# The Measure of Fuzzy Filters onBL-Algebras

Authors: Xiang Zhu; Md Gapar Md Johar; Lilysuriazna Binti Raya; Zu-hua Liao
Journal: Advances in Fuzzy Systems
 (2019)

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## Abstract

The new concept of the fuzzy filter degree was given by means of the implication operator, which enables to measure a degree to which a fuzzy subset of aBL-algebra is a fuzzy filter. In this paper, we put forward several equivalent characterizations of the fuzzy filter degree by studying its properties and the relationship with level cut sets. Furthermore, we study the fuzzy filter degrees of the intersection and fuzzy direct products of fuzzy subsets and investigate the fuzzy filter degrees of the image and the preimage of a fuzzy subset under a homomorphism.


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## Body

## 1. Introduction

As we know, nonclassical logic algebras are one of the focuses in the basic study of artificial intelligence. Among these logic algebras,BL-algebra is one of the most fundamental algebraic structures, which was put forward by Hájek [1] in 1998. MV-algebras, Gödel algebras, and product algebras are the most known classes of BL-algebras.Filters theory plays an important role in the theoretical study of these algebras. From logical point of view, a kind of filter corresponds to a set of provable formula. In [2], Liu proposed the notion of fuzzy filters in BL-algebras and investigated some of their properties and showed that fuzzy filters are a useful tool to obtain results on classical filters of BL-algebras.In the application of fuzzy theory, a series of concepts of degree were introduced to depict the level of similarity among objects. When these thoughts are applied in the research of fuzzy logic algebras, the question is how to measure the degree to which a fuzzy subset is a fuzzy logic algebra. In 2010, Shi gave an idea in [3]. Shi proposed the concept of fuzzy subgroup degree in the paper to depict the degree to which a fuzzy subset is a fuzzy subgroup. Using this idea, Wang, Shi, Liao, and others have done a lot of work in recent years and got a lot of good results [4–15].Inspired by the ideas mentioned above, we present a new concept of fuzzy filter, by which the degree to which a fuzzy subset of latticeL is a fuzzy filter is depicted and some academic results are obtained.This paper is organized as follows: In Section2, the basic knowledge required in the paper is proposed. In Section 3, we put forward the concept of fuzzy filter degree and its equivalent characterizations. Furthermore, the fuzzy filter degrees of the intersection and fuzzy direct products of fuzzy subsets are investigated and the fuzzy filter degrees of the image and the preimage of a fuzzy subset under a homomorphism are also discussed.In what follows, letL denote a BL-algebra.


## 2. Preliminaries

In this section, we recollect some basic definitions and results which will be used in the following.Definition 1 (see [1]).
ABL-algebra is a structure L,∧,∨,⊗,⟶,0,1 such that(i)
L,∧,∨,0,1 is a bounded lattice(ii)
L,⊗,1 is an abelian monoid, i.e., ⊗ is commutative and associative and x⊗1=1⊗x=x(iii)
The following conditions hold for allx,y,z∈L:(B1)
x⊗y≤z if and only if x≤y⟶z (residuation)(B2)
x∧y=x⊗x⟶y (divisibility)(B3)
x⟶y∨y⟶x=1 (prelinearity)Definition 2 (see [1]).
LetL be a BL-algebra. A subset F of L is called a filter of L if it satisfies(1)
1∈F(2)
x∈F and x⟶y∈F imply y∈FDefinition 3 (see [1]).
LetA be a fuzzy set in L. A is called a fuzzy filter if it satisfies(1)
A1≥Ax for all x∈L(2)
Ay≥Ax∧Ax⟶y for all x, y∈LDefinition 4 (see [1]).
A mappingf: L⟶M from a BL-algebra L into a BL-algebra M is called a BL homomorphism if for any x, y∈L:(1)
f0L=0M(2)
fx⊗y=fx⊗fy(3)
fx⟶y=fx⟶fywhere 0L and 0M are the least elements of L and M, respectively.
Iff is a surjection, then f is called an epimorphism. If f is an injective, then f is called a monomorphism. If f is a bijective, then f is called an isomorphism.
It is easy to verify thatf1L=1M, where 1L and 1M are the largest elements of L and M, respectively.Definition 5 (see [16]).
A mappingI: [0,1] × [0,1] ⟶ [0,1] is called a fuzzy implication operator, if it satisfies(1)
x≤y implies Ix,z≥Iy,z for all x, y, z∈0,1(2)
y≤z implies Ix,y≤Iy,z for all x, y, z∈0,1(3)
I1,0=0, I0,0=I1,1=1
For anyx, y∈0,1, let Ix,y=∨t∈0,1x∧t≤y. Obviously, I is an implication, which is called the R-implication generated by “min,” and is usually denoted as x ⟶ y.
The implication operator has the following properties.Proposition 1 (see [17]).
For anya, aii∈I∈0,1, the following hold:(1)
a⟶∧i∈Iai=∧i∈Ia⟶ai(2)
∨i∈Iai⟶a=∧i∈Iai⟶a(3)
∧i∈Iai∧∧j∈Ibj=∧i,j∈Iai∧bjDefinition 6 (see [18]).
LetA be a fuzzy set in L, b∈0,1, then the set Ab=x∈LAx≥b is called a level subset of A; the set Ab=x∈LAx>b is called a strong level subset of A.Definition 7 (see [16]) (extension principle).
With the mappingf:X⟶Y, two mappings can be induced by this mapping, respectively, denoted as f and f−1 as follows:(1)f:FX⟶FY,A⟼fA,f−1:FY⟶FX,B⟼f−1B,where the membership function of fA and f−1B is defined as(2)fAy=∨fx=yAx,f−1Bx=Bfx.Definition 8 (see [19]).
LetLii∈I be a family of BL-algebras. ∏i∈ILi is defined as xx:I⟶∪i∈LLi,xi≜xi∈Li,i∈I. The operations “∨, ∧, ⊗, and ⟶” are given as follows:(3)x∨yi=xi∨yi=xi∨yi,x∧yi=xi∧yi=xi∧yi,x⊗yi=xi⊗yi=xi⊗yi,x⟶yi=xi⟶yi=xi⟶yi,Oi=0i,Ii=1i,where 1i and 0i are the largest element and the least element of Li, respectively. (∏i∈ILi, ∨, ∧) is called the BL direct product.
It is easy to verify that (∏i∈ILi, ∨, ∧, ⊗, ⟶, O, I) is a BL-algebra.Definition 9 (see [20]).
LetAi be a fuzzy subset of Li (i ∈ I). The fuzzy subset ∏i∈IAi of ∏i∈ILi is defined as(4)∀x=∏i∈Ixi∈∏i∈ILixi∈Li,∏i∈IAix=∧i∈IAixi,where ∏i∈IAi is called the fuzzy direct product of Aii∈I.Definition 10 (see [21]).
Letf be a mapping of a BL-algebra L into a BL-algebra M. Then, a fuzzy subset A of L is called f-invariant if for all x,y∈L, and fx=fy implies Ax=Ay.


## 3. Fuzzy Filters Degree

Definition 11.
LetA be a fuzzy subset in L. The fuzzy filter degree mLA of A is defined as(5)mLA=∧x,y∈LAx∧Ax⟶y⟶Ay∧∧x∈LAx⟶A1=∧x,y∈L∨t∈0,1Ax∧Ax⟶y∧t≤Ay∧∧x∈L∨t∈0,1Ax∧t≤A1.
For convenience, we denote(6)B1x,y=t∈0,1Ax∧Ax⟶y∧t≤Ay,B2x=t∈0,1Ax∧t≤A1.
So,mLA=∧x,y∈L∨B1x,y∧∧x∈L∨B2x.Remark 1.
In Definition11, mLA can be used to measure the degree of a fuzzy subset A to be a fuzzy filter of L.Example 1.
LetL = {0, a, b, 1} be a lattice, and the order “≤” on L be determined by using Figure 1. For all x, y ∈ L, the two binary operations ⟶ and ⊗ are defined by Table 1 and 2. Then, (L, ∨, ∧, ⊗, ⟶, 0, 1) is a BL-algebra.
LetA1, A2, and A3 be the fuzzy sets in L given by(7)A1=0.20,0.2a,0.6b,0.71,A2=0.20,0.4a,0.4b,0.71,A3=0.30,0.4a,0.4b,0.71.
One can easily check thatA1 is a fuzzy filter of L. Neither A2 nor A3 is a fuzzy filter of L. From Definition 11, we have mLA1=1, mLA2=0.2, and mLA3=0.3.
SinceI (x, y) is a continuous t-norm on [0,1], it is not difficult to have the following.Figure 1
The order “≤” onL.Table 1
The binary operation “⊗”.
⊗
0
a
b
1
0
0
0
0
0
a
0
a
0
a
b
0
0
b
b
1
0
a
b
1Table 2
The binary operation “⟶”.
⟶
0
a
b
1
0
1
1
1
1
a
b
1
b
1
b
a
a
1
1
1
0
a
b
1Theorem 1.
LetA be a fuzzy subset of L. A is a fuzzy filter of L if and only if mLA=1.The properties ofmLA are discussed below, and their equivalent characterizations are also given.Lemma 1.
LetA be a fuzzy subset of L. mLA≥a if and only if for all x, y ∈ L, Ax∧Ax⟶y∧a≤Ay and Ax∧a≤A1.Proof.
SupposemLA≥a and then ∧x,y∈L∨B1x,y∧∧x∈L∨B2x≥a by Definition 11. Thus, ∧x,y∈L∨B1x,y≥a and ∧x∈L∨B2x≥a.
Noteb1x,y=∨B1x,y. There exists tε∈B1x,y for any ε>0 such that tε>b1x,y−ε. Therefore, since tε∈B1x,y, we have Ax∧Ax⟶y∧tε≤Ay. So, Ax∧Ax⟶y∧b1x,y−ε≤Ax∧Ay∧tε≤Ay.
By the arbitrariness ofε, Ax∧Ax⟶y∧b1x,y≤Ay as well as b1x,y≥a; therefore, Ax∧Ax⟶y∧a≤Ay. Similarly, we can prove Ax∧a≤A1.
Conversely, ifAx∧Ax⟶y∧a≤Ay and Ax∧a≤A1 for any x, y ∈ L, then a∈B1x,y and a∈B2x. Hence, we have(8)mLA∧x,y∈L∨B1x,y∧∧x∈L∨B2x≥a.
The proof is completed.Theorem 2.
SupposeA be a fuzzy subset of L and(9)K1=a∈0,1Ax∧Ax⟶y∧a≤Ay,∀x,y∈L,K2=a∈0,1Ax∧a≤A1,∀x∈L,K=a∈0,1Ax∧Ax⟶y∧a≤Ay,Ax∧a≤A1,∀x,y∈L.
Then,mLA=∨K1∧∨K2=∨K.Proof.
We first show thatmLA=∨K.
LetmLA=c. By Lemma 1, we get that for any x, y ∈ L, Ax∧Ax⟶y∧c≤Ay and Ax∧c≤A1.
Letb = ∨K, and it is easy to get b≥c. There exists tε∈K for any ε>0 such that tε>b−ε. Since tε∈K, we have Ax∧Ax⟶y∧tε≤Ay and Ax∧tε≤A1 for any x, y ∈ L. So, Ax∧Ax⟶y∧b−ε≤Ay and Ax∧b−ε≤A1.
By the arbitrariness ofε, we have(10)Ax∧Ax⟶y∧b≤Ay,Ax∧b≤A1.
Moreover,mLA≥b and c ≥ b by Lemma 1. So, b = c.
Now we prove∨K1∧∨K2=∨K.
AsAx∧Ax⟶y∧a≤Ay and Ax∧a≤A1 for any a ∈ K and x, y ∈ L, we have a ∈ K1 and a ∈ K2. Then, a≤∨K1 and a≤∨K2. Thus, a≤∨K1∧∨K2. Therefore, ∨K≤∨K1∧∨K2.
Letai=∨K1i=1,2. There exists ti∈Kii=1,2, such that ti>ai−ε for any ε>0.
Letb=∨K1∧∨K2. Then, ai≥bi=1,2. So, ti>b−ε.
Thus,Ax∧Ax⟶y∧b−εa≤Ay and Ax∧b−ε≤A1.
By the arbitrariness ofε, we have(11)Ax∧Ax⟶y∧b≤Ay,Ax∧b≤A1.
By Lemma1, we get that mLA≥b, that is, ∨K≥b
Finally, we have∨K1∧∨K2=∨K.
We know there is a close relationship between the fuzzy filter of aBL-algebra and its level set. Next, we give some characterizations of the fuzzy filter degree of a fuzzy set by means of its level sets.Lemma 2.
LetA be a fuzzy subset of L. If mLA=c>0, then nonvoid A[b] is a filter of L for any b ∈ (0, c].Proof.
IfAb≠∅ for any b∈0,c; then, for any x, x⟶y∈Ab, we have Ax≥b and Ax⟶y≥b. Since mLA=c, the followings are obtained by Lemma 1:(12)Ay≥Ax∧Ax⟶y∧c≥b∧c=b,A1≥Ax∧c≥b∧c=b.
So,y∈Ab and 1∈Ab. Hence, Ab is a filter of L.
For strong level sets, we can get the similar conclusions such as Lemma3 by suitable modification to the proof of Lemma 2.Lemma 3.
LetA be a fuzzy subset of L. If mLA=c>0, then nonvoid A(b) is a filter of L for any b ∈ (0, c].Theorem 3.
LetA be a fuzzy subset of L. If mLA=c>0, then mLA=∨a∈0,1∀b∈0,a,Ab≠∅is a filter ofL.Proof.
Suppose thatmLA=c and B=a∈0,1∀b∈0,a,Ab≠∅is a filter ofL. According to Lemma 2, we have c ∈ B; thus, ∨B≥c, i.e., ∨B≥mLA
Next, we prove∨B≤mLA.
In fact, for anya ∈ B and x, y ∈ L, we have Ax∧Ax⟶y∧a≤Ay. Otherwise, there exists x0, y0 ∈ L, such that Ax0∧Ax0⟶y0∧a>Ay0.
LetAx0∧Ax0⟶y0∧a=b. Then, b∈0,a, Ax0∧Ax0⟶y0≥b, and Ay0<b. That is, x0, x0⟶y0∈Ab and Ay0<b.
Sincea ∈ B and, b∈0,a Ab is a filter of L. This together with x0, x0⟶y0∈Ab implies that y0∈Ab, i.e., Ay0≥b. This is contrary to Ay0<b
Similarly, for anya ∈ B, x ∈ L, we have Ax∧a≤A1. By Lemma 1, it follows that mLA≥a. Thus, ∨B≤mLA.
From the above, we havemLA=∨B=∨a∈0,1∀b∈0,a,Ab≠∅is a filter ofL.
With Lemma1 and Lemma 3, proceeding similar as in the proof of Theorem 3, we have the following theorem.Theorem 4.
LetA be a fuzzy subset of L. Then, mLA=∨a∈0,1∀b∈0,a,Ab≠∅is a filter ofL.Next, we discuss the fuzzy filter degree under fuzzy subset operations.Firstly, the relation of the fuzzy filter degree between the intersection of a family of fuzzy subsets and each fuzzy subset is given.Theorem 5.
Let {Ai}i∈I be a family of fuzzy subsets of L. Then,(13)mL∩i∈IAi≥∧i∈ImLAi.Proof.
By Proposition 1 and Definition5, we have(14)∧x,y∈L∩i∈IAix∧∩i∈IAix⟶y⟶∩i∈IAiy=∧x,y∈L∧i∈IAix∧∧i∈IAix⟶y⟶∧i∈IAiy=∧x,y∈L∧i∈I∧i∈IAix∧∧i∈IAix⟶y⟶Aiyby Proposition1≥∧x,y∈L∧i∈IAix∧Aix⟶y⟶Aiyby Definition5=∧i∈I∧x,y∈LAix∧Aix⟶y⟶Aiy≥∧i∈I∧x,y∈LAix∧Aix⟶y⟶Aiy∧∧i∈LAix⟶Ai1=∧i∈ImLAi.
Following in a similar manner, we can get(15)∧x,y∈L∩i∈IAix⟶∩i∈IAi1≥∧i∈ImLAi.
From Definition11, it follows that(16)mL∩i∈IAi=∧x,y∈L∩i∈IAix∧∩i∈IAix⟶y⟶∩i∈IAiy∧∧x,y∈L∩i∈IAix⟶∩i∈IAi1≥∧i∈ImLAi.
Secondly, the fuzzy filter degree of fuzzy subsets under the fuzzy direct product is discussed.Theorem 6.
LetAi be a fuzzy subset of Lii∈I; then, mL∏i∈IAi≥∧i∈ImLiAi.Proof.
Supposex, y∈∏i∈ILi=L. Then, x=∏i∈Ixi and y=∏i∈Iyi, where xi, yi ∈ Li.(17)∧x,y∈L∏i∈IAix∧∏i∈IAix⟶y⟶∏i∈IAiy=∧xi,yi∈Li∧i∈IAixi∧∧i∈IAixi⟶yi⟶∧i∈IAiyi=∧xi,yi∈Li∧i∈I∧i∈IAixi∧∧i∈IAixi⟶yi⟶Aiyiby Proposition1≥∧xi,yi∈Li∧i∈IAixi∧Aixi⟶yi⟶Aiyiby Definition5=∧i∈I∧xi,yi∈LiAixi∧Aixi⟶yi⟶Aiyi≥∧i∈I∧xi,yi∈LiAixi∧Aixi⟶yi⟶Aiyi∧∧xi∈LiAixi⟶Ai1=∧i∈ImLiAi.
Following in a similar manner, we can get(18)∧x,y∈L∏i∈IAix⟶∏i∈IAi1≥∧i∈ImLiAi.
So, by Definition11,(19)mL∏i∈IAi=∧x,y∈L∏i∈IAix∧∏i∈IAix⟶y⟶∏i∈IAiy∧∧x,y∈L∏i∈IAix⟶∏i∈IAi1≥∧i∈ImLiAi.
Finally, we give the relation between the fuzzy ideal degrees of the image and the preimage of a fuzzy subset under a homomorphism.Theorem 7.
Letf: L1⟶ L2 be a BL homomorphism, and A and B be the fuzzy subsets of L1 and L2, respectively. The following assertions hold:(1)
Iff is a surjective and f-invariant, then mL1A≤mL2fA(2)
mL2B≤mL1f−1B(3)
Iff is a surjective, then mL2B≤mL1f−1B.Proof.
(1)
By Theorem2 and Definition 7, we can have that
(20)
m
L
2
f
A
=
∨
a
∈
0,1
f
A
x
′
∧
f
A
x
′
⟶
y
′
∧
a
≤
f
A
y
′
,
f
A
x
′
∧
a
≤
f
A
1
,
∀
x
′
,
y
′
∈
L
2
=
∨
a
∈
0,1
∨
f
x
=
x
′
A
x
∧
∨
f
x
=
x
′
⟶
y
′
A
x
∧
a
≤
∨
f
x
=
y
′
A
x
,
∨
f
x
=
x
′
A
x
∧
a
≤
∨
f
x
=
1
A
x
,
∀
x
′
,
y
′
∈
L
2
.
Sincef is an epimorphism, there exists x0, y0 ∈ L1 such that fx0=x′ and fy0y′.
Thus,fx0⟶y0=fx0⟶fy0=x′⟶y′. Now that f is an f-invariant, which together with f1=1, we have
(21)
∨
f
x
=
x
′
A
x
=
A
x
0
,
∨
f
x
=
y
′
A
x
=
A
y
0
,
∨
f
x
=
x
′
⟶
y
′
A
x
=
A
x
0
⟶
y
0
,
∨
f
x
=
1
A
x
=
A
1
.
It follows that
(22)
m
L
2
f
A
=
∨
a
∈
0,1
A
x
0
∧
A
x
0
⟶
y
0
∧
a
≤
A
y
0
,
A
x
0
∧
a
≤
A
1
≥
∨
a
∈
0,1
A
x
∧
A
x
⟶
y
∧
a
≤
A
y
,
A
x
∧
a
≤
A
1
,
∀
x
,
y
∈
L
1
=
m
L
1
A
.
(2)
By Definition11 and Definition 7, we can have that
(23)
m
L
1
f
−
1
B
=
∧
x
,
y
∈
L
1
f
−
1
B
x
∧
f
−
1
B
x
⟶
y
⟶
f
−
1
B
y
∧
∧
x
∈
L
1
f
−
1
B
x
⟶
f
−
1
B
1
=
∧
x
,
y
∈
L
1
B
f
x
∧
B
f
x
⟶
y
⟶
B
f
y
∧
∧
x
∈
L
1
B
f
x
⟶
B
f
1
=
∧
x
,
y
∈
L
1
B
f
x
∧
B
f
x
⟶
f
y
⟶
B
f
y
∧
∧
x
∈
L
1
B
f
x
⟶
B
1
by Definition
4
≥
∧
x
′
,
y
′
∈
L
2
B
x
′
∧
B
x
′
⟶
y
′
⟶
B
y
′
∧
∧
x
′
∈
L
2
B
x'
⟶
B
1
=
m
L
2
B
.
(3)
Iff is an epimorphism, then from the proof of (2), we have
(24)
m
L
1
f
−
1
B
=
∧
x
,
y
∈
L
1
B
f
x
∧
B
f
x
⟶
f
y
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f
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L
1
B
f
x
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B
1
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y
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∧
x
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∈
L
2
B
x
′
⟶
B
1
=
m
L
2
B
.


## 4. Conclusions

In order to measure the degree to which a fuzzy subset is a fuzzy filter, we put forward the concept of the fuzzy filter degree ofL. Through the research on the fuzzy filter degree of L, we have a deeper understanding about the fuzzy filter, further enriching the theory of fuzzy filter of BL-algebras. Using this idea, we can also measure other fuzzy algebraic structures and it also provides theoretical basis for fuzzy pattern recognition and other applications.



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