# Generalizations of(∈,∈∨q)-Fuzzy Filters in R0-Algebras

Authors: Young Bae Jun; Seok Zun Song; Jianming Zhan
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

Generalizations of a part of the paper (Ma et al., 2009) are considered. As a generalization of an(∈,∈∨q)-fuzzy filter, the notion of an (∈,∈∨qk)-fuzzy filter is introduced, and its characterizations are
provided. The implication-based fuzzy filters of an R0-algebra are discussed.


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## 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 and randomness. 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 R0-algebras was first introduced by Wang in [2] by providing an algebraic proof of the completeness theorem of a formal deductive system [3]. Obviously, R0-algebras are different from the BL-algebras. Jun and Lianzhen [4] studied filters of R0-algebras. Lianzhen and Kaitai [5] discussed the fuzzy set theory of filters in R0-algebras. As a generalization of the notion of fuzzy filters, Ma et al. [6] dealt with the notion of (∈,∈∨q)-fuzzy filters in R0-algebras.In this article, we try to get more general form of the notion of(∈,∈∨q)-fuzzy filters. We introduce the notion of (∈,∈∨qk)-fuzzy filters and investigate related properties. We establish characterizations of an (∈,∈∨qk)-fuzzy filter and finally consider the implication-based fuzzy filters of an R0-algebra. The important achievement of the study with an (∈,∈∨qk)-fuzzy filter is that the notion of an (∈,∈∨q)-fuzzy filter is a special case of an (∈,∈∨qk)-fuzzy filter, and thus the related results obtained in the paper [6] are a corollary of our results obtained in this paper.


## 2. Preliminaries

Definition 2.1 (see [2]).
LetL be a bounded distributive lattice with order-reversing involution ¬ and a binary operation →. Then (L,⋀,∨,¬,→) is called an R0-algebra 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 R0-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.Example 2.2 (see [5]).
LetL=[0,1]. For any x,y∈L, we define x⋀y=min{x,y},x∨y=max{x,y},¬x=1-x, and
(2.1)x→y:={1ifx≤y,¬x∨yifx>y.
Then (L,⋀,∨,¬,→) is an R0-algebra which is neither a BL-algebra nor a lattice implication algebra.AnR0-algebra has the following useful properties.Proposition 2.3 (see [7]).
For any elementsx,  y, and z of an R0-algebra L, one has the following properties: (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 R0-algebra L is called a filter of L if it satisfies the following two conditions: (b1)
1∈A,(b2)
(forallx∈A)(forally∈L)(x→y∈A⇒y∈A).It can be easily verified that a nonempty subsetA of an R0-algebra L is a filter of L if and only if it satisfies the following conditions: (b3)
(forallx,y∈A)(x⊙y∈A),(b4)
(forally∈L)(∃x∈A)(x≤y⇒y∈A).Definition 2.4.
A fuzzy setμ in an R0-algebra L is called a fuzzy filter of L if it satisfies the following: (c1)
(forallx,y∈L)(μ(x⊙y)≥min{μ(x),μ(y)}),(c2)
μ is order-preserving, that is, (forallx,y∈L)(x≤y⇒μ(x)≤μ(y)).Theorem 2.5.
A fuzzy setμ in an R0-algebra L is a fuzzy filter of L if and only if it satisfies the following: (c3)
(forallx∈L)(μ(1)≥μ(x)),(c4)
(forallx,y∈L)(μ(y)≥min{μ(x→y),μ(x)}).For any fuzzy setμ in L and t∈(0,1], the set (2.2)U(μ;t)={x∈L∣μ(x)≥t}
is called a level subset of L. A fuzzy set μ in a set L of the form(2.3)μ(y):={t∈(0,1]ify=x,0ify≠x
is said to be a fuzzy point with support x and value t and is denoted by (x,t).For a fuzzy point(x,t) and a fuzzy set μ in a set L, Pu and Liu [8] introduced the symbol (x,t)αμ, where α∈{∈,q,∈∨q,∈⋀q}. To say that (x,t)∈μ (resp. (x,t)qμ), we mean μ(x)≥t (resp. μ(x)+t>1), and in this case, (x,t) is said to belong to (resp. be quasi-coincident with) a fuzzy set μ. To say that (x,t)∈∨qμ (resp. (x,t)∈⋀qμ), we mean that (x,t)∈μ or (x,t)qμ (resp. (x,t)∈μ and (x,t)qμ).


## 3. Generalizations of(∈,∈∨q)-Fuzzy Filters

In what follows,L is an R0-algebra and let k denote an arbitrary element of [0,1) unless otherwise specified. To say that (x,t)qkμ, we mean μ(x)+t+k>1. To say that (x,t)∈∨qkμ, we mean (x,t)∈μ or (x,t)qkμ. For α∈{∈,∈∨qk}, to say that (x,t)α¯μ, we mean (x,t)αμ does not hold.Definition 3.1.
A fuzzy setμ in L is said to be an (∈,∈∨qk)-fuzzy filter of L if it satisfies the following: (d1)
(x,t)∈μ&(y,r)∈μ⇒(x⊙y,min{t,r})∈∨qkμ,(d2)
(x,t)∈μ&x≤y⇒(y,t)∈∨qkμ for all x,y∈L and t,r∈(0,1].An(∈,∈∨qk)-fuzzy filter of L with k=0 is called an (∈,∈∨q)-fuzzy filter of L.Example 3.2.
LetL={0,a,b,c,1} be a set with Hasse diagram and Cayley tables which are given in Table 1. Then (L,⋀,∨,¬,→,0,1) is an R0-algebra (see [5]), where x⋀y=min{x,y} and x∨y=max{x,y}. Define a fuzzy set μ in L by
(3.1)μ=(0abc10.30.30.30.80.45).
It is routine to verify that μ is an (∈,∈∨q0.2)-fuzzy filter of L. But it is neither a fuzzy filter nor an (∈,∈∨q)-fuzzy filter of L since μ(1)=0.45≱0.8=μ(c), and c≤1 and (c,0.5)∈μ but (1,0.5)∈∨q¯μ.Table 1
Hasse diagram and Cayley tables.Theorem 3.3.
Every fuzzy filter is an(∈,∈∨qk)-fuzzy filter.Proof.
It is straightforward.Example3.2 shows that the converse of Theorem 3.3 may not be true and shows that an (∈,∈∨qk)-fuzzy filter may not be an (∈,∈∨q)-fuzzy filter in general.We establish characterizations of an(∈,∈∨qk)-fuzzy filter.Theorem 3.4.
A fuzzy setμ in L is an (∈,∈∨qk)-fuzzy filter of L if and only if it satisfies the following: (d3)
(forallx,y∈L)(μ(x⊙y)≥min{μ(x),μ(y),(1-k)/2}),(d4)
(forallx,y∈L)(x≤y⇒μ(y)≥min{μ(x),(1-k)/2}).Proof.
Letμ be an (∈,∈∨qk)-fuzzy filter of L. Assume that (d3) is not valid. Then there exist a,b∈L such that
(3.2)μ(a⊙b)<min{μ(a),μ(b),1-k2}.
If min{μ(a),μ(b)}<(1-k)/2, then μ(a⊙b)<min{μ(a),μ(b)}. Hence
(3.3)μ(a⊙b)<t≤min{μ(a),μ(b)}
for some t∈(0,(1-k)/2]. It follows that (a,t)∈μ and (b,t)∈μ, but (a⊙b,t)∈¯μ. Moreover, μ(a⊙b)+t<2t<1-k, and so (a⊙b,t)qk¯μ. Consequently (a⊙b,t)∈∨qk¯μ, a contradiction. If min{μ(a),μ(b)}≥(1-k)/2, then μ(a)≥(1-k)/2,μ(b)≥(1-k)/2 and μ(a⊙b)<(1-k)/2. Thus (a,(1-k)/2)∈μ and (b,(1-k)/2)∈μ, but (a⊙b,(1-k)/2)∈¯μ. Also,
(3.4)μ(a⊙b)+1-k2<1-k2+1-k2=1-k,
that is, (a⊙b,(1-k)/2)qk¯μ. Hence (a⊙b,(1-k)/2)∈∨qk¯μ, again, a contradiction. Therefore (d3) is valid. Let x,y∈L be such that x≤y. Assume that μ(y)<min{μ(x),(1-k)/2}. Then
(3.5)μ(y)<r≤min{μ(x),1-k2}
for some r∈(0,(1-k)/2]. If μ(x)<(1-k)/2, then μ(y)<r≤μ(x) by (3.5). Hence (x,r)∈μ and (y,r)∈¯μ. Furthermore, μ(y)+r<2r≤1-k, that is, (y,r)qk¯μ. Thus (y,r)∈∨qk¯μ, a contradiction. If μ(x)≥(1-k)/2, then μ(y)<r≤(1-k)/2 by (3.5). Hence (x,(1-k)/2)∈μ and (y,(1-k)/2)∈¯μ. Also, μ(y)+(1-k)/2≤1-k, that is, (y,(1-k)/2)qk¯μ. Thus (y,(1-k)/2)∈∨qk¯μ which is also a contradiction. Therefore μ(y)≥min{μ(x),(1-k)/2} for all x,y∈L with x≤y; that is, (d4) is valid.
Conversely, letμ be a fuzzy set in L satisfying two conditions (d3) and (d4). Let x,y∈L and t,r∈(0,1] be such that (x,t)∈μ and (y,r)∈μ. Then μ(x)≥t and μ(y)≥r. It follows from (d3) that
(3.6)μ(x⊙y)≥min{μ(x),μ(y),1-k2}≥min{t,r,1-k2}={min{t,r}ift≤1-k2orr≤1-k2,1-k2ift>1-k2andr>1-k2.
The case μ(x⊙y)≥min{t,r} implies that (x⊙y,min{t,r})∈μ. From the case μ(x⊙y)≥(1-k)/2, we have
(3.7)μ(x⊙y)+min{t,r}>1-k2+1-k2=1-k,
that is, (x⊙y,min{t,r})qkμ. Hence (x⊙y,min{t,r})∈∨qkμ. Finally let x,y∈L and t∈(0,1] be such that x≤y and (x,t)∈μ. Then μ(x)≥t, and so
(3.8)μ(y)≥min{μ(x),1-k2}≥min{t,1-k2}
by (d4). If t≤(1-k)/2, then μ(y)≥t, and thus (y,t)∈μ. If t>(1-k)/2, then μ(y)≥(1-k)/2 which implies that μ(y)+t>(1-k)/2+(1-k)/2=1-k, that is, (y,t)qkμ. Thus (y,t)∈∨qkμ. Consequently, μ is an (∈,∈∨qk)-fuzzy filter of L.If we takek=0 in Theorem 3.4, then we have the following corollary.Corollary 3.5 (see [6]).
A fuzzy setμ in L is an (∈,∈∨q)-fuzzy filter of L if and only if it satisfies the following: (1)
(forallx,y∈L)(μ(x⊙y)≥min{μ(x),μ(y),0.5}),(2)
(forallx,y∈L)(x≤y⇒μ(y)≥min{μ(x),0.5}).Theorem 3.6.
A fuzzy setμ in L is an (∈,∈∨qk)-fuzzy filter of L if and only if it satisfies the following: (d5)
(forallx∈L)(μ(1)≥min{μ(x),(1-k)/2}),(d6)
(forallx,y∈L)(μ(y)≥min{μ(x),μ(x→y),(1-k)/2}).Proof.
Letμ be an (∈,∈∨qk)-fuzzy filter of L. Since x≤1 for all x∈L, it follows from (d4) that μ(1)≥min{μ(x),(1-k)/2} for all x∈L. Let x,y∈L. Since x≤(x→y)→y, we have x⊙(x→y)≤y by (a13). Using (d4) and (d3), we obtain
(3.9)μ(y)≥min{μ(x⊙(x→y)),1-k2}≥min{μ(x),μ(x→y),1-k2}.
Conversely, letμ be a fuzzy set in L satisfying two conditions (d5) and (d6). Let x,y∈L be such that x≤y. Then x→y=1, and so
(3.10)μ(y)≥min{μ(x),μ(x→y),1-k2}=min{μ(x),μ(1),1-k2}=min{μ(x),1-k2}
by (d6) and (d5). Note from (a11) that
(3.11)x→(y→(x⊙y))=(x⊙y)→(x⊙y)=1
for all x,y∈L. It follows from (d5) and (d6) that
(3.12)μ(x⊙y)≥min{μ(y),μ(y→(x⊙y)),1-k2}≥min{μ(y),min{μ(x),μ(x→(y→(x⊙y))),1-k2},1-k2}=min{μ(y),min{μ(x),μ(1),1-k2},1-k2}=min{μ(x),μ(y),1-k2}.
Using Theorem 3.4, we conclude that μ is an (∈,∈∨qk)-fuzzy filter of L.Corollary 3.7 (see [6]).
A fuzzy setμ in L is an (∈,∈∨q)-fuzzy filter of L if and only if it satisfies the following: (1)
(forallx∈L)(μ(1)≥min{μ(x),0.5}),(2)
(forallx,y∈L)(μ(y)≥min{μ(x),μ(x→y),0.5}).Proof.
It is straightforward by takingk=0 in Theorem 3.6.Corollary 3.8.
Ifμ is an (∈,∈∨qk)-fuzzy filter of L with μ(1)<(1-k)/2, then μ is a fuzzy filter.If we takek=0 in Corollary 3.8, then we obtain the following corollary.Corollary 3.9 (see [6]).
Ifμ is an (∈,∈∨q)-fuzzy filter of L with μ(1)<0.5, then μ is a fuzzy filter.Theorem 3.10.
A fuzzy setμ in L is an (∈,∈∨qk)-fuzzy filter of L if and only if it satisfies the following: (d7)
(forallx,y,z∈L)(x≤y→z⇒μ(z)≥min{μ(x),μ(y),(1-k)/2}).Proof.
Assume thatμ is an (∈,∈∨qk)-fuzzy filter of L. Let x,y,z∈L be such that x≤y→z. It follows from (d4) that μ(y→z)≥min{μ(x),(1-k)/2} and so from (d6) that
(3.13)μ(z)≥min{μ(y),μ(y→z),1-k2}≥min{μ(x),μ(y),1-k2}.
Conversely, letμ be a fuzzy set in L satisfying (d7). Since x≤x→1 for all x∈L, we have μ(1)≥min{μ(x),(1-k)/2} by (d7). Note that x→y≤x→y for all x,y∈L. It follows from (d7) that
(3.14)μ(y)≥min{μ(x),μ(x→y),1-k2}.
Using Theorem 3.6, we conclude that μ is an (∈,∈∨qk)-fuzzy filter of L.Corollary 3.11 (see [6]).
A fuzzy setμ in L is an (∈,∈∨q)-fuzzy filter of L if and only if it satisfies the following:
(3.15)(forallx,y,z∈L)(x≤y→z⇒μ(z)≥min{μ(x),μ(y),0.5}).Proof.
It is obvious by takingk=0 in Theorem 3.10.Theorem 3.12.
For an(∈,∈∨qk)-fuzzy filter μ of L, the followings are equivalent: (1)
(forallx,y,z∈L)(μ(x→z)≥min{μ(x→(y→z)),μ(x→y),(1-k)/2}),(2)
(forallx,y∈L)(μ(x→y)≥min{μ(x→(x→y)),(1-k)/2}),(3)
(forallx,y,z∈L)(μ((x→y)→(x→z))≥min{μ(x→(y→z)),(1-k)/2}).Proof.
(1)⇒(2): Suppose that μ satisfies the condition (1). If we take z=y and y=x in (1), then
(3.16)μ(x→y)≥min{μ(x→(x→y)),μ(x→x),1-k2}=min{μ(x→(x→y)),μ(1),1-k2}=min{μ(x→(x→y)),1-k2}
for all x,y∈L by (d5).
(2)⇒(3): Assume that μ satisfies the condition (2). Note that
(3.17)x→(y→z)≤x→((x→y)→(x→z))
for all x,y,z∈L. It follows from (R4), (2), and (d4) that
(3.18)μ((x→y)→(x→z))=μ(x→((x→y)→z))≥min{μ(x→(x→((x→y)→z))),1-k2}=min{μ(x→((x→y)→(x→z))),1-k2}≥min{min{μ(x→(y→z)),1-k2},1-k2}=min{μ(x→(y→z)),1-k2}
for all x,y,z∈L.
(3)⇒(1): Suppose that μ satisfies the condition (3). Using (d6), we have
(3.19)μ(x→z)≥min{μ(x→y),μ((x→y)→(x→z)),1-k2}≥min{μ(x→y),min{μ(x→(y→z)),1-k2},1-k2}=min{μ(x→y),μ(x→(y→z)),1-k2}
for all x,y,z∈L.Corollary 3.13 (see [6]).
For an(∈,∈∨q)-fuzzy filter μ of L, the followings are equivalent: (1)
(forallx,y,z∈L)(μ(x→z)≥min{μ(x→(y→z)),μ(x→y),0.5}),(2)
(forallx,y∈L)(μ(x→y)≥min{μ(x→(x→y)),0.5}),(3)
(forallx,y,z∈L)(μ((x→y)→(x→z))≥min{μ(x→(y→z)),0.5}).Theorem 3.14.
A fuzzy setμ in L is an (∈,∈∨qk)-fuzzy filter of L if and only if it satisfies the following:
(3.20)(∀t∈(0,1-k2])(U(μ;t)≠∅⇒U(μ;t)isafilterofL).Proof.
Assume thatμ is an (∈,∈∨qk)-fuzzy filter of L. Let t∈(0,(1-k)/2] be such that U(μ;t)≠∅. Let x,y∈U(μ;t). Then μ(x)≥t and μ(y)≥t. Hence μ(x⊙y)≥min{μ(x),μ(y),(1-k)/2}≥min{t,(1-k)/2}=t by (d3), and so x⊙y∈U(μ;t). Let x,y∈L be such that x≤y. If x∈U(μ;t), then
(3.21)μ(y)≥min{μ(x),1-k2}≥min{t,1-k2}=t
by (d4). This shows that y∈U(μ;t). Therefore U(μ;t) is a filter of L.
Conversely, letμ be a fuzzy set in L satisfying (3.20). If there exist a,b∈L such that μ(a⊙b)<min{μ(a),μ(b),(1-k)/2}, then
(3.22)μ(a⊙b)<t≤min{μ(a),μ(b),1-k2}
for some t∈(0,(1-k)/2]. Thus a,b∈U(μ;t) and a⊙b∉U(μ;t), which is a contradiction. Hence μ(x⊙y)≥min{μ(x),μ(y),(1-k)/2} for all x,y∈L. Let x,y∈L be such that x≤y. If μ(y)<min{μ(x),(1-k)/2}, then
(3.23)μ(y)<r≤min{μ(x),1-k2}
for some r∈(0,(1-k)/2]. It follows that x∈U(μ;r) and y∉U(μ;r), a contradiction. Therefore μ(y)≥min{μ(x),(1-k)/2} for all x,y∈L with x≤y. Using Theorem 3.4, we conclude that μ is an (∈,∈∨qk)-fuzzy filter of L.Corollary 3.15 (see [6]).
A fuzzy setμ in L is an (∈,∈∨q)-fuzzy filter of L if and only if it satisfies the following:
(3.24)(∀t∈(0,0.5])(U(μ;t)≠∅⇒U(μ;t)isafilterofL).Theorem 3.16.
IfA is a filter of L, then a fuzzy set μ in L such that μ(x)=t1 for x∈A and μ(x)=t2 otherwise, where t1∈[(1-k)/2,1] and t2∈(0,(1-k)/2), is an (∈,∈∨qk)-fuzzy filter of L.Proof.
Note that(3.25)U(μ;r)={Aifr∈(t2,1-k2],Lifr∈(0,t2],
which is a filter of L. It follows from Theorem 3.14 that μ is an (∈,∈∨qk)-fuzzy filter of L.Corollary 3.17.
IfA is a filter of L, then a fuzzy set μ in L such that μ(x)=t1 for x∈A and μ(x)=t2 otherwise, where t1∈[0.5,1] and t2∈(0,0.5), is an (∈,∈∨q)-fuzzy filter of L.For any fuzzy setμ in L and any t∈(0,1], we consider four subsets: Q(μ;t):={x∈L∣(x,t)qμ},[μ]t:={x∈L∣(x,t)∈∨qμ},Qk(μ;t):={x∈L∣(x,t)qkμ},[μ]tk:={x∈L∣(x,t)∈∨qkμ}.It is clear that[μ]t=U(μ;t)∪Q(μ;t) and [μ]tk=U(μ;t)∪Qk(μ;t).Theorem 3.18.
Ifμ is an (∈,∈∨qk)-fuzzy filter of L, then
(3.26)(∀t∈(1-k2,1])(Qk(μ;t)≠∅⇒Qk(μ;t)isafilterofL).Proof.
Assume thatμ is an (∈,∈∨qk)-fuzzy filter of L and let t∈((1-k)/2,1] be such that Qk(μ;t)≠∅. Let x,y∈Qk(μ;t). Then (x,t)qkμ and (y,t)qkμ, that is, μ(x)+t+k>1 and μ(y)+t+k>1. Using (d3), we have
(3.27)μ(x⊙y)≥min{μ(x),μ(y),1-k2}={min{μ(x),μ(y)}ifmin{μ(x),μ(y)}<1-k2,1-k2ifmin{μ(x),μ(y)}≥1-k2>1-t-k,
that is, (x⊙y,t)qkμ. Hence x⊙y∈Qk(μ;t). Let x,y∈L be such that x≤y. If x∈Qk(μ;t), then (x,t)qkμ, that is, μ(x)+t+k>1. It follows from (d4) that
(3.28)μ(y)≥min{μ(x),1-k2}={1-k2ifμ(x)≥1-k2,μ(x)ifμ(x)<1-k2,>1-t-k,
that is, (y,t)qkμ. Hence y∈Qk(μ;t). Therefore Qk(μ;t) is a filter of L.Corollary 3.19.
Ifμ is an (∈,∈∨q)-fuzzy filter of L, then
(3.29)(∀t∈(0.5,1])(Q(μ;t)≠∅⇒Q(μ;t)isafilterofL).Proof.
It is clear by takingk=0 in Theorem 3.18.Theorem 3.20.
For any fuzzy setμ in L, the followings are equivalent: (1)
μ is an (∈,∈∨qk)-fuzzy filter of L,(2)
(forallt∈(0,1])([μ]tk≠∅⇒[μ]tk is a filter of L).Proof.
Assume thatμ is an (∈,∈∨qk)-fuzzy filter of L and let t∈(0,1] be such that [μ]tk≠∅. Let x,y∈[μ]tk. Then μ(x)≥t or μ(x)+t+k>1, and μ(y)≥t or μ(y)+t+k>1. We can consider four cases:
(3.30)μ(x)≥t,μ(y)≥t,(3.31)μ(x)≥t,μ(y)+t+k>1,(3.32)μ(x)+t+k>1,μ(y)≥t,(3.33)μ(x)+t+k>1,μ(y)+t+k>1.
For the first case, (d3) implies that
(3.34)μ(x⊙y)≥min{μ(x),μ(y),1-k2}≥min{t,1-k2}={1-k2ift>1-k2,tift≤1-k2,
and so μ(x⊙y)+t+k>(1-k)/2+(1-k)/2+k=1, that is, (x⊙y)tqkμ, or x⊙y∈U(μ;t). Therefore x⊙y∈U(μ;t)∪Qk(μ;t)=[μ]tk. For the case (3.31), assume that t>(1-k)/2. Then 1-t-k≤1-t<(1-k)/2, and so
(3.35)μ(x⊙y)≥min{μ(x),μ(y),1-k2}={min{μ(y),1-k2}>1-t-kifmin{μ(y),1-k2}≤μ(x),μ(x)≥tifmin{μ(y),1-k2}>μ(x).
Thus x⊙y∈U(μ;t)∪Qk(μ;t)=[μ]tk. Suppose that t≤(1-k)/2. Then 1-t≥(1-k)/2, which implies that
(3.36)μ(x⊙y)≥min{μ(x),μ(y),1-k2}={min{μ(x),1-k2}≥tifmin{μ(x),1-k2}≤μ(y),μ(y)>1-t-kifmin{μ(x),1-k2}>μ(y),
and thus x⊙y∈U(μ;t)∪Qk(μ;t)=[μ]tk. We have similar result for the case (3.32). For the final case, if t>(1-k)/2, then 1-t-k≤1-t<(1-k)/2. Hence
(3.37)μ(x⊙y)≥min{μ(x),μ(y),1-k2}={1-k2>1-t-kifmin{μ(x),μ(y)}≥1-k2,min{μ(x),μ(y)}>1-t-kifmin{μ(x),μ(y)}<1-k2,
and so x⊙y∈Qk(μ;t)⊆[μ]tk. If t≤(1-k)/2, then
(3.38)μ(x⊙y)≥min{μ(x),μ(y),1-k2}={1-k2≥tifmin{μ(x),μ(y)}≥1-k2,min{μ(x),μ(y)}>1-t-kifmin{μ(x),μ(y)}<1-k2,
which implies that x⊙y∈U(μ;t)∪Qk(μ;t)=[μ]tk. Let x,y∈L be such that x≤y. If x∈[μ]tk, then μ(x)≥t or μ(x)+t+k>1. Assume that μ(x)≥t. (d4) implies that
(3.39)μ(y)≥min{μ(x),1-k2}≥min{t,1-k2}={tift≤1-k2,1-k2>1-t-kift>1-k2
so that y∈U(μ;t)∪Qk(μ;t)=[μ]tk. Suppose that μ(x)+t+k>1. If t>(1-k)/2, then
(3.40)μ(y)≥min{μ(x),1-k2}={1-k2>1-t-kifμ(x)≥1-k2,μ(x)>1-t-kifμ(x)<1-k2,
and thus y∈Qk(μ;t)⊆[μ]tk. If t≤(1-k)/2, then
(3.41)μ(y)≥min{μ(x),1-k2}={1-k2≥tifμ(x)≥1-k2,μ(x)>1-t-kifμ(x)<1-k2,
which implies that y∈U(μ;t)∪Qk(μ;t)=[μ]tk. Therefore [μ]tk is a filter of L.
Conversely, letμ be a fuzzy set in L such that [μ]tk is a filter of L whenever it is nonempty for all t∈(0,1]. If there exists a∈L such that μ(1)<min{μ(a),(1-k)/2}, then μ(1)<ta≤min{μ(a),(1-k)/2} for some ta∈(0,(1-k)/2]. It follows that a∈U(μ;ta)⊆[μ]tak but 1∉U(μ;ta). Also, μ(1)+ta<2ta≤1-k, and so (1,ta)qk¯μ, that is, 1∉Qk(μ;ta). Therefore 1∉[μ]tak, a contradiction. Hence μ(1)≥min{μ(x),(1-k)/2} for all x∈L. Suppose that there exist a,b∈L such that μ(b)<min{μ(a),μ(a→b),(1-k)/2}. Then
(3.42)μ(b)<tb≤min{μ(a),μ(a→b),1-k2}
for some tb∈(0,(1-k)/2], which implies that a,a→b∈U(μ;tb)⊆[μ]tbk and so from (b2) that b∈[μ]tbk=U(μ;tb)∪Qk(μ;tb) since [μ]tbk is a filter of L. But, (3.42) implies that b∉U(μ;tb) and μ(b)+tb<2tb≤1-k, that is, b∉Qk(μ;tb). This is a contradiction, and therefore μ(y)≥min{μ(x),μ(x→y),(1-k)/2} for all x,y∈L. Using Theorem 3.6, we conclude that μ is an (∈,∈∨qk)-fuzzy filter of L.If we takek=0 in Theorem 3.20, then we have the following corollary.Corollary 3.21.
For any fuzzy setμ in L, the followings are equivalent: (1)
μ is an (∈,∈∨q)-fuzzy filter of L,(2)
(forallt∈(0,1])([μ]t≠∅⇒[μ]t is a filter of L).


## 4. Implication-Based Fuzzy 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:
(4.1)[x∈μ]=μ(x),(4.2)[Φ⋀Ψ]=min{[Φ],[Ψ]},(4.3)[Φ→Ψ]=min{1,1-[Φ]+[Ψ]},(4.4)[∀xΦ(x)]=infx∈U[Φ(x)],(4.5)⊨Φiff[Φ]=1forallvaluations.
The truth valuation rules given in (4.3) 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 (IGR) is
(4.6)IGR(a,b)={1ifa≤b,0otherwise.(b)
Gödel implication operator (IG) is
(4.7)IG(a,b)={1ifa≤b,botherwise.(c)
The contraposition of Gödel implication operator (IcG) is
(4.8)IcG(a,b)={1ifa≤b,1-aotherwise.Ying [9] introduced the concept of fuzzifying topology. We can expand his/her idea to R0-algebras, and we define a fuzzifying filter as follows.Definition 4.1.
A fuzzy subsetμ of L is called a fuzzifying filter of L if it satisfies the following conditions: (1)
for allx∈L, we have
(4.9)⊨[x∈μ]→[1∈μ];(2)
for allx,y∈R, we get
(4.10)⊨[x∈μ]⋀[x→y∈μ]→[y∈μ].Obviously, conditions (4.9) and (4.10) are equivalent to (c3) and (c4), respectively. Therefore a fuzzifying filter is an ordinary fuzzy filter.In [10], the concept of t-tautology is introduced, that is,(4.11)⊨tΦiff[Φ]≥tforallvaluations.Definition 4.2.
Letμ be a fuzzy set in L and t∈(0,1]. Then μ is called a t-implication-based fuzzy filter of L if it satisfies the following conditions: (1)
for allx∈L, we have
(4.12)⊨t[x∈μ]→[1∈μ],(2)
for allx,y∈R, we get
(4.13)⊨t[x∈μ]⋀[x→y∈μ]→[y∈μ].LetI be an implication operator. Clearly, μ is a t-implication-based fuzzy filter of L if and only if it satisfies the following: (1)
(forallx∈L)(I(μ(x),μ(1))≥t),(2)
(forallx,y∈L)(I(min{μ(x),μ(x→y)},μ(y))≥t).Theorem 4.3.
For any fuzzy setμ in L, one has the following. (1)
IfI=IGR, then μ is a 0.5-implication-based fuzzy filter of L if and only if μ is a fuzzy filter of L.(2)
IfI=IG, then μ is a ((1-k)/2)-implication-based fuzzy filter of L if and only if μ is an (∈,∈∨qk)-fuzzy filter of L.(3)
IfI=IcG, then μ is a ((1-k)/2)-implication-based fuzzy filter of L if and only if μ satisfies the following conditions:
(3.1)
(forallx∈L)(max{μ(1),(1-k)/2}≥min{μ(x),1}).(3.2)
(forallx,y∈L)(max{μ(y),(1-k)/2}≥min{μ(x),μ(x→y),1}).Proof.
(1) It is straightforward.
(2) Assume that μ is a ((1-k)/2)-implication-based fuzzy filter of L. Then (i)
(forallx∈L)(IG(μ(x),μ(1))≥(1-k)/2),(ii)
(forallx,y∈L)(IG(min{μ(x),μ(x→y)},μ(y))≥(1-k)/2).From (i), we have μ(1)≥μ(x) or μ(x)≥μ(1)≥(1-k)/2, and so μ(1)≥min{μ(x),(1-k)/2}. The second case implies that μ(y)≥min{μ(x),μ(x→y)} or min{μ(x),μ(x→y)}>μ(y)≥(1-k)/2. It follows that
(4.14)μ(y)≥min{μ(x),μ(x→y),1-k2}.
Using Theorem 3.6, we know that μ is an (∈,∈∨qk)-fuzzy filter of L.
Conversely, suppose thatμ is an (∈,∈∨qk)-fuzzy filter of L. From (d5), if min{μ(x),(1-k)/2}=μ(x), then IG(μ(x),μ(1))=1≥(1-k)/2. Otherwise, IG(μ(x),μ(1))≥(1-k)/2. From (d6), if
(4.15)min{μ(x),μ(x→y),1-k2}=min{μ(x),μ(x→y)},
then μ(y)≥min{μ(x),μ(x→y)} and so
(4.16)IG(min{μ(x),μ(x→y)},μ(y))=1≥1-k2.
If min{μ(x),μ(x→y),(1-k)/2}=(1-k)/2, then μ(y)≥(1-k)/2 and thus
(4.17)IG(min{μ(x),μ(x→y)},μ(y))≥1-k2.
Consequently, μ is a ((1-k)/2)-implication-based fuzzy filter of L.
(3) Suppose that μ satisfies (3.5) and (3.20). In (3.5), if μ(x)=1, then max{μ(1),(1-k)/2}=1 and hence IcG(μ(x),μ(1))=1≥(1-k)/2. If μ(x)<1, then
(4.18)max{μ(1),1-k2}≥μ(x).
If max{μ(1),(1-k)/2}=μ(1) in (4.18), then μ(1)≥μ(x). Hence
(4.19)IcG(μ(x),μ(1))=1≥1-k2.
If max{μ(1),(1-k)/2}=(1-k)/2 in (4.18), then μ(x)≤(1-k)/2 which implies that
(4.20)IcG(μ(x),μ(1))={1≥1-k2ifμ(1)≥μ(x),1-μ(x)≥1-k2otherwise.
In (3.20), if min{μ(x),μ(x→y),1}=1, then max{μ(y),(1-k)/2}=1 and so μ(y)=1≥min{μ(x),μ(x→y)}. Therefore
(4.21)IcG(min{μ(x),μ(x→y)},μ(y))=1≥1-k2.
If min{μ(x),μ(x→y),1}=min{μ(x),μ(x→y)}, then
(4.22)max{μ(y),1-k2}≥min{μ(x),μ(x→y)}.
Thus, if max{μ(y),(1-k)/2}=(1-k)/2 in (4.22), then μ(y)≤(1-k)/2 and
(4.23)min{μ(x),μ(x→y)}≤1-k2.
Therefore
(4.24)IcG(min{μ(x),μ(x→y)},μ(y))={1≥1-k2ifμ(y)≥min{μ(x),μ(x→y)},1-min{μ(x),μ(x→y)}≥1-k2otherwise.
If max{μ(y),(1-k)/2}=μ(y) in (4.22), then μ(y)≥min{μ(x),μ(x→y)} and so
(4.25)IcG(min{μ(x),μ(x→y)},μ(y))=1≥1-k2.
Consequently, μ is a ((1-k)/2)-implication-based fuzzy filter of L.
Conversely assume thatμ is a ((1-k)/2)-implication-based fuzzy filter of L. Then (iii)
(forallx∈L)(IcG(μ(x),μ(1))≥(1-k)/2),(iv)
(forallx,y∈L)(IcG(min{μ(x),μ(x→y)},μ(y))≥(1-k)/2).
The case (iii) implies thatIcG(μ(x),μ(1))=1, that is, μ(x)≤μ(1), or 1-μ(x)≥(1-k)/2 and so μ(x)≤(1-k)/2. It follows that
(4.26)max{μ(1),1-k2}≥μ(x)=min{μ(x),1}.
From (iv), we have
(4.27)IcG(min{μ(x),μ(x→y)},μ(y))=1,
that is, min{μ(x),μ(x→y)}≤μ(y), or 1-min{μ(x),μ(x→y)}≥(1-k)/2. Hence
(4.28)max{μ(y),1-k2}≥min{μ(x),μ(x→y)}=min{μ(x),μ(x→y),1}.
This completes the proof.Corollary 4.4.
For any fuzzy setμ in L, one has the following. (1)
IfI=IG, then μ is a 0.5-implication-based fuzzy filter of L if and only if μ is an (∈,∈∨qk)-fuzzy filter of L.(2)
IfI=IcG, then μ is a 0.5-implication-based fuzzy filter of L if and only if μ satisfies the following conditions:
(2.1)
(forallx∈L)(max{μ(1),0.5}≥min{μ(x),1}).(2.2)
(forallx,y∈L)(max{μ(y),0.5}≥min{μ(x),μ(x→y),1}).


## 5. Conclusions

We introduced the notion of an(∈,∈∨qk)-fuzzy filter which is a generalization of an (∈,∈∨q)-fuzzy filter in R0-algebras. We investigated related properties and provided characterizations of an (∈,∈∨qk)-fuzzy filter in R0-algebras. We discussed the implication-based fuzzy filters of an R0-algebra. Using our notions/results, we know that the related results in the paper [6] are induced from the notions/results in this paper.



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*Source: 918656-2010-05-09.xml*