbe a fuzzy filter of a pseudo-
BCK algebra X. Then the following conditions are
equivalent:
(i) µis a fuzzy 2-type positive implicative filter of X;
(ii) µis both a fuzzy normal filter and a fuzzy 1-type
positive implicative filter of X.
Theorem 10. Let µbe a fuzzy set in a pseudo-BCK
algebra X. Then µis a fuzzy 1-type positive implicative
filter of Xif and only if it satisfies (FF1) and
(C1) µ(x→z)≥min{µ(y→(x→z)),µ
(x→
y)},∀x, y, z ∈X;
(C2) µ(x→z)≥min{µ(y→(x→z)),µ(x→
y)},∀x, y, z ∈X.
Proof. Assume that µis a fuzzy 1-type positive implica-
tive filter of X. For any x, y, z ∈X, by Proposition 1 (5),
y→(x→z)≤(x→y)→(x→(x→z)).
From this, using Proposition 3 we get
µ(x→(x→z))
≥min{µ(y→(x→z)),µ
(x→y)}.
On the other hand, by Definition 11 (FPIF11), µ(x→
z)≥µ(x→(x→z)). Thus,
µ(x→z)
≥µ(x→(x→z))
≥min{µ(y→(x→z)),µ(x→y)}.
This means that (C1) holds. Similarly, we can get (C2).
Conversely, assume that µsatisfies (FF1), (C1) and
(C2). For any x, y ∈X, by (C1) and (C2) we have
µ(y)=µ(1 →y)
≥min{µ(x→(1 →y)),µ(1 →x)}
=min{µ(x→y),µ(x)}.
µ(y)=µ(1 →y)
≥min{µ(x→(1 →y)),µ(1 →x)}
=min{µ(x→y),µ(x)}.
By Definition 8 we know that µis a fuzzy filter of X.
Moreover, for any x, y ∈X, by (C1) and (C2) we have
µ(x→y)≥
min{µ((x→y)→(x→y)),µ(x→(x→y))}
=min{µ(1),µ(x→(x→y))}
=µ(x→(x→y));
µ(x→y)≥

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2314 X. Zhang / Fuzzy 1-type and 2-type positive implicative filters of pseudo-BCK algebras
min{µ((x→y)→(x→y)),µ(x→(x→y))}
=min{µ(1),µ(x→(x→y))}
=µ(x→(x→y)).
This means that (FPIF11) and (FPIF12) hold. By Def-
inition 11 we know that µis a fuzzy 1-type positive
implicative filter of X.
By Definition 11, Theorem 1 and Theorem 9 we can
prove the following results (the proofs are omitted).
Theorem 11. Let Xbe a pseudo-BCK algebra. Then
(1) if Fis a 1-type positive implicative filter of X,
then the character function χFof Fis a fuzzy
1-type positive implicative filter of X;
(2) if µis a fuzzy set in X, then µis a fuzzy 1-
type positive implicative filter of Xif and only
if the level set µt={x∈X|µ(x)≥t}is 1-type
positive implicative filter of Xfor all t∈Im(µ).
Theorem 12. A pseudo-BCK algebra Xis 1-type pos-
itive implicative if and only if every fuzzy filter of X
is a fuzzy 1-type positive implicative filter. Moreover, a
pseudo-BCK algebra Xis 2-type positive implicative
(i.e., Xis a positive implicative BCK-algebra) if and
only if every fuzzy filter of Xis a fuzzy 2-type positive
implicative filter (i.e., both a fuzzy normal filter and a
fuzzy 1-type positive implicative filter of X).
4. Fuzzy commutative filters and fuzzy
implicative Filters
In this section, we discuss two new kinds of fuzzy
filters of pseudo-BCK algebras, they are closely related
with fuzzy 1-type (2-type) positive implicative filters.
4.1. Fuzzy commutative filters
In 2008, as a generalization of BCI-algebra, W.A.
Dudek and Y.B. Jun [3] introduced the notion of
pseudo-BCI algebra which is also generalization of
pseudo-BCK algebra introducing by G. Georgescu and
A. Iorgulescu in [4]. Some fuzzy filters of pseudo-BCI
are studied in [15]. In the recent paper [16], we investi-
gated fuzzy commutative filters and fuzzy closed filters
of pseudo-BCI algebras. Since every pseudo-BCK
algebra is a pseudo-BCI algebra, then as an applica-
tion we propose the notion of fuzzy commutative filter
of pseudo-BCK algebra.
Definition 12. A fuzzy set µ:X→[0,1] is called a
fuzzy commutative filter of pseudo-BCK algebra Xif
it is a fuzzy filter of Xsuch that:
(FCF1) µ(((x→y)→y)→x)≥µ(y→x),
∀x, y ∈X;
(FCF2) µ(((x→y)→y)→x)≥µ(y→x),
∀x, y ∈X.
Example 5. Let (X;→,→,≤,1) be the pseudo-BCK
algebra in Example 1. Define fuzzy set µ:X→[0,1]
as following:
µ(1) =µ(c)=µ(b)=µ(d)=0.87,µ(a)=0.19.
Then µis fuzzy commutative filter of X.
Definition 13. [11] A pseudo-BCK algebra Xis called
commutative pseudo-BCK algebra if it satisfies the
equations
(C1) (x→y)→y=(y→x)→x, ∀x, y ∈X;
(C2) (x→y)→y=(y→x)→x, ∀x, y ∈X.
Theorem 13. Let Xbe a commutative pseudo-BCK
algebra and µa fuzzy filter of X. Then µis a fuzzy
commutative filter of X.
Proof. For any x, y, z ∈X, by Definition 13 (C1),
µ(((x→y)→y)→x)=µ(((y→x)→x)→x).
On the other hand, using Proposition 1 (10),
((y→x)→x)→x=y→x.
Thus,
µ(((x→y)→y)→x)
=µ(((y→x)→x)→x)
=µ(y→x)≥µ(y→x).
This means that Definition 12 (FCF1) holds. Similarly,
we can get (FCF2). Therefore, µis a fuzzy commutative
filter of X.
The following example shows that there exists fuzzy
commutative filter in general pseudo-BCK algebra
which is not commutative.
Example 6. Let (X;→,→,≤,1) be the pseudo-BCK
algebra in Example 1. Define fuzzy set µ:X→[0,1]
as following:
µ(1) =µ(b)=µ(c)=µ(d)=0.85,µ(a)=0.33.
Then µis a fuzzy commutative filter of X. But (X;→,
→,≤,1) is not commutative, since
(a→b)→b=b/=d=(b→a)→a.

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X. Zhang / Fuzzy 1-type and 2-type positive implicative filters of pseudo-BCK algebras 2315
4.2. Fuzzy implicative filters
Definition 14. A fuzzy set µ:X→[0,1] is called a
fuzzy implicative filter of pseudo-BCK algebra Xif it
is a fuzzy filter of Xsuch that:
(FIF1) µ(x)≥µ((x→y)→x),∀x, y ∈X;
(FIF2) µ(x)≥µ((x→y)→x),∀x, y ∈X.
Example 7. Let (X;→,→,≤,1) be the pseudo-BCK
algebra and µthe fuzzy filter in Example 4. Then µis
a fuzzy implicative filter of X.
Theorem 14. Let µbe a fuzzy implicative filter of a
pseudo-BCK algebra X. Then µsatisfies
(1) µ((y→x)→x)≥µ((x→y)→y),∀x, y ∈
X;
(2) µ((y→x)→x)≥µ((x→y)→y),∀x, y ∈
X;
(3) µ(((x→y)→y)→x)≥µ(y→x),∀x, y ∈
X;
(4) µ(((x→y)→y)→x)≥µ(y→x),∀x, y ∈
X.
Proof. (1) For any x, y ∈X, by Definition 1 (1) and
Proposition 1 (3), we have
(x→y)→y
≤(y→x)→((x→y)→x)
=(x→y)→((y→x)→x).
On the other hand, using Proposition 1 (6) and (9) we
get
x≤(y→x)→x,
((y→x)→x)→y≤x→y,
(x→y)→((y→x)→x)
≤(((y→x)→x)→y)→((y→x)→x).
Hence,
(x→y)→y
≤(((y→x)→x)→y)→((y→x)→x).
Applying Proposition 3,
µ((x→y)→y)
≤µ((((y→x)→x)→y)→((y→x)→x)).
By Definition 14 (FIF1) we have
µ((((y→x)→x)→y)→((y→x)→x))
≤µ((y→x)→x).
Therefore,
µ((x→y)→y)≤µ((y→x)→x).
(2) It is similar to (1).
(3) For any x, y ∈X, by Definition 1 (1) and Propo-
sition 1 (5), we have
y→x≤
(((x→y)→y)→y)→(((x→y)→y)→x).
On the other hand, using Definition 1 (2) and Proposi-
tion 1 (9) we get
x→y≤((x→y)→y)→y,
(((x→y)→y)→y)→(((x→y)→y)→x)
≤(x→y)→(((x→y)→y)→x).
Hence,
y→x≤(x→y)→(((x→y)→y)→x).
Moreover, applying Proposition 1 (6) and (9) we have
x≤((x→y)→y)→x,
(((x→y)→y)→x)→y≤x→y,
(x→y)→(((x→y)→y)→x)≤((((x→
y)→y)→x)→y)→(((x→y)→y)→x).
Thus,
y→x≤((((x→y)→y)→x)→y)→(((x→
y)→y)→x).
From this, using Proposition 3, we get
µ(y→x)≤µ(((((x→y)→y)→x)→y)→
(((x→y)→y)→x)).
By Definition 14 (FIF1) we have
µ(((((x→y)→y)→x)→y)→(((x→y)→
y)→x)) ≤µ(((x→y)→y)→x)).
Therefore,
µ(y→x)≤µ(((x→y)→y)→x)).
(4) It is similar to (3).
By Theorem 14 (3), (4) and Definition 14 we get
Corollary 1. Every fuzzy implicative filter of a pseudo-
BCK algebra Xis a fuzzy commutative filter of X.
Theorem 15. Let Xbe a pseudo-BCK algebra and µa
fuzzy normal filter of X.Ifµis a fuzzy implicative filter
of X, then µis a fuzzy 1-type positive implicative filter
of X.
Proof. For any x, y ∈X, by Definition 1 (2), Proposi-
tion 1 (8) and (3), we have
x→y≤((x→y)→y)→y,
x→(x→y)
≤x→(((x→y)→y)→y)
=((x→y)→y)→(x→y).
From this, using Proposition 3, we get
µ(x→(x→y))
≤µ(((x→y)→y)→(x→y)).
Since µis normal, by Definition 10,
µ(((x→y)→y)→(x→y))
=µ(((x→y)→y)→(x→y)).
Moreover, since µis implicative, Definition 14 (FIF1),
µ(((x→y)→y)→(x→y)) ≤µ(x→y).
Thus, we get
µ(x→(x→y)) ≤µ(x→y).

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2316 X. Zhang / Fuzzy 1-type and 2-type positive implicative filters of pseudo-BCK algebras
Similarly, we can get that
µ(x→(x→y)) ≤µ(x→y).
By Definition 11 we know that µis a fuzzy 1-type
positive implicative filter of X.
Theorem 16. Let Xbe a pseudo-BCK algebra and µa
fuzzy normal filter of X.Ifµis both fuzzy commutative
filter and fuzzy 1-type positive implicative of X, then µ
is a fuzzy implicative filter of X.
Proof. For any x, y ∈X, by Definition 8 (FF2), we have
µ(x)≥min{µ(((x→y)→y)→x),µ((x→
y)→y)}.
Since µis commutative, by Definition 12 (FCF1),
µ(((x→y)→y)→x)≥µ(y→x).
Then
µ(x)≥min{µ(((x→y)→y)→x),µ((x→
y)→y)}≥min{µ(y→x),µ((x→y)→y)}.
On the other hand, applying Definition 1 (1),
(x→y)→x≤(x→y)→((x→y)→y).
From this, using Proposition 3, we get
µ((x→y)→x)
≤µ((x→y)→((x→y)→y)).
Moreover, since µis normal and 1-type positive
implicative, by Definition 10 and 11,
µ((x→y)→((x→y)→y))
=µ((x→y)→((x→y)→y))
=µ((x→y)→y).
Thus,
µ((x→y)→x)
≤µ((x→y)→y)
=µ((x→y)→y)}.
Hence,
µ(x)
≥min{µ(y→x),µ
((x→y)→y)}
≥min{µ(y→x),µ((x→y)→x)}.
Moreover, by Proposition 1 (6), (9) and Proposition
3wehave
y≤x→y,
(x→y)→x≤y→x,
µ((x→y)→x)≤µ(y→x).
Therefore,
µ(x)
≥min{µ(y→x),µ((x→y)→x)}
=µ((x→y)→x).
Similarly, we can get that
µ(x)≥µ((x→y)→x).
By Definition 14 we know that µis a fuzzy implicative
filter of X.
By Theorem 15, 16, Corollary 1 and Theorem 9 we
get
Corollary 2. Let Xbe a pseudo-BCK algebra and µa
fuzzy normal filter of X. Then µis a fuzzy implicative
filter of Xif and only if µis both commutaive and 1-type
(or 2-type) positive implicative filter of X.
5. Conclusions
In this paper, we investigated some kinds of fuzzy filters
of pseudo-BCK algebras. Particularly, we introduced
the new notions of fuzzy 1-type positive implicative
filter, fuzzy normal filter, fuzzy commutative filter
and fuzzy implicative filter, and examined the rela-
tionships among them. The results show that the
fuzzy filters play important roles in characterization
of pseudo-BCK algebras. Moreover, relative to the
fuzzy filters of BCK-algebras, various fuzzy filters of
pseudo-BCK algebras have more complex relation-
ships.
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