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

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 (∈,∈∨q_𝑘)-fuzzy filter is introduced, and its characterizations are provided. The implication-based fuzzy filters of an 𝑅₀-algebra are discussed.

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 (∈,∈∨q_𝑘)-fuzzy filters and investigate related properties. We establish characterizations of an (∈,∈∨q_𝑘)-fuzzy filter and finally consider the implication-based fuzzy filters of an 𝑅₀-algebra. The important achievement of the study with an (∈,∈∨q_𝑘)-fuzzy filter is that the notion of an (∈,∈∨q)-fuzzy filter is a special case of an (∈,∈∨q_𝑘)-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]). Let 𝐿 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.
Let 𝐿 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]). Let 𝐿=[0,1]. For any 𝑥,𝑦∈𝐿, we define 𝑥⋀𝑦=min{𝑥,𝑦},𝑥∨𝑦=max{𝑥,𝑦},¬𝑥=1−𝑥, and 1𝑥→𝑦∶=if𝑥≤𝑦,¬𝑥∨𝑦if𝑥>𝑦. (2.1) Then (𝐿,⋀,∨,¬,→) is an 𝑅₀-algebra which is neither a BL-algebra nor a lattice implication algebra.

An 𝑅₀-algebra has the following useful properties.

Proposition 2.3 (see [7]). For any elements 𝑥,  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 subset 𝐴 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 subset 𝐴 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 set 𝜇 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 set 𝜇 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 set 𝜇 in 𝐿 and 𝑡∈(0,1], the set 𝑈(𝜇;𝑡)={𝑥∈𝐿∣𝜇(𝑥)≥𝑡} (2.2) is called a level subset of 𝐿. A fuzzy set 𝜇 in a set 𝐿 of the form𝜇(𝑦)∶=𝑡∈(0,1]if0𝑦=𝑥,if𝑦≠𝑥 (2.3) 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)q_𝑘𝜇, we mean 𝜇(𝑥)+𝑡+𝑘>1. To say that (𝑥,𝑡)∈∨q_𝑘𝜇, we mean (𝑥,𝑡)∈𝜇 or (𝑥,𝑡)q_𝑘𝜇. For 𝛼∈{∈,∈∨q_𝑘}, to say that (𝑥,𝑡)𝛼𝜇, we mean (𝑥,𝑡)𝛼𝜇 does not hold.

Definition 3.1. A fuzzy set 𝜇 in 𝐿 is said to be an (∈,∈∨q_𝑘)-fuzzy filter of 𝐿 if it satisfies the following: (d1)(𝑥,𝑡)∈𝜇&(𝑦,𝑟)∈𝜇⇒(𝑥⊙𝑦,min{𝑡,𝑟})∈∨q_𝑘𝜇,(d2)(𝑥,𝑡)∈𝜇&𝑥≤𝑦⇒(𝑦,𝑡)∈∨q_𝑘𝜇 for all 𝑥,𝑦∈𝐿 and 𝑡,𝑟∈(0,1].

An (∈,∈∨q_𝑘)-fuzzy filter of 𝐿 with 𝑘=0 is called an (∈,∈∨q)-fuzzy filter of 𝐿.

Example 3.2. Let 𝐿={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 𝜇=.0𝑎𝑏𝑐10.30.30.30.80.45 (3.1) It is routine to verify that 𝜇 is an (∈,∈∨q_0.2)-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)∈∨q𝜇.

Table 1 

Hasse diagram and Cayley tables.

Theorem 3.3. Every fuzzy filter is an (∈,∈∨q_𝑘)-fuzzy filter.

Proof. It is straightforward.

Example 3.2 shows that the converse of Theorem 3.3 may not be true and shows that an (∈,∈∨q_𝑘)-fuzzy filter may not be an (∈,∈∨q)-fuzzy filter in general.

We establish characterizations of an (∈,∈∨q_𝑘)-fuzzy filter.

Theorem 3.4. A fuzzy set 𝜇 in 𝐿 is an (∈,∈∨q_𝑘)-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. Let 𝜇 be an (∈,∈∨q_𝑘)-fuzzy filter of 𝐿. Assume that (d3) is not valid. Then there exist 𝑎,𝑏∈𝐿 such that 𝜇(𝑎⊙𝑏)<min{𝜇(𝑎),𝜇(𝑏),1−𝑘2}. (3.2) If min{𝜇(𝑎),𝜇(𝑏)}<(1−𝑘)/2, then 𝜇(𝑎⊙𝑏)<min{𝜇(𝑎),𝜇(𝑏)}. Hence 𝜇(𝑎⊙𝑏)<𝑡≤min{𝜇(𝑎),𝜇(𝑏)} (3.3) 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)q_𝑘𝜇. Consequently (𝑎⊙𝑏,𝑡)∈∨q_𝑘𝜇, 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, 𝜇(𝑎⊙𝑏)+1−𝑘2<1−𝑘2+1−𝑘2=1−𝑘, (3.4) that is, (𝑎⊙𝑏,(1−𝑘)/2)q_𝑘𝜇. Hence (𝑎⊙𝑏,(1−𝑘)/2)∈∨q_𝑘𝜇, 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 𝜇(𝑦)<𝑟≤min{𝜇(𝑥),1−𝑘2} (3.5) 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)q_𝑘𝜇. Thus (𝑦,𝑟)∈∨q_𝑘𝜇, 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)q_𝑘𝜇. Thus (𝑦,(1−𝑘)/2)∈∨q_𝑘𝜇 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, let 𝜇 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 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. (3.6) 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 𝜇(𝑥⊙𝑦)+min{𝑡,𝑟}>1−𝑘2+1−𝑘2=1−𝑘, (3.7) that is, (𝑥⊙𝑦,min{𝑡,𝑟})q_𝑘𝜇. Hence (𝑥⊙𝑦,min{𝑡,𝑟})∈∨q_𝑘𝜇. Finally let 𝑥,𝑦∈𝐿 and 𝑡∈(0,1] be such that 𝑥≤𝑦 and (𝑥,𝑡)∈𝜇. Then 𝜇(𝑥)≥𝑡, and so 𝜇(𝑦)≥min{𝜇(𝑥),1−𝑘2}≥min{𝑡,1−𝑘2} (3.8) 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)q_𝑘𝜇. Thus (𝑦,𝑡)∈∨q_𝑘𝜇. Consequently, 𝜇 is an (∈,∈∨q_𝑘)-fuzzy filter of 𝐿.

If we take 𝑘=0 in Theorem 3.4, then we have the following corollary.

Corollary 3.5 (see [6]). A fuzzy set 𝜇 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 set 𝜇 in 𝐿 is an (∈,∈∨q_𝑘)-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. Let 𝜇 be an (∈,∈∨q_𝑘)-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 𝜇(𝑦)≥min{𝜇(𝑥⊙(𝑥→𝑦)),1−𝑘2}≥min{𝜇(𝑥),𝜇(𝑥→𝑦),1−𝑘2}. (3.9)
Conversely, let 𝜇 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 𝜇(𝑦)≥min{𝜇(𝑥),𝜇(𝑥→𝑦),1−𝑘2}=min{𝜇(𝑥),𝜇(1),1−𝑘2}=min{𝜇(𝑥),1−𝑘2} (3.10) by (d6) and (d5). Note from (a11) that 𝑥→(𝑦→(𝑥⊙𝑦))=(𝑥⊙𝑦)→(𝑥⊙𝑦)=1 (3.11) for all 𝑥,𝑦∈𝐿. It follows from (d5) and (d6) that 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}. (3.12) Using Theorem 3.4, we conclude that 𝜇 is an (∈,∈∨q_𝑘)-fuzzy filter of 𝐿.

Corollary 3.7 (see [6]). A fuzzy set 𝜇 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 taking 𝑘=0 in Theorem 3.6.

Corollary 3.8. If 𝜇 is an (∈,∈∨q_𝑘)-fuzzy filter of 𝐿 with 𝜇(1)<(1−𝑘)/2, then 𝜇 is a fuzzy filter.

If we take 𝑘=0 in Corollary 3.8, then we obtain the following corollary.

Corollary 3.9 (see [6]). If 𝜇 is an (∈,∈∨q)-fuzzy filter of 𝐿 with 𝜇(1)<0.5, then 𝜇 is a fuzzy filter.

Theorem 3.10. A fuzzy set 𝜇 in 𝐿 is an (∈,∈∨q_𝑘)-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 that 𝜇 is an (∈,∈∨q_𝑘)-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 𝜇(𝑧)≥min{𝜇(𝑦),𝜇(𝑦→𝑧),1−𝑘2}≥min{𝜇(𝑥),𝜇(𝑦),1−𝑘2}. (3.13)
Conversely, let 𝜇 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 𝜇(𝑦)≥min{𝜇(𝑥),𝜇(𝑥→𝑦),1−𝑘2}. (3.14) Using Theorem 3.6, we conclude that 𝜇 is an (∈,∈∨q_𝑘)-fuzzy filter of 𝐿.

Corollary 3.11 (see [6]). A fuzzy set 𝜇 in 𝐿 is an (∈,∈∨q)-fuzzy filter of 𝐿 if and only if it satisfies the following: (forall𝑥,𝑦,𝑧∈𝐿)(𝑥≤𝑦→𝑧⇒𝜇(𝑧)≥min{𝜇(𝑥),𝜇(𝑦),0.5}). (3.15)

Proof. It is obvious by taking 𝑘=0 in Theorem 3.10.

Theorem 3.12. For an (∈,∈∨q_𝑘)-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 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} (3.16) for all 𝑥,𝑦∈𝐿 by (d5).
(2)⇒(3): Assume that 𝜇 satisfies the condition (2). Note that 𝑥→(𝑦→𝑧)≤𝑥→((𝑥→𝑦)→(𝑥→𝑧)) (3.17) for all 𝑥,𝑦,𝑧∈𝐿. It follows from (R4), (2), and (d4) that 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} (3.18) for all 𝑥,𝑦,𝑧∈𝐿.
(3)⇒(1): Suppose that 𝜇 satisfies the condition (3). Using (d6), we have 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} (3.19) 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 set 𝜇 in 𝐿 is an (∈,∈∨q_𝑘)-fuzzy filter of 𝐿 if and only if it satisfies the following: (∀𝑡∈(0,1−𝑘2])(𝑈(𝜇;𝑡)≠∅⇒𝑈(𝜇;𝑡)isafilterof𝐿). (3.20)

Proof. Assume that 𝜇 is an (∈,∈∨q_𝑘)-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 𝜇(𝑦)≥min{𝜇(𝑥),1−𝑘2}≥min{𝑡,1−𝑘2}=𝑡 (3.21) by (d4). This shows that 𝑦∈𝑈(𝜇;𝑡). Therefore 𝑈(𝜇;𝑡) is a filter of 𝐿.
Conversely, let 𝜇 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 𝜇(𝑎⊙𝑏)<𝑡≤min{𝜇(𝑎),𝜇(𝑏),1−𝑘2} (3.22) 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 𝜇(𝑦)<𝑟≤min{𝜇(𝑥),1−𝑘2} (3.23) 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 (∈,∈∨q_𝑘)-fuzzy filter of 𝐿.

Corollary 3.15 (see [6]). A fuzzy set 𝜇 in 𝐿 is an (∈,∈∨q)-fuzzy filter of 𝐿 if and only if it satisfies the following: (∀𝑡∈(0,0.5])(𝑈(𝜇;𝑡)≠∅⇒𝑈(𝜇;𝑡)isafilterof𝐿). (3.24)

Theorem 3.16. If 𝐴 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 (∈,∈∨q_𝑘)-fuzzy filter of 𝐿.

Proof. Note that 𝐴𝑈(𝜇;𝑟)=if𝑟∈(𝑡₂,1−𝑘2𝐿],if𝑟∈(0,𝑡₂], (3.25) which is a filter of 𝐿. It follows from Theorem 3.14 that 𝜇 is an (∈,∈∨q_𝑘)-fuzzy filter of 𝐿.

Corollary 3.17. If 𝐴 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 set 𝜇 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},𝑄^𝑘(𝜇;𝑡)∶={𝑥∈𝐿∣(𝑥,𝑡)q_𝑘𝜇},[𝜇]^𝑘_𝑡∶={𝑥∈𝐿∣(𝑥,𝑡)∈∨q_𝑘𝜇}.

It is clear that [𝜇]_𝑡=𝑈(𝜇;𝑡)∪𝑄(𝜇;𝑡) and [𝜇]^𝑘_𝑡=𝑈(𝜇;𝑡)∪𝑄^𝑘(𝜇;𝑡).

Theorem 3.18. If 𝜇 is an (∈,∈∨q_𝑘)-fuzzy filter of 𝐿, then (∀𝑡∈(1−𝑘2,1])(𝑄^𝑘(𝜇;𝑡)≠∅⇒Q^k(𝜇;t)isafilterof𝐿). (3.26)

Proof. Assume that 𝜇 is an (∈,∈∨q_𝑘)-fuzzy filter of 𝐿 and let 𝑡∈((1−𝑘)/2,1] be such that 𝑄^𝑘(𝜇;𝑡)≠∅. Let 𝑥,𝑦∈𝑄^𝑘(𝜇;𝑡). Then (𝑥,𝑡)q_𝑘𝜇 and (𝑦,𝑡)q_𝑘𝜇, that is, 𝜇(𝑥)+𝑡+𝑘>1 and 𝜇(𝑦)+𝑡+𝑘>1. Using (d3), we have 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, (3.27) that is, (𝑥⊙𝑦,𝑡)q_𝑘𝜇. Hence 𝑥⊙𝑦∈𝑄^𝑘(𝜇;𝑡). Let 𝑥,𝑦∈𝐿 be such that 𝑥≤𝑦. If 𝑥∈𝑄^𝑘(𝜇;𝑡), then (𝑥,𝑡)q_𝑘𝜇, that is, 𝜇(𝑥)+𝑡+𝑘>1. It follows from (d4) that 𝜇(𝑦)≥min{𝜇(𝑥),1−𝑘2}=1−𝑘2if𝜇(𝑥)≥1−𝑘2,𝜇(𝑥)if𝜇(𝑥)<1−𝑘2,>1−𝑡−𝑘, (3.28) that is, (𝑦,𝑡)q_𝑘𝜇. Hence 𝑦∈𝑄^𝑘(𝜇;𝑡). Therefore 𝑄^𝑘(𝜇;𝑡) is a filter of 𝐿.

Corollary 3.19. If 𝜇 is an (∈,∈∨q)-fuzzy filter of 𝐿, then (∀𝑡∈(0.5,1])(𝑄(𝜇;𝑡)≠∅⇒Q(𝜇;t)isafilterof𝐿). (3.29)

Proof. It is clear by taking 𝑘=0 in Theorem 3.18.

Theorem 3.20. For any fuzzy set 𝜇 in 𝐿, the followings are equivalent: (1)𝜇 is an (∈,∈∨q_𝑘)-fuzzy filter of 𝐿,(2)(forall𝑡∈(0,1])([𝜇]^𝑘_𝑡≠∅⇒[𝜇]^𝑘_𝑡 is a filter of 𝐿).

Proof. Assume that 𝜇 is an (∈,∈∨q_𝑘)-fuzzy filter of 𝐿 and let 𝑡∈(0,1] be such that [𝜇]^𝑘_𝑡≠∅. Let 𝑥,𝑦∈[𝜇]^𝑘_𝑡. 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: 𝜇(𝑥)≥𝑡,𝜇(𝑦)≥𝑡,𝜇(𝑥)≥𝑡,𝜇(𝑦)+𝑡+𝑘>1,𝜇(𝑥)+𝑡+𝑘>1,𝜇(𝑦)≥𝑡,𝜇(𝑥)+𝑡+𝑘>1,𝜇(𝑦)+𝑡+𝑘>1. (3.30)(3.31)(3.32)(3.33) For the first case, (d3) implies that 𝜇(𝑥⊙𝑦)≥min{𝜇(𝑥),𝜇(𝑦),1−𝑘2}≥min{𝑡,1−𝑘2}=1−𝑘2if𝑡>1−𝑘2,𝑡if𝑡≤1−𝑘2, (3.34) and so 𝜇(𝑥⊙𝑦)+𝑡+𝑘>(1−𝑘)/2+(1−𝑘)/2+𝑘=1, that is, (𝑥⊙𝑦)_𝑡q_𝑘𝜇, or 𝑥⊙𝑦∈𝑈(𝜇;𝑡). Therefore 𝑥⊙𝑦∈𝑈(𝜇;𝑡)∪𝑄^𝑘(𝜇;𝑡)=[𝜇]^𝑘_𝑡. For the case (3.31), assume that 𝑡>(1−𝑘)/2. Then 1−𝑡−𝑘≤1−𝑡<(1−𝑘)/2, and so 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). (3.35) Thus 𝑥⊙𝑦∈𝑈(𝜇;𝑡)∪𝑄^𝑘(𝜇;𝑡)=[𝜇]^𝑘_𝑡. Suppose that 𝑡≤(1−𝑘)/2. Then 1−𝑡≥(1−𝑘)/2, which implies that 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), (3.36) and thus 𝑥⊙𝑦∈𝑈(𝜇;𝑡)∪𝑄^𝑘(𝜇;𝑡)=[𝜇]^𝑘_𝑡. 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 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, (3.37) and so 𝑥⊙𝑦∈𝑄^𝑘(𝜇;𝑡)⊆[𝜇]^𝑘_𝑡. If 𝑡≤(1−𝑘)/2, then 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, (3.38) which implies that 𝑥⊙𝑦∈𝑈(𝜇;𝑡)∪𝑄^𝑘(𝜇;𝑡)=[𝜇]^𝑘_𝑡. Let 𝑥,𝑦∈𝐿 be such that 𝑥≤𝑦. If 𝑥∈[𝜇]^𝑘_𝑡, then 𝜇(𝑥)≥𝑡 or 𝜇(𝑥)+𝑡+𝑘>1. Assume that 𝜇(𝑥)≥𝑡. (d4) implies that 𝜇(𝑦)≥min{𝜇(𝑥),1−𝑘2}≥min{𝑡,1−𝑘2}=𝑡if𝑡≤1−𝑘2,1−𝑘2>1−𝑡−𝑘if𝑡>1−𝑘2 (3.39) so that 𝑦∈𝑈(𝜇;𝑡)∪𝑄^𝑘(𝜇;𝑡)=[𝜇]^𝑘_𝑡. Suppose that 𝜇(𝑥)+𝑡+𝑘>1. If 𝑡>(1−𝑘)/2, then 𝜇(𝑦)≥min{𝜇(𝑥),1−𝑘2}=1−𝑘2>1−𝑡−𝑘if𝜇(𝑥)≥1−𝑘2,𝜇(𝑥)>1−𝑡−𝑘if𝜇(𝑥)<1−𝑘2, (3.40) and thus 𝑦∈𝑄^𝑘(𝜇;𝑡)⊆[𝜇]^𝑘_𝑡. If 𝑡≤(1−𝑘)/2, then 𝜇(𝑦)≥min{𝜇(𝑥),1−𝑘2}=1−𝑘2≥𝑡if𝜇(𝑥)≥1−𝑘2,𝜇(𝑥)>1−𝑡−𝑘if𝜇(𝑥)<1−𝑘2, (3.41) which implies that 𝑦∈𝑈(𝜇;𝑡)∪𝑄^𝑘(𝜇;𝑡)=[𝜇]^𝑘_𝑡. Therefore [𝜇]^𝑘_𝑡 is a filter of 𝐿.
Conversely, let 𝜇 be a fuzzy set in 𝐿 such that [𝜇]^𝑘_𝑡 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)<𝑡_𝑎≤min{𝜇(𝑎),(1−𝑘)/2} for some 𝑡_𝑎∈(0,(1−𝑘)/2]. It follows that 𝑎∈𝑈(𝜇;𝑡_𝑎)⊆[𝜇]^𝑘_𝑡𝑎 but 1∉𝑈(𝜇;𝑡_𝑎). Also, 𝜇(1)+𝑡_𝑎<2𝑡_𝑎≤1−𝑘, and so (1,𝑡_𝑎)q_𝑘𝜇, that is, 1∉𝑄^𝑘(𝜇;𝑡_𝑎). Therefore 1∉[𝜇]^𝑘_𝑡𝑎, 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 𝜇(𝑏)<𝑡_𝑏≤min{𝜇(𝑎),𝜇(𝑎→𝑏),1−𝑘2} (3.42) for some 𝑡_𝑏∈(0,(1−𝑘)/2], which implies that 𝑎,𝑎→𝑏∈𝑈(𝜇;𝑡_𝑏)⊆[𝜇]^𝑘_𝑡𝑏 and so from (b2) that 𝑏∈[𝜇]^𝑘_𝑡𝑏=𝑈(𝜇;𝑡_𝑏)∪𝑄^𝑘(𝜇;𝑡_𝑏) since [𝜇]^𝑘_𝑡𝑏 is a filter of 𝐿. But, (3.42) implies that 𝑏∉𝑈(𝜇;𝑡_𝑏) and 𝜇(𝑏)+𝑡_𝑏<2𝑡_𝑏≤1−𝑘, that is, 𝑏∉𝑄^𝑘(𝜇;𝑡_𝑏). 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 (∈,∈∨q_𝑘)-fuzzy filter of 𝐿.

If we take 𝑘=0 in Theorem 3.20, then we have the following corollary.

Corollary 3.21. For any fuzzy set 𝜇 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: =𝜇(𝑥),[ΦΨ]=min{[Φ],[Ψ]},[Φ→Ψ]=min{1,1−[Φ]+[Ψ]},[∀𝑥Φ(𝑥)]=inf_𝑥∈𝑈[Φ(𝑥)],⊨Φiff[Φ]=1forallvaluations. (4.1)(4.2)(4.3)(4.4)(4.5) 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 (𝐼_GR) is 𝐼_GR1(𝑎,𝑏)=if0𝑎≤𝑏,otherwise. (4.6)(b)Gödel implication operator (𝐼_G) is 𝐼_G1(𝑎,𝑏)=if𝑏𝑎≤𝑏,otherwise. (4.7)(c)The contraposition of Gödel implication operator (𝐼_cG) is 𝐼_cG1(𝑎,𝑏)=if𝑎≤𝑏,1−𝑎otherwise. (4.8)

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 subset 𝜇 of 𝐿 is called a fuzzifying filter of 𝐿 if it satisfies the following conditions: (1)for all 𝑥∈𝐿, we have ⊨[𝑥∈𝜇]→[1∈𝜇]; (4.9)(2)for all 𝑥,𝑦∈𝑅, we get ⊨[𝑥∈𝜇][𝑥→𝑦∈𝜇]→[𝑦∈𝜇]. (4.10)

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,⊨_𝑡Φiff[Φ]≥𝑡forallvaluations. (4.11)

Definition 4.2. Let 𝜇 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 all 𝑥∈𝐿, we have ⊨_𝑡[𝑥∈𝜇]→[1∈𝜇], (4.12)(2)for all 𝑥,𝑦∈𝑅, we get ⊨_𝑡[𝑥∈𝜇][𝑥→𝑦∈𝜇]→[𝑦∈𝜇]. (4.13)

Let 𝐼 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 set 𝜇 in 𝐿, one has the following. (1)If 𝐼=𝐼_GR, then 𝜇 is a 0.5-implication-based fuzzy filter of 𝐿 if and only if 𝜇 is a fuzzy filter of 𝐿.(2)If 𝐼=𝐼_G, then 𝜇 is a ((1−𝑘)/2)-implication-based fuzzy filter of 𝐿 if and only if 𝜇 is an (∈,∈∨q_𝑘)-fuzzy filter of 𝐿.(3)If 𝐼=I_cG, 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)(𝐼_G(𝜇(𝑥),𝜇(1))≥(1−𝑘)/2),(ii)(forall𝑥,𝑦∈𝐿)(𝐼_G(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 𝜇(𝑦)≥min{𝜇(𝑥),𝜇(𝑥→𝑦),1−𝑘2}. (4.14) Using Theorem 3.6, we know that 𝜇 is an (∈,∈∨q_𝑘)-fuzzy filter of 𝐿.
Conversely, suppose that 𝜇 is an (∈,∈∨q_𝑘)-fuzzy filter of 𝐿. From (d5), if min{𝜇(𝑥),(1−𝑘)/2}=𝜇(𝑥), then 𝐼_G(𝜇(𝑥),𝜇(1))=1≥(1−𝑘)/2. Otherwise, 𝐼_G(𝜇(𝑥),𝜇(1))≥(1−𝑘)/2. From (d6), if min{𝜇(𝑥),𝜇(𝑥→𝑦),1−𝑘2}=min{𝜇(𝑥),𝜇(𝑥→𝑦)}, (4.15) then 𝜇(𝑦)≥min{𝜇(𝑥),𝜇(𝑥→𝑦)} and so 𝐼_G(min{𝜇(𝑥),𝜇(𝑥→𝑦)},𝜇(𝑦))=1≥1−𝑘2. (4.16) If min{𝜇(𝑥),𝜇(𝑥→𝑦),(1−𝑘)/2}=(1−𝑘)/2, then 𝜇(𝑦)≥(1−𝑘)/2 and thus 𝐼_G(min{𝜇(𝑥),𝜇(𝑥→𝑦)},𝜇(𝑦))≥1−𝑘2. (4.17) 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 𝐼_cG(𝜇(𝑥),𝜇(1))=1≥(1−𝑘)/2. If 𝜇(𝑥)<1, then max{𝜇(1),1−𝑘2}≥𝜇(𝑥). (4.18) If max{𝜇(1),(1−𝑘)/2}=𝜇(1) in (4.18), then 𝜇(1)≥𝜇(𝑥). Hence 𝐼_cG(𝜇(𝑥),𝜇(1))=1≥1−𝑘2. (4.19) If max{𝜇(1),(1−𝑘)/2}=(1−𝑘)/2 in (4.18), then 𝜇(𝑥)≤(1−𝑘)/2 which implies that 𝐼_cG(𝜇(𝑥),𝜇(1))=1≥1−𝑘2if𝜇(1)≥𝜇(𝑥),1−𝜇(𝑥)≥1−𝑘2otherwise. (4.20) 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 𝐼_cG(min{𝜇(𝑥),𝜇(𝑥→𝑦)},𝜇(𝑦))=1≥1−𝑘2. (4.21) If min{𝜇(𝑥),𝜇(𝑥→𝑦),1}=min{𝜇(𝑥),𝜇(𝑥→𝑦)}, then max{𝜇(𝑦),1−𝑘2}≥min{𝜇(𝑥),𝜇(𝑥→𝑦)}. (4.22) Thus, if max{𝜇(𝑦),(1−𝑘)/2}=(1−𝑘)/2 in (4.22), then 𝜇(𝑦)≤(1−𝑘)/2 and min{𝜇(𝑥),𝜇(𝑥→𝑦)}≤1−𝑘2. (4.23) Therefore 𝐼_cG=(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. (4.24) If max{𝜇(𝑦),(1−𝑘)/2}=𝜇(𝑦) in (4.22), then 𝜇(𝑦)≥min{𝜇(𝑥),𝜇(𝑥→𝑦)} and so 𝐼_cG(min{𝜇(𝑥),𝜇(𝑥→𝑦)},𝜇(𝑦))=1≥1−𝑘2. (4.25) Consequently, 𝜇 is a ((1−𝑘)/2)-implication-based fuzzy filter of 𝐿.
Conversely assume that 𝜇 is a ((1−𝑘)/2)-implication-based fuzzy filter of 𝐿. Then (iii)(forall𝑥∈𝐿)(𝐼_cG(𝜇(𝑥),𝜇(1))≥(1−𝑘)/2),(iv)(forall𝑥,𝑦∈𝐿)(𝐼_cG(min{𝜇(𝑥),𝜇(𝑥→𝑦)},𝜇(𝑦))≥(1−𝑘)/2).
The case (iii) implies that 𝐼_cG(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 max{𝜇(1),1−𝑘2}≥𝜇(𝑥)=min{𝜇(𝑥),1}. (4.26) From (iv), we have 𝐼_cG(min{𝜇(𝑥),𝜇(𝑥→𝑦)},𝜇(𝑦))=1, (4.27) that is, min{𝜇(𝑥),𝜇(𝑥→𝑦)}≤𝜇(𝑦), or 1−min{𝜇(𝑥),𝜇(𝑥→𝑦)}≥(1−𝑘)/2. Hence max{𝜇(𝑦),1−𝑘2}≥min{𝜇(𝑥),𝜇(𝑥→𝑦)}=min{𝜇(𝑥),𝜇(𝑥→𝑦),1}. (4.28) This completes the proof.

Corollary 4.4. For any fuzzy set 𝜇 in 𝐿, one has the following. (1)If 𝐼=𝐼_G, then 𝜇 is a 0.5-implication-based fuzzy filter of 𝐿 if and only if 𝜇 is an (∈,∈∨q_𝑘)-fuzzy filter of 𝐿.(2)If 𝐼=𝐼_𝑐G, 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 (∈,∈∨q_𝑘)-fuzzy filter which is a generalization of an (∈,∈∨q)-fuzzy filter in 𝑅₀-algebras. We investigated related properties and provided characterizations of an (∈,∈∨q_𝑘)-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.

Acknowledgment

The authors wish to thank the anonymous reviewers for their valuable suggestions.