International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 918656, 19 pages
doi:10.1155/2010/918656
Research Article

Generalizations of ( ∈ , ∈ ∨ 𝑞 ) -Fuzzy Filters in 𝑅 0 -Algebras

Young Bae Jun,1 Seok Zun Song,2 and Jianming Zhan3

1Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju 660-701, South Korea
2Department of Mathematics, Jeju National University, Jeju 690-756, South Korea
3Department of Mathematics, Hubei Institute for Nationalities, Enshi, Hubei 445000, China

Received 4 November 2009; Accepted 9 February 2010

Academic Editor: Pentti Haukkanen

Copyright © 2010 Young Bae Jun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝑅 0 -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 𝑅 0 -algebras was first introduced by Wang in [2] by providing an algebraic proof of the completeness theorem of a formal deductive system [3]. Obviously, 𝑅 0 -algebras are different from the BL-algebras. Jun and Lianzhen [4] studied filters of 𝑅 0 -algebras. Lianzhen and Kaitai [5] discussed the fuzzy set theory of filters in 𝑅 0 -algebras. As a generalization of the notion of fuzzy filters, Ma et al. [6] dealt with the notion of ( ∈ , ∈ ∨ q ) -fuzzy filters in 𝑅 0 -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 𝑅 0 -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 𝑅 0 -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 𝑅 0 -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 = m i n { 𝑥 , 𝑦 } , 𝑥 ∨ 𝑦 = m a x { 𝑥 , 𝑦 } , ¬ 𝑥 = 1 − 𝑥 , and 1 𝑥 ⟶ 𝑦 ∶ = i f 𝑥 ≤ 𝑦 , ¬ 𝑥 ∨ 𝑦 i f 𝑥 > 𝑦 . ( 2 . 1 ) Then ⋀ ( 𝐿 , , ∨ , ¬ , → ) is an 𝑅 0 -algebra which is neither a B L -algebra nor a lattice implication algebra.

An 𝑅 0 -algebra has the following useful properties.

Proposition 2.3 (see [7]). For any elements 𝑥 ,    𝑦 , and 𝑧 of an 𝑅 0 -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) ⋀ 𝑦 𝑥 ⊙ 𝑦 ≤ 𝑥 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 𝑅 0 -algebra 𝐿 is called a filter of 𝐿 if it satisfies the following two conditions: (b1) 1 ∈ 𝐴 , (b2) ( f o r a l l 𝑥 ∈ 𝐴 ) ( f o r a l l 𝑦 ∈ 𝐿 ) ( 𝑥 → 𝑦 ∈ 𝐴 ⇒ 𝑦 ∈ 𝐴 ) .

It can be easily verified that a nonempty subset 𝐴 of an 𝑅 0 -algebra 𝐿 is a filter of 𝐿 if and only if it satisfies the following conditions: (b3) ( f o r a l l 𝑥 , 𝑦 ∈ 𝐴 ) ( 𝑥 ⊙ 𝑦 ∈ 𝐴 ) , (b4) ( f o r a l l 𝑦 ∈ 𝐿 ) ( ∃ 𝑥 ∈ 𝐴 ) ( 𝑥 ≤ 𝑦 ⇒ 𝑦 ∈ 𝐴 ) .

Definition 2.4. A fuzzy set 𝜇 in an 𝑅 0 -algebra 𝐿 is called a fuzzy filter of 𝐿 if it satisfies the following: (c1) ( f o r a l l 𝑥 , 𝑦 ∈ 𝐿 ) ( 𝜇 ( 𝑥 ⊙ 𝑦 ) ≥ m i n { 𝜇 ( 𝑥 ) , 𝜇 ( 𝑦 ) } ) , (c2) 𝜇 is order-preserving, that is, ( f o r a l l 𝑥 , 𝑦 ∈ 𝐿 ) ( 𝑥 ≤ 𝑦 ⇒ 𝜇 ( 𝑥 ) ≤ 𝜇 ( 𝑦 ) ) .

Theorem 2.5. A fuzzy set 𝜇 in an 𝑅 0 -algebra 𝐿 is a fuzzy filter of 𝐿 if and only if it satisfies the following: (c3) ( f o r a l l 𝑥 ∈ 𝐿 ) ( 𝜇 ( 1 ) ≥ 𝜇 ( 𝑥 ) ) , (c4) ( f o r a l l 𝑥 , 𝑦 ∈ 𝐿 ) ( 𝜇 ( 𝑦 ) ≥ m i n { 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 form ] 𝜇 ( 𝑦 ) ∶ = 𝑡 ∈ ( 0 , 1 i f 0 𝑦 = 𝑥 , i f 𝑦 ≠ 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 ) 𝛼 𝜇 , where 𝛼 ∈ { ∈ , q , ∈ ∨ q ⋀ , ∈ q } . To say that ( 𝑥 , 𝑡 ) ∈ 𝜇 (resp. ( 𝑥 , 𝑡 ) q 𝜇 ), 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 ) ∈ ∨ q 𝜇 (resp. ⋀ ( 𝑥 , 𝑡 ) ∈ q 𝜇 ), we mean that ( 𝑥 , 𝑡 ) ∈ 𝜇 or ( 𝑥 , 𝑡 ) q 𝜇 (resp. ( 𝑥 , 𝑡 ) ∈ 𝜇 and ( 𝑥 , 𝑡 ) q 𝜇 ).

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

In what follows, 𝐿 is an 𝑅 0 -algebra and let 𝑘 denote an arbitrary element of [ 0 , 1 ) unless otherwise specified. To say that ( 𝑥 , 𝑡 ) q 𝑘 𝜇 , we mean 𝜇 ( 𝑥 ) + 𝑡 + 𝑘 > 1 . To say that ( 𝑥 , 𝑡 ) ∈ ∨ q 𝑘 𝜇 , we mean ( 𝑥 , 𝑡 ) ∈ 𝜇 or ( 𝑥 , 𝑡 ) q 𝑘 𝜇 . For 𝛼 ∈ { ∈ , ∈ ∨ q 𝑘 } , to say that ( 𝑥 , 𝑡 ) 𝛼 𝜇 , we mean ( 𝑥 , 𝑡 ) 𝛼 𝜇 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 , m i n { 𝑡 , 𝑟 } ) ∈ ∨ q 𝑘 𝜇 , (d2) ( 𝑥 , 𝑡 ) ∈ 𝜇 & 𝑥 ≤ 𝑦 ⇒ ( 𝑦 , 𝑡 ) ∈ ∨ q 𝑘 𝜇 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 𝑅 0 -algebra (see [5]), where 𝑥 ⋀ 𝑦 = m i n { 𝑥 , 𝑦 } and 𝑥 ∨ 𝑦 = m a x { 𝑥 , 𝑦 } . Define a fuzzy set 𝜇 in 𝐿 by . 𝜇 = 0 𝑎 𝑏 𝑐 1 0 . 3 0 . 3 0 . 3 0 . 8 0 . 4 5 ( 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 . 4 5 ≥ 0 . 8 = 𝜇 ( 𝑐 ) , and 𝑐 ≤ 1 and ( 𝑐 , 0 . 5 ) ∈ 𝜇 but ( 1 , 0 . 5 ) ∈ ∨ q 𝜇 .

tab1
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) ( f o r a l l 𝑥 , 𝑦 ∈ 𝐿 ) ( 𝜇 ( 𝑥 ⊙ 𝑦 ) ≥ m i n { 𝜇 ( 𝑥 ) , 𝜇 ( 𝑦 ) , ( 1 − 𝑘 ) / 2 } ) , (d4) ( f o r a l l 𝑥 , 𝑦 ∈ 𝐿 ) ( 𝑥 ≤ 𝑦 ⇒ 𝜇 ( 𝑦 ) ≥ m i n { 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 ) < m i n 𝜇 ( 𝑎 ) , 𝜇 ( 𝑏 ) , 1 − 𝑘 2 . ( 3 . 2 ) If m i n { 𝜇 ( 𝑎 ) , 𝜇 ( 𝑏 ) } < ( 1 − 𝑘 ) / 2 , then 𝜇 ( 𝑎 ⊙ 𝑏 ) < m i n { 𝜇 ( 𝑎 ) , 𝜇 ( 𝑏 ) } . Hence 𝜇 ( 𝑎 ⊙ 𝑏 ) < 𝑡 ≤ m i n { 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 < 2 𝑡 < 1 − 𝑘 , and so ( 𝑎 ⊙ 𝑏 , 𝑡 ) q 𝑘 𝜇 . Consequently ( 𝑎 ⊙ 𝑏 , 𝑡 ) ∈ ∨ q 𝑘 𝜇 , a contradiction. If m i n { 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 − 𝑘 2 < 1 − 𝑘 2 + 1 − 𝑘 2 = 1 − 𝑘 , ( 3 . 4 ) that is, ( 𝑎 ⊙ 𝑏 , ( 1 − 𝑘 ) / 2 ) q 𝑘 𝜇 . Hence ( 𝑎 ⊙ 𝑏 , ( 1 − 𝑘 ) / 2 ) ∈ ∨ q 𝑘 𝜇 , again, a contradiction. Therefore (d3) is valid. Let 𝑥 , 𝑦 ∈ 𝐿 be such that 𝑥 ≤ 𝑦 . Assume that 𝜇 ( 𝑦 ) < m i n { 𝜇 ( 𝑥 ) , ( 1 − 𝑘 ) / 2 } . Then 𝜇 ( 𝑦 ) < 𝑟 ≤ m i n 𝜇 ( 𝑥 ) , 1 − 𝑘 2 ( 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 < 2 𝑟 ≤ 1 − 𝑘 , that is, ( 𝑦 , 𝑟 ) q 𝑘 𝜇 . Thus ( 𝑦 , 𝑟 ) ∈ ∨ q 𝑘 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 ) q 𝑘 𝜇 . Thus ( 𝑦 , ( 1 − 𝑘 ) / 2 ) ∈ ∨ q 𝑘 𝜇 which is also a contradiction. Therefore 𝜇 ( 𝑦 ) ≥ m i n { 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 ) ≥ m i n 𝜇 ( 𝑥 ) , 𝜇 ( 𝑦 ) , 1 − 𝑘 2 ≥ m i n 𝑡 , 𝑟 , 1 − 𝑘 2 = ⎧ ⎪ ⎨ ⎪ ⎩ m i n { 𝑡 , 𝑟 } i f 𝑡 ≤ 1 − 𝑘 2 o r 𝑟 ≤ 1 − 𝑘 2 , 1 − 𝑘 2 i f 𝑡 > 1 − 𝑘 2 a n d 𝑟 > 1 − 𝑘 2 . ( 3 . 6 ) The case 𝜇 ( 𝑥 ⊙ 𝑦 ) ≥ m i n { 𝑡 , 𝑟 } implies that ( 𝑥 ⊙ 𝑦 , m i n { 𝑡 , 𝑟 } ) ∈ 𝜇 . From the case 𝜇 ( 𝑥 ⊙ 𝑦 ) ≥ ( 1 − 𝑘 ) / 2 , we have 𝜇 ( 𝑥 ⊙ 𝑦 ) + m i n { 𝑡 , 𝑟 } > 1 − 𝑘 2 + 1 − 𝑘 2 = 1 − 𝑘 , ( 3 . 7 ) that is, ( 𝑥 ⊙ 𝑦 , m i n { 𝑡 , 𝑟 } ) q 𝑘 𝜇 . Hence ( 𝑥 ⊙ 𝑦 , m i n { 𝑡 , 𝑟 } ) ∈ ∨ q 𝑘 𝜇 . 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 ) ≥ m i n 𝜇 ( 𝑥 ) , 1 − 𝑘 2 ≥ m i n 𝑡 , 1 − 𝑘 2 ( 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 𝑘 𝜇 . Thus ( 𝑦 , 𝑡 ) ∈ ∨ q 𝑘 𝜇 . 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) ( f o r a l l 𝑥 , 𝑦 ∈ 𝐿 ) ( 𝜇 ( 𝑥 ⊙ 𝑦 ) ≥ m i n { 𝜇 ( 𝑥 ) , 𝜇 ( 𝑦 ) , 0 . 5 } ) , (2) ( f o r a l l 𝑥 , 𝑦 ∈ 𝐿 ) ( 𝑥 ≤ 𝑦 ⇒ 𝜇 ( 𝑦 ) ≥ m i n { 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) ( f o r a l l 𝑥 ∈ 𝐿 ) ( 𝜇 ( 1 ) ≥ m i n { 𝜇 ( 𝑥 ) , ( 1 − 𝑘 ) / 2 } ) , (d6) ( f o r a l l 𝑥 , 𝑦 ∈ 𝐿 ) ( 𝜇 ( 𝑦 ) ≥ m i n { 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 ) ≥ m i n { 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 ) ≥ m i n 𝜇 ( 𝑥 ⊙ ( 𝑥 ⟶ 𝑦 ) ) , 1 − 𝑘 2 ≥ m i n 𝜇 ( 𝑥 ) , 𝜇 ( 𝑥 ⟶ 𝑦 ) , 1 − 𝑘 2 . ( 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 ) ≥ m i n 𝜇 ( 𝑥 ) , 𝜇 ( 𝑥 ⟶ 𝑦 ) , 1 − 𝑘 2 = m i n 𝜇 ( 𝑥 ) , 𝜇 ( 1 ) , 1 − 𝑘 2 = m i n 𝜇 ( 𝑥 ) , 1 − 𝑘 2 ( 3 . 1 0 ) by (d6) and (d5). Note from (a11) that 𝑥 ⟶ ( 𝑦 ⟶ ( 𝑥 ⊙ 𝑦 ) ) = ( 𝑥 ⊙ 𝑦 ) ⟶ ( 𝑥 ⊙ 𝑦 ) = 1 ( 3 . 1 1 ) for all 𝑥 , 𝑦 ∈ 𝐿 . It follows from (d5) and (d6) that 𝜇 ( 𝑥 ⊙ 𝑦 ) ≥ m i n 𝜇 ( 𝑦 ) , 𝜇 ( 𝑦 ⟶ ( 𝑥 ⊙ 𝑦 ) ) , 1 − 𝑘 2 ≥ m i n 𝜇 ( 𝑦 ) , m i n 𝜇 ( 𝑥 ) , 𝜇 ( 𝑥 ⟶ ( 𝑦 ⟶ ( 𝑥 ⊙ 𝑦 ) ) ) , 1 − 𝑘 2 , 1 − 𝑘 2 = m i n 𝜇 ( 𝑦 ) , m i n 𝜇 ( 𝑥 ) , 𝜇 ( 1 ) , 1 − 𝑘 2 , 1 − 𝑘 2 = m i n 𝜇 ( 𝑥 ) , 𝜇 ( 𝑦 ) , 1 − 𝑘 2 . ( 3 . 1 2 ) 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) ( f o r a l l 𝑥 ∈ 𝐿 ) ( 𝜇 ( 1 ) ≥ m i n { 𝜇 ( 𝑥 ) , 0 . 5 } ) , (2) ( f o r a l l 𝑥 , 𝑦 ∈ 𝐿 ) ( 𝜇 ( 𝑦 ) ≥ m i n { 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) ( f o r a l l 𝑥 , 𝑦 , 𝑧 ∈ 𝐿 ) ( 𝑥 ≤ 𝑦 → 𝑧 ⇒ 𝜇 ( 𝑧 ) ≥ m i n { 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 ) ≥ m i n { 𝜇 ( 𝑥 ) , ( 1 − 𝑘 ) / 2 } and so from (d6) that 𝜇 ( 𝑧 ) ≥ m i n 𝜇 ( 𝑦 ) , 𝜇 ( 𝑦 ⟶ 𝑧 ) , 1 − 𝑘 2 ≥ m i n 𝜇 ( 𝑥 ) , 𝜇 ( 𝑦 ) , 1 − 𝑘 2 . ( 3 . 1 3 )
Conversely, let 𝜇 be a fuzzy set in 𝐿 satisfying (d7). Since 𝑥 ≤ 𝑥 → 1 for all 𝑥 ∈ 𝐿 , we have 𝜇 ( 1 ) ≥ m i n { 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 ) ≥ m i n 𝜇 ( 𝑥 ) , 𝜇 ( 𝑥 ⟶ 𝑦 ) , 1 − 𝑘 2 . ( 3 . 1 4 ) 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: ( f o r a l l 𝑥 , 𝑦 , 𝑧 ∈ 𝐿 ) ( 𝑥 ≤ 𝑦 ⟶ 𝑧 ⟹ 𝜇 ( 𝑧 ) ≥ m i n { 𝜇 ( 𝑥 ) , 𝜇 ( 𝑦 ) , 0 . 5 } ) . ( 3 . 1 5 )

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) ( f o r a l l 𝑥 , 𝑦 , 𝑧 ∈ 𝐿 ) ( 𝜇 ( 𝑥 → 𝑧 ) ≥ m i n { 𝜇 ( 𝑥 → ( 𝑦 → 𝑧 ) ) , 𝜇 ( 𝑥 → 𝑦 ) , ( 1 − 𝑘 ) / 2 } ) , (2) ( f o r a l l 𝑥 , 𝑦 ∈ 𝐿 ) ( 𝜇 ( 𝑥 → 𝑦 ) ≥ m i n { 𝜇 ( 𝑥 → ( 𝑥 → 𝑦 ) ) , ( 1 − 𝑘 ) / 2 } ) , (3) ( f o r a l l 𝑥 , 𝑦 , 𝑧 ∈ 𝐿 ) ( 𝜇 ( ( 𝑥 → 𝑦 ) → ( 𝑥 → 𝑧 ) ) ≥ m i n { 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 ) ≥ m i n 𝜇 ( 𝑥 ⟶ ( 𝑥 ⟶ 𝑦 ) ) , 𝜇 ( 𝑥 ⟶ 𝑥 ) , 1 − 𝑘 2 = m i n 𝜇 ( 𝑥 ⟶ ( 𝑥 ⟶ 𝑦 ) ) , 𝜇 ( 1 ) , 1 − 𝑘 2 = m i n 𝜇 ( 𝑥 ⟶ ( 𝑥 ⟶ 𝑦 ) ) , 1 − 𝑘 2 ( 3 . 1 6 ) 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 . 1 7 ) for all 𝑥 , 𝑦 , 𝑧 ∈ 𝐿 . It follows from (R4), (2), and (d4) that 𝜇 ( ( 𝑥 ⟶ 𝑦 ) ⟶ ( 𝑥 ⟶ 𝑧 ) ) = 𝜇 ( 𝑥 ⟶ ( ( 𝑥 ⟶ 𝑦 ) ⟶ 𝑧 ) ) ≥ m i n 𝜇 ( 𝑥 ⟶ ( 𝑥 ⟶ ( ( 𝑥 ⟶ 𝑦 ) ⟶ 𝑧 ) ) ) , 1 − 𝑘 2 𝜇 = m i n ( 𝑥 ⟶ ( ( 𝑥 ⟶ 𝑦 ) ⟶ ( 𝑥 ⟶ 𝑧 ) ) ) , 1 − 𝑘 2 ≥ m i n m i n 𝜇 ( 𝑥 ⟶ ( 𝑦 ⟶ 𝑧 ) ) , 1 − 𝑘 2 , 1 − 𝑘 2 = m i n 𝜇 ( 𝑥 ⟶ ( 𝑦 ⟶ 𝑧 ) ) , 1 − 𝑘 2 ( 3 . 1 8 ) for all 𝑥 , 𝑦 , 𝑧 ∈ 𝐿 .
( 3 ) ⇒ ( 1 ): Suppose that 𝜇 satisfies the condition (3). Using (d6), we have 𝜇 ( 𝑥 ⟶ 𝑧 ) ≥ m i n 𝜇 ( 𝑥 ⟶ 𝑦 ) , 𝜇 ( ( 𝑥 ⟶ 𝑦 ) ⟶ ( 𝑥 ⟶ 𝑧 ) ) , 1 − 𝑘 2 ≥ m i n 𝜇 ( 𝑥 ⟶ 𝑦 ) , m i n 𝜇 ( 𝑥 ⟶ ( 𝑦 ⟶ 𝑧 ) ) , 1 − 𝑘 2 , 1 − 𝑘 2 = m i n 𝜇 ( 𝑥 ⟶ 𝑦 ) , 𝜇 ( 𝑥 ⟶ ( 𝑦 ⟶ 𝑧 ) ) , 1 − 𝑘 2 ( 3 . 1 9 ) for all 𝑥 , 𝑦 , 𝑧 ∈ 𝐿 .

Corollary 3.13 (see [6]). For an ( ∈ , ∈ ∨ q ) -fuzzy filter 𝜇 of 𝐿 , the followings are equivalent: (1) ( f o r a l l 𝑥 , 𝑦 , 𝑧 ∈ 𝐿 ) ( 𝜇 ( 𝑥 → 𝑧 ) ≥ m i n { 𝜇 ( 𝑥 → ( 𝑦 → 𝑧 ) ) , 𝜇 ( 𝑥 → 𝑦 ) , 0 . 5 } ) ,(2) ( f o r a l l 𝑥 , 𝑦 ∈ 𝐿 ) ( 𝜇 ( 𝑥 → 𝑦 ) ≥ m i n { 𝜇 ( 𝑥 → ( 𝑥 → 𝑦 ) ) , 0 . 5 } ) , (3) ( f o r a l l 𝑥 , 𝑦 , 𝑧 ∈ 𝐿 ) ( 𝜇 ( ( 𝑥 → 𝑦 ) → ( 𝑥 → 𝑧 ) ) ≥ m i n { 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 − 𝑘 2 ( 𝑈 ( 𝜇 ; 𝑡 ) ≠ ∅ ⟹ 𝑈 ( 𝜇 ; 𝑡 ) i s a fi l t e r o f 𝐿 ) . ( 3 . 2 0 )

Proof. Assume that 𝜇 is an ( ∈ , ∈ ∨ q