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Discrete Dynamics in Nature and Society
VolumeΒ 2011, Article IDΒ 980315, 19 pages
http://dx.doi.org/10.1155/2011/980315
Research Article

Fuzzy-Fated Filters of 𝑅0-Algebras

1Department of Applied Mathematics, Hanyang University, Ansan 426-791, Republic of Korea
2Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju 660-701, Republic of Korea
3Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Republic of Korea

Received 5 January 2011; Revised 18 April 2011; Accepted 30 May 2011

Academic Editor: JindeΒ Cao

Copyright Β© 2011 Jeong Soon Han 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

Fuzzy set theory of fated filters in 𝑅0-algebras is considered. A characterization of a fuzzy-fated filter is established, and conditions for a fuzzy filter to be a fuzzy-fated filter are provided. The notion of an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter is introduced. Characterizations of an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter are provided. Implication-based fuzzy-fated filters 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 β€œsacred” (resp., a β€œprofane”) form which is dominant in proof theory (resp., model theory). The role of logic in mathematics and computer science is twofoldβ€”as a tool for applications in both areasβ€”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 randomness and 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 (fated) filters of 𝑅0-algebras. Lianzhen and Kaitai [5] discussed the fuzzy set theory of filters in 𝑅0-algebras. The algebraic structure of set theories dealing with uncertainties has been studied by some authors. The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh [6]. Murali [7] proposed a definition of a fuzzy point belonging to fuzzy subset under a natural equivalence on fuzzy subset. The idea of quasicoincidence of a fuzzy point with a fuzzy set, which is mentioned in [8], played a vital role to generate some different types of fuzzy subsets. It is worth pointing out that Bhakat and Das [9, 10] initiated the concepts of (𝛼,𝛽)-fuzzy subgroups by using the β€œbelongs to” relation (∈) and β€œquasicoincident with” relation (π‘ž) between a fuzzy point and a fuzzy subgroup and introduced the concept of an (∈,βˆˆβˆ¨π‘ž)-fuzzy subgroup. In particular, an (∈,βˆˆβˆ¨π‘ž)-fuzzy subgroup is an important and useful generalization of Rosenfeld's fuzzy subgroup. As a generalization of the notion of fuzzy filters in 𝑅0-algebras, Ma et al. [11] dealt with the notion of (∈,βˆˆβˆ¨π‘ž)-fuzzy filters in 𝑅0-algebras.

In this paper, we deal with the fuzzy set theory of fated filters in 𝑅0-algebras. We provide conditions for a fuzzy filter to be a fuzzy-fated filter. We also introduce the notion of (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filters and investigate related properties. We establish a relation between an (∈,βˆˆβˆ¨π‘ž)-fuzzy filter and an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter and provide conditions for an (∈,βˆˆβˆ¨π‘ž)-fuzzy filter to be an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter. We deal with characterizations of an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter. Finally, we discuss the implication-based fuzzy-fated filters of an 𝑅0-algebra.

2. Preliminaries

Let 𝐿 be a bounded distributive lattice with order-reversing involution Β¬ and a binary operation β†’, then (𝐿,∧,∨,Β¬,β†’) is called an 𝑅0-algebra (see [2]) if it satisfies the following axioms: (R1)π‘₯→𝑦=¬𝑦→¬π‘₯, (R2)1β†’π‘₯=π‘₯, (R3)(𝑦→𝑧)∧((π‘₯→𝑦)β†’(π‘₯→𝑧))=𝑦→𝑧, (R4)π‘₯β†’(𝑦→𝑧)=𝑦→(π‘₯→𝑧), (R5)π‘₯β†’(π‘¦βˆ¨π‘§)=(π‘₯→𝑦)∨(π‘₯→𝑧), (R6)(π‘₯→𝑦)∨((π‘₯→𝑦)β†’(Β¬π‘₯βˆ¨π‘¦))=1.

Let 𝐿 be an 𝑅0-algebra. For any π‘₯,π‘¦βˆˆπΏ, we define π‘₯βŠ™π‘¦=Β¬(π‘₯→¬𝑦) and π‘₯βŠ•π‘¦=Β¬π‘₯→𝑦. It is proved that βŠ™ and βŠ• are commutative, associative, and π‘₯βŠ•π‘¦=Β¬(Β¬π‘₯βŠ™Β¬π‘¦), and (𝐿,∧,∨,βŠ™,β†’,0,1) is a residuated lattice.

For any elements π‘₯,𝑦, and 𝑧 of an 𝑅0-algebra 𝐿, we have the following properties (see [12]): (a1)π‘₯≀𝑦 if and only if π‘₯→𝑦=1,(a2)π‘₯≀𝑦→π‘₯, (a3)Β¬π‘₯=π‘₯β†’0, (a4)(π‘₯→𝑦)∨(𝑦→π‘₯)=1, (a5)π‘₯≀𝑦 implies 𝑦→𝑧≀π‘₯→𝑧,(a6)π‘₯≀𝑦 implies 𝑧→π‘₯≀𝑧→𝑦,(a7)((π‘₯→𝑦)→𝑦)→𝑦=π‘₯→𝑦, (a8)π‘₯βˆ¨π‘¦=((π‘₯→𝑦)→𝑦)∧((𝑦→π‘₯)β†’π‘₯), (a9)π‘₯βŠ™Β¬π‘₯=0 and π‘₯βŠ•Β¬π‘₯=1,(a10)π‘₯βŠ™π‘¦β‰€π‘₯βˆ§π‘¦ and π‘₯βŠ™(π‘₯→𝑦)≀π‘₯βˆ§π‘¦,(a11)(π‘₯βŠ™π‘¦)→𝑧=π‘₯β†’(𝑦→𝑧), (a12)π‘₯≀𝑦→(π‘₯βŠ™π‘¦), (a13)π‘₯βŠ™π‘¦β‰€π‘§ if and only if π‘₯≀𝑦→𝑧,(a14)π‘₯≀𝑦 implies π‘₯βŠ™π‘§β‰€π‘¦βŠ™π‘§,(a15)π‘₯→𝑦≀(𝑦→𝑧)β†’(π‘₯→𝑧), (a16)(π‘₯→𝑦)βŠ™(𝑦→𝑧)≀π‘₯→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)(βˆ€π‘₯∈𝐴)(βˆ€π‘¦βˆˆπΏ)(π‘₯β†’π‘¦βˆˆπ΄β‡’π‘¦βˆˆπ΄).

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)(βˆ€π‘₯,π‘¦βˆˆπ΄)(π‘₯βŠ™π‘¦βˆˆπ΄), (b4)(βˆ€π‘¦βˆˆπΏ)(βˆ€π‘₯∈𝐴)(π‘₯β‰€π‘¦β‡’π‘¦βˆˆπ΄).

Definition 2.1. A fuzzy subset πœ‡ of an 𝑅0-algebra 𝐿 is called a fuzzy filter of 𝐿 if it satisfies: (c1)(βˆ€π‘₯,π‘¦βˆˆπΏ)(πœ‡(π‘₯βŠ™π‘¦)β‰₯min{πœ‡(π‘₯),πœ‡(𝑦)}), (c2)πœ‡ is order preserving, that is, (βˆ€π‘₯,π‘¦βˆˆπΏ)(π‘₯β‰€π‘¦β‡’πœ‡(π‘₯)β‰€πœ‡(𝑦)).

Denote by 𝐹(𝐿) the set of all filters of 𝐿, and by ℱ𝐹(𝐿) the set of all fuzzy filters of 𝐿.

Theorem 2.2. A fuzzy subset πœ‡ of an 𝑅0-algebra 𝐿 is a fuzzy filter of 𝐿 if and only if it satisfies the following: (c3)(βˆ€π‘₯∈𝐿)(πœ‡(1)β‰₯πœ‡(π‘₯)), (c4)(βˆ€π‘₯,π‘¦βˆˆπΏ)(πœ‡(𝑦)β‰₯min{πœ‡(π‘₯→𝑦),πœ‡(π‘₯)}).

For any fuzzy subset πœ‡ of 𝐿 and π‘‘βˆˆ(0,1], the set

π‘ˆ(πœ‡;𝑑)={π‘₯βˆˆπΏβˆ£πœ‡(π‘₯)β‰₯𝑑} is called a level subset of 𝐿. It is well known that a fuzzy subset πœ‡ of 𝐿 is a fuzzy filter of 𝐿 if and only if the nonempty level subset π‘ˆ(πœ‡;𝑑),π‘‘βˆˆ(0,1], of πœ‡ is a filter of 𝐿.

A fuzzy subset πœ‡ of a set 𝐿 of the form ξ‚»]πœ‡(𝑦)∢=π‘‘βˆˆ(0,1if0𝑦=π‘₯,if𝑦≠π‘₯,(2.1) is said to be a fuzzy point with support π‘₯ and value 𝑑 and is denoted by (π‘₯,𝑑).

3. Fuzzy-Fated Filters

In what follows, 𝐿 is an 𝑅0-algebra unless otherwise specified. In [4], the notion of a fated filter of 𝐿 is introduced as follows.

A nonempty subset 𝐴 of 𝐿 is called a fated filter of 𝐿 (see [4]) if it satisfies (b1) and(βˆ€π‘₯,π‘¦βˆˆπΏ)(βˆ€π‘Žβˆˆπ΄)(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)∈𝐴⟹π‘₯∈𝐴).(3.1)

Denote by 𝐹𝐹(𝐿) the set of all fated filters of 𝐿. Note that 𝐹𝐹(𝐿) is a complete lattice under the set inclusion with the largest element 𝐿 and the least element {1}. Now, we consider the fuzzy form of a fated filter of 𝐿.

Lemma 3.1 (see [4]). A filter 𝐹 of 𝐿 is fated if and only if the following assertion is valid: (βˆ€π‘₯,𝑦,π‘§βˆˆπΏ)(π‘₯⟢(π‘¦βŸΆπ‘§)∈𝐹,π‘₯βŸΆπ‘¦βˆˆπΉβŸΉπ‘₯βŸΆπ‘§βˆˆπΉ).(3.2)

Lemma 3.2 (see [4]). A filter 𝐹 of 𝐿 is fated if and only if the following assertion is valid: (βˆ€π‘₯,π‘¦βˆˆπΏ)((π‘₯βŸΆπ‘¦)⟢π‘₯∈𝐹⟹π‘₯∈𝐹).(3.3)

Definition 3.3. A fuzzy subset πœ‡ of 𝐿 is called a fuzzy-fated filter of 𝐿 if it satisfies the following assertion: [](βˆ€π‘‘βˆˆ0,1)(π‘ˆ(πœ‡;𝑑)∈𝐹𝐹(𝐿)βˆͺ{βˆ…}).(3.4)

Denote by ℱ𝐹𝐹(𝐿) the set of all fuzzy-fated filters of 𝐿.

Example 3.4. Let 𝐿={0,π‘Ž,𝑏,𝑐,𝑑,1} be a set with Hasse diagram and Cayley tables which are given in Table 1, then (𝐿,∧,∨,Β¬,β†’,0,1) is an 𝑅0-algebra (see [5]), where π‘₯βˆ§π‘¦=min{π‘₯,𝑦} and π‘₯βˆ¨π‘¦=max{π‘₯,𝑦}. Define a fuzzy subset πœ‡ of 𝐿 by []ξ‚»πœ‡βˆΆπΏβŸΆ0,1,π‘₯⟼0.7ifπ‘₯∈{𝑐,𝑑,1},0.2otherwise.(3.5) Then ⎧βŽͺ⎨βŽͺβŽ©βˆ…π‘ˆ(πœ‡;𝑑)=if0.7<𝑑≀1,{𝑐,𝑑,1}if𝐿0.2<𝑑≀0.7,if0≀𝑑≀0.2,(3.6) which is a fated filter of 𝐿. Therefore, πœ‡ is a fuzzy-fated filter of 𝐿.

tab1
Table 1: Hasse diagram and Cayley tables.

We provide a characterization of a fuzzy-fated filter.

Theorem 3.5. For a fuzzy subset πœ‡ of 𝐿,πœ‡βˆˆβ„±πΉπΉ(𝐿) if and only if it satisfies the following conditions: (1)(βˆ€π‘₯∈𝐿)(πœ‡(1)β‰₯πœ‡(π‘₯)), (2)(βˆ€π‘₯,π‘Ž,π‘¦βˆˆπΏ)(πœ‡(π‘₯)β‰₯min{πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)),πœ‡(π‘Ž)}).

Proof. Suppose that πœ‡ is a fuzzy-fated filter of 𝐿. For any π‘₯∈𝐿, let πœ‡(π‘₯)=𝑑, then π‘₯βˆˆπ‘ˆ(πœ‡;𝑑), that is, π‘ˆ(πœ‡;𝑑)β‰ βˆ…, and so π‘ˆ(πœ‡;𝑑) is a fated filter of 𝐿. Thus, 1βˆˆπ‘ˆ(πœ‡;𝑑), and hence πœ‡(1)β‰₯𝑑=πœ‡(π‘₯) for all π‘₯∈𝐿. For any π‘₯,π‘Ž,π‘¦βˆˆπΏ, let π‘‘π‘ŽβˆΆ=min{πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),πœ‡(π‘Ž)}.(3.7) Then π‘Žβˆˆπ‘ˆ(πœ‡;π‘‘π‘Ž) and π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)βˆˆπ‘ˆ(πœ‡;π‘‘π‘Ž). Since π‘ˆ(πœ‡;π‘‘π‘Ž) is a fated filter of 𝐿, it follows from (3.1) that π‘₯βˆˆπ‘ˆ(πœ‡;π‘‘π‘Ž), so that πœ‡(π‘₯)β‰₯π‘‘π‘Ž,=min{πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),πœ‡(π‘Ž)}(3.8) for all π‘₯,π‘Ž,π‘¦βˆˆπΏ.
Conversely, let πœ‡ satisfy two conditions (1) and (2), and let π‘‘βˆˆ[0,1] be such that π‘ˆ(πœ‡;𝑑)β‰ βˆ…, then there exists π‘₯0βˆˆπ‘ˆ(πœ‡;𝑑) such that πœ‡(π‘₯0)β‰₯𝑑. Using (1), we have πœ‡(1)β‰₯πœ‡(π‘₯0)β‰₯𝑑, and so 1βˆˆπ‘ˆ(πœ‡;𝑑). Let π‘₯,π‘Ž,π‘¦βˆˆπΏ be such that π‘Žβˆˆπ‘ˆ(πœ‡;𝑑) and π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)βˆˆπ‘ˆ(πœ‡;𝑑), then πœ‡(π‘Ž)β‰₯𝑑 and πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯))β‰₯𝑑. It follows from (2) that πœ‡(π‘₯)β‰₯min{πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),πœ‡(π‘Ž)}β‰₯𝑑,(3.9) so that π‘₯βˆˆπ‘ˆ(πœ‡;𝑑). Hence, π‘ˆ(πœ‡;𝑑) is a fated filter of 𝐿, and therefore πœ‡ is a fuzzy-fated filter of 𝐿.

Theorem 3.6. For any fuzzy subset πœ‡ of 𝐿, one has ℱ𝐹𝐹(𝐿)βŠ†β„±πΉ(𝐿).

Proof. Let πœ‡ be a fuzzy-fated filter of 𝐿. Replacing π‘Ž and π‘₯ by π‘₯ and 𝑦, respectively, in Theorem 3.5(2) and using (R2), we have πœ‡(𝑦)β‰₯min{πœ‡(π‘₯⟢((π‘¦βŸΆπ‘¦)βŸΆπ‘¦)),πœ‡(π‘₯)}=min{πœ‡(π‘₯⟢(1βŸΆπ‘¦)),πœ‡(π‘₯)}=min{πœ‡(π‘₯βŸΆπ‘¦),πœ‡(π‘₯)},(3.10) for all π‘₯,π‘¦βˆˆπΏ. Using Theorem 2.2, πœ‡ is a fuzzy filter of 𝐿.

The following example shows that the converse of Theorem 3.6 may not be true.

Example 3.7. Let 𝐿={0,π‘Ž,𝑏,𝑐,1} be a set with Hasse diagram and Cayley tables which are given in Table 2, then (𝐿,∧,∨,Β¬,β†’,0,1) is an 𝑅0-algebra (see [5]), where π‘₯βˆ§π‘¦=min{π‘₯,𝑦} and π‘₯βˆ¨π‘¦=max{π‘₯,𝑦}. Define a fuzzy subset πœ‡ of 𝐿 by []ξ‚»πœ‡βˆΆπΏβŸΆ0,1,π‘₯⟼0.6ifπ‘₯∈{𝑐,1},0.2otherwise.(3.11) Then πœ‡ is a fuzzy filter of 𝐿, but it is not a fuzzy-fated filter of 𝐿 since πœ‡ΜΈ(𝑏)=0.2β‰₯0.6=min{πœ‡(π‘βŸΆ((π‘βŸΆπ‘Ž)βŸΆπ‘)),πœ‡(𝑐)}.(3.12)

tab2
Table 2: Hasse diagram and Cayley tables.

Theorem 3.8. For any fuzzy filter πœ‡ of 𝐿, the following assertions are equivalent: (1)πœ‡ is a fuzzy-fated filter of 𝐿,(2)πœ‡ satisfies the following inequality:(βˆ€π‘₯,π‘¦βˆˆπΏ)(πœ‡(π‘₯)β‰₯πœ‡((π‘₯βŸΆπ‘¦)⟢π‘₯)).(3.13)

Proof. Assume that πœ‡ is a fuzzy-fated filter of 𝐿. Putting π‘Ž=1 in Theorem 3.5(2) and using (R2) and Theorem 3.5(1), we have πœ‡(π‘₯)β‰₯min{πœ‡(1⟢((π‘₯βŸΆπ‘¦)⟢π‘₯)),πœ‡(1)}=min{πœ‡((π‘₯βŸΆπ‘¦)⟢π‘₯),πœ‡(1)}=πœ‡((π‘₯βŸΆπ‘¦)⟢π‘₯),(3.14) for all π‘₯,π‘¦βˆˆπΏ.
Conversely, suppose that πœ‡ satisfies (3.13), it follows from (c4) that πœ‡(π‘₯)β‰₯πœ‡((π‘₯βŸΆπ‘¦)⟢π‘₯)β‰₯min{πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),πœ‡(π‘Ž)},(3.15) for all π‘₯,π‘Ž,π‘¦βˆˆπΏ, so from Theorem 3.5 that πœ‡ is a fuzzy-fated filter of 𝐿.

Theorem 3.9. Let πœ‡ be a fuzzy filter of 𝐿, then πœ‡βˆˆβ„±πΉπΉ(𝐿) if and only if it satisfies ,πœ‡(π‘₯βŸΆπ‘§)β‰₯min{πœ‡(π‘₯⟢(π‘¦βŸΆπ‘§)),πœ‡(π‘₯βŸΆπ‘¦)}(3.16) for all π‘₯,𝑦,π‘§βˆˆπΏ.

Proof. Assume that πœ‡βˆˆβ„±πΉπΉ(𝐿). If π‘ˆ(πœ‡;𝑑)β‰ βˆ… for all π‘‘βˆˆ[0,1], then π‘ˆ(πœ‡;𝑑)∈𝐹𝐹(𝐿). Suppose that πœ‡(π‘₯βŸΆπ‘§)<min{πœ‡(π‘₯⟢(π‘¦βŸΆπ‘§)),πœ‡(π‘₯βŸΆπ‘¦)},(3.17) for some π‘₯,𝑦,π‘§βˆˆπΏ, then there exists π‘‘βˆˆ(0,1] such that πœ‡(π‘₯βŸΆπ‘§)<𝑑≀min{πœ‡(π‘₯⟢(π‘¦βŸΆπ‘§)),πœ‡(π‘₯βŸΆπ‘¦)}.(3.18) It follows that π‘₯β†’(𝑦→𝑧)βˆˆπ‘ˆ(πœ‡;𝑑) and π‘₯β†’π‘¦βˆˆπ‘ˆ(πœ‡;𝑑), so from Lemma 3.1 that π‘₯β†’π‘§βˆˆπ‘ˆ(πœ‡;𝑑), that is, πœ‡(π‘₯→𝑧)β‰₯𝑑. This is a contradiction, and so πœ‡(π‘₯→𝑧)β‰₯min{πœ‡(π‘₯β†’(𝑦→𝑧)),πœ‡(π‘₯→𝑦)} for all π‘₯,𝑦,π‘§βˆˆπΏ.
Conversely, let πœ‡ be a fuzzy filter of 𝐿 that satisfies (3.16). Let π‘‘βˆˆ[0,1] be such that π‘ˆ(πœ‡;𝑑)β‰ βˆ…, then π‘ˆ(πœ‡;𝑑)∈𝐹(𝐿) by Theorem 3.5 in [5]. Assume that π‘₯β†’(𝑦→𝑧)βˆˆπ‘ˆ(πœ‡;𝑑) and π‘₯β†’π‘¦βˆˆπ‘ˆ(πœ‡;𝑑) for all π‘₯,𝑦,π‘§βˆˆπΏ, then πœ‡(π‘₯β†’(𝑦→𝑧))β‰₯𝑑 and πœ‡(π‘₯→𝑦)β‰₯𝑑. Using (3.16), we have πœ‡(π‘₯βŸΆπ‘§)β‰₯min{πœ‡(π‘₯⟢(π‘¦βŸΆπ‘§)),πœ‡(π‘₯βŸΆπ‘¦)}β‰₯𝑑,(3.19) and so π‘₯β†’π‘§βˆˆπ‘ˆ(πœ‡;𝑑). Therefore, π‘ˆ(πœ‡;𝑑)∈𝐹𝐹(𝐿), and thus πœ‡ is a fuzzy-fated filter of 𝐿.

Remark 3.10. Based on Theorem 3.9 and [5, Definition 4.1], we know that the notion of a fuzzy-fated filter is equivalent to the notion of a fuzzy implicative filter.

4. Fuzzy-Fated Filters Based on Fuzzy Points

For a fuzzy point (π‘₯,𝑑) and a fuzzy subset πœ‡ of 𝐿, Pu and Liu [8] introduced the symbol (π‘₯,𝑑)π›Όπœ‡, where π›Όβˆˆ{∈,π‘ž,βˆˆβˆ¨π‘ž}. We say that (i)(π‘₯,𝑑)belong to πœ‡, denoted by (π‘₯,𝑑)βˆˆπœ‡ if πœ‡(π‘₯)β‰₯𝑑,(ii)(π‘₯,𝑑) is quasicoincident with πœ‡, denoted by (π‘₯,𝑑)π‘žπœ‡, if πœ‡(π‘₯)+𝑑>1,(iii)(π‘₯,𝑑)βˆˆβˆ¨π‘žπœ‡ if (π‘₯,𝑑)βˆˆπœ‡ or (π‘₯,𝑑)π‘žπœ‡,(iv)(π‘₯,𝑑)π›Όπœ‡ if (π‘₯,𝑑)π›Όπœ‡ does not hold for π›Όβˆˆ{∈,π‘ž,βˆˆβˆ¨π‘ž}.

Definition 4.1 (see [11]). A fuzzy subset πœ‡ of 𝐿 is said to be an (∈,βˆˆβˆ¨π‘ž)-fuzzy filter of 𝐿 if it satisfies (1)(π‘₯,𝑑)βˆˆπœ‡and(𝑦,π‘Ÿ)βˆˆπœ‡β‡’(π‘₯βŠ™π‘¦,min{𝑑,π‘Ÿ})βˆˆβˆ¨π‘žπœ‡, (2)(π‘₯,𝑑)βˆˆπœ‡andπ‘₯≀𝑦⇒(𝑦,𝑑)βˆˆβˆ¨π‘žπœ‡, for all π‘₯,π‘¦βˆˆπΏ and 𝑑,π‘Ÿβˆˆ(0,1].

Theorem 4.2 (see [11]). A fuzzy subset πœ‡ of 𝐿 is an (∈,βˆˆβˆ¨π‘ž)-fuzzy filter of 𝐿 if and only if the following conditions are valid: (1)(βˆ€π‘₯∈𝐿)(πœ‡(1)β‰₯min{πœ‡(π‘₯),0.5}), (2)(βˆ€π‘₯,π‘¦βˆˆπΏ)(πœ‡(𝑦)β‰₯min{πœ‡(π‘₯),πœ‡(π‘₯→𝑦),0.5}).

Definition 4.3. A fuzzy subset πœ‡ of 𝐿 is said to be an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿 if it satisfies (1)(π‘₯,𝑑)βˆˆπœ‡β‡’(1,𝑑)βˆˆβˆ¨π‘žπœ‡, (2)(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯),𝑑)βˆˆπœ‡and(π‘Ž,π‘Ÿ)βˆˆπœ‡β‡’(π‘₯,min{𝑑,π‘Ÿ})βˆˆβˆ¨π‘žπœ‡, for all π‘₯,π‘Ž,π‘¦βˆˆπΏ and 𝑑,π‘Ÿβˆˆ(0,1].

If a fuzzy subset πœ‡ of 𝐿 satisfies (c3) and Definition 4.3(2), then we say that πœ‡ is a strong (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿.

Example 4.4. Consider an 𝑅0-algebra 𝐿={0,π‘Ž,𝑏,𝑐,𝑑,1} which appeared in Example 3.4. Define a fuzzy subset πœ‡ of 𝐿 by .πœ‡=0π‘Žπ‘π‘π‘‘10.30.30.30.70.60.8(4.1) It is routine to verify that πœ‡ is a strong (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿. A fuzzy subset 𝜈 of 𝐿 defined by ξ‚€ξ‚πœˆ=0π‘Žπ‘π‘π‘‘10.40.40.40.80.80.7(4.2) is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿, but it is not a strong (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿.

Obviously, every strong (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter, but not converse as seen in Example 4.4.

We provided characterizations of an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter.

Theorem 4.5. A fuzzy subset πœ‡ of 𝐿 is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿 if and only if it satisfies the following inequalities: (1)(βˆ€π‘₯∈𝐿)(πœ‡(1)β‰₯min{πœ‡(π‘₯),0.5}), (2)(βˆ€π‘₯,π‘Ž,π‘¦βˆˆπΏ)(πœ‡(π‘₯)β‰₯min{πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)),πœ‡(π‘Ž),0.5}).

Proof. Let πœ‡ be an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿. Assume that there exists π‘ŽβˆˆπΏ such that πœ‡(1)<min{πœ‡(π‘Ž),0.5}, then πœ‡(1)<𝑑≀min{πœ‡(π‘Ž),0.5} for some π‘‘βˆˆ(0,0.5], and so (π‘Ž,𝑑)βˆˆπœ‡. It follows from Definition 4.3(1) that (1,𝑑)βˆˆβˆ¨π‘žπœ‡, that is, (1,𝑑)βˆˆπœ‡ or (1,𝑑)π‘žπœ‡, so that πœ‡(1)β‰₯𝑑 or πœ‡(1)+𝑑>1. This is a contradiction. Hence, πœ‡(1)β‰₯min{πœ‡(π‘₯),0.5} for all π‘₯∈𝐿. Suppose that there exist π‘Ž,𝑏,π‘βˆˆπΏ such that πœ‡(𝑏)<min{πœ‡(π‘ŽβŸΆ((π‘βŸΆπ‘)βŸΆπ‘)),πœ‡(π‘Ž),0.5},(4.3) then πœ‡(𝑏)<𝑑𝑏≀min{πœ‡(π‘Žβ†’((𝑏→𝑐)→𝑏)),πœ‡(π‘Ž),0.5} for some π‘‘π‘βˆˆ(0,0.5]. Thus (π‘Žβ†’((𝑏→𝑐)→𝑏),𝑑𝑏)βˆˆπœ‡ and (π‘Ž,𝑑𝑏)βˆˆπœ‡. Using Definition 4.3(2), we have (𝑏,𝑑𝑏)=(𝑏,min{𝑑𝑏,𝑑𝑏})βˆˆβˆ¨π‘žπœ‡, which implies that πœ‡(𝑏)β‰₯𝑑𝑏 or πœ‡(𝑏)+𝑑𝑏>1. This is a contradiction, and therefore ,πœ‡(π‘₯)β‰₯min{πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),πœ‡(π‘Ž),0.5}(4.4) for all π‘₯,π‘Ž,π‘¦βˆˆπΏ. Conversely, let πœ‡ be a fuzzy subset of 𝐿 that satisfies two conditions (1) and (2). Let π‘₯∈𝐿 and π‘‘βˆˆ(0,1] be such that (π‘₯,𝑑)βˆˆπœ‡, then πœ‡(π‘₯)β‰₯𝑑, which implies from (1) that πœ‡(1)β‰₯min{πœ‡(π‘₯),0.5}β‰₯min{𝑑,0.5}. If 𝑑≀0.5, then πœ‡(1)β‰₯𝑑, that is, (1,𝑑)βˆˆπœ‡. If 𝑑>0.5, then πœ‡(1)β‰₯0.5 and so πœ‡(1)+𝑑>0.5+0.5=1, that is, (1,𝑑)π‘žπœ‡. Hence, (1,𝑑)βˆˆβˆ¨π‘žπœ‡. Let π‘₯,π‘Ž,π‘¦βˆˆπΏ and 𝑑,π‘Ÿβˆˆ(0,1] be such that (π‘Žβ†’((π‘₯→𝑦)β†’π‘₯),𝑑)βˆˆπœ‡ and (π‘Ž,π‘Ÿ)βˆˆπœ‡, then πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯))β‰₯𝑑 and πœ‡(π‘Ž)β‰₯π‘Ÿ. It follows from (2) that πœ‡(π‘₯)β‰₯min{πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),πœ‡(π‘Ž),0.5}β‰₯min{𝑑,π‘Ÿ,0.5}.(4.5) If min{𝑑,π‘Ÿ}≀0.5, then πœ‡(π‘₯)β‰₯min{𝑑,π‘Ÿ}, which shows that (π‘₯,min{𝑑,π‘Ÿ})βˆˆπœ‡. If min{𝑑,π‘Ÿ}>0.5, then πœ‡(π‘₯)β‰₯0.5, and thus πœ‡(π‘₯)+min{𝑑,π‘Ÿ}>1, that is, (π‘₯,min{𝑑,π‘Ÿ})π‘žπœ‡. Hence, (π‘₯,min{𝑑,π‘Ÿ})βˆˆβˆ¨π‘žπœ‡. Consequently, πœ‡ is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿.

Corollary 4.6. Every strong (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter πœ‡ of 𝐿 satisfies the following inequalities: (1)(βˆ€π‘₯∈𝐿)(πœ‡(1)β‰₯min{πœ‡(π‘₯),0.5}), (2)(βˆ€π‘₯,π‘Ž,π‘¦βˆˆπΏ)(πœ‡(π‘₯)β‰₯min{πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)),πœ‡(π‘Ž),0.5}).

Theorem 4.7. A fuzzy subset πœ‡ of 𝐿 is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿 if and only if it satisfies the following assertion: ](βˆ€π‘‘βˆˆ(0,0.5)(π‘ˆ(πœ‡;𝑑)∈𝐹𝐹(𝐿)βˆͺ{βˆ…}).(4.6)

Proof. Assume that πœ‡ is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿. Let π‘‘βˆˆ(0,0.5] be such that π‘ˆ(πœ‡;𝑑)β‰ βˆ…, then there exists π‘₯βˆˆπ‘ˆ(πœ‡;𝑑), and so πœ‡(π‘₯)β‰₯𝑑. Using Theorem 4.5(1), we get πœ‡(1)β‰₯min{πœ‡(π‘₯),0.5}β‰₯min{𝑑,0.5}=𝑑,(4.7) that is, 1βˆˆπ‘ˆ(πœ‡;𝑑). Assume that π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)βˆˆπ‘ˆ(πœ‡;𝑑) for all π‘₯,π‘¦βˆˆπΏ and π‘Žβˆˆπ‘ˆ(πœ‡;𝑑), then πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯))β‰₯𝑑 and πœ‡(π‘Ž)β‰₯𝑑. It follows from Theorem 4.5 (2) that πœ‡(π‘₯)β‰₯min{πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),πœ‡(π‘Ž),0.5}β‰₯min{𝑑,0.5}=𝑑,(4.8) so that π‘₯βˆˆπ‘ˆ(πœ‡;𝑑). Therefore, π‘ˆ(πœ‡;𝑑) is a fated filter of 𝐿.
Conversely, let πœ‡ be a fuzzy subset of 𝐿 satisfying the assertion (4.6). Assume that πœ‡(1)<min{πœ‡(π‘Ž),0.5} for some π‘ŽβˆˆπΏ. Putting π‘‘π‘ŽβˆΆ=min{πœ‡(π‘Ž),0.5}, we have π‘Žβˆˆπ‘ˆ(πœ‡;π‘‘π‘Ž) and so π‘ˆ(πœ‡;π‘‘π‘Ž)β‰ βˆ…. Hence, π‘ˆ(πœ‡;π‘‘π‘Ž) is a fated filter of 𝐿 by (4.6), which implies that 1βˆˆπ‘ˆ(πœ‡;π‘‘π‘Ž). Thus, πœ‡(1)β‰₯π‘‘π‘Ž, which is a contradiction. Therefore, πœ‡(1)β‰₯min{πœ‡(π‘₯),0.5} for all π‘₯∈𝐿. Suppose that πœ‡(π‘₯)<min{πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),πœ‡(π‘Ž),0.5},(4.9) for some π‘₯,π‘Ž,π‘¦βˆˆπΏ. Taking 𝑑π‘₯∢=min{πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)),πœ‡(π‘Ž),0.5}, we get π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)βˆˆπ‘ˆ(πœ‡;𝑑π‘₯) and π‘Žβˆˆπ‘ˆ(πœ‡;𝑑π‘₯). It follows from (3.1) that π‘₯βˆˆπ‘ˆ(πœ‡;𝑑π‘₯), that is, πœ‡(π‘₯)β‰₯𝑑π‘₯. This is a contradiction. Hence, ,πœ‡(π‘₯)β‰₯min{πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),πœ‡(π‘Ž),0.5}(4.10) for all π‘₯,π‘Ž,π‘¦βˆˆπΏ. Using Theorem 4.5, we conclude that πœ‡ is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿.

Proposition 4.8. Every (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter πœ‡ of 𝐿 satisfies the following inequalities: (1)πœ‡(π‘₯→𝑧)β‰₯min{πœ‡(π‘₯β†’(𝑦→𝑧)),πœ‡(π‘₯→𝑦),0.5}, (2)πœ‡(π‘₯)β‰₯min{πœ‡((π‘₯→𝑦)β†’π‘₯),0.5}, for all π‘₯,𝑦,π‘§βˆˆπΏ.

Proof. (1) Suppose that there exist π‘Ž,𝑏,π‘βˆˆπΏ such that πœ‡(π‘ŽβŸΆπ‘)<min{πœ‡(π‘ŽβŸΆ(π‘βŸΆπ‘)),πœ‡(π‘ŽβŸΆπ‘),0.5}.(4.11) Taking π‘‘βˆΆ=min{πœ‡(π‘Žβ†’(𝑏→𝑐)),πœ‡(π‘Žβ†’π‘),0.5} implies that π‘‘βˆˆ(0,0.5],π‘Žβ†’(𝑏→𝑐)βˆˆπ‘ˆ(πœ‡;𝑑) and π‘Žβ†’π‘βˆˆπ‘ˆ(πœ‡;𝑑). Since π‘ˆ(πœ‡;𝑑)∈𝐹𝐹(𝐿) by Theorem 4.7, it follows from Lemma 3.1 that π‘Žβ†’π‘βˆˆπ‘ˆ(πœ‡;𝑑), that is, πœ‡(π‘Žβ†’π‘)β‰₯𝑑. This is a contradiction, and therefore πœ‡ satisfies (1).
(2) If πœ‡ is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿, then π‘ˆ(πœ‡;𝑑)∈𝐹𝐹(𝐿)βˆͺ{βˆ…} for all π‘‘βˆˆ(0,0.5] by Theorem 4.7. Hence, π‘ˆ(πœ‡;𝑑)∈𝐹(𝐿)βˆͺ{βˆ…} for all π‘‘βˆˆ(0,0.5]. Suppose that πœ‡(π‘₯)<𝑑≀min{πœ‡((π‘₯βŸΆπ‘¦)⟢π‘₯),0.5},(4.12) for some π‘₯,π‘¦βˆˆπΏ and π‘‘βˆˆ(0,0.5] then (π‘₯→𝑦)β†’π‘₯βˆˆπ‘ˆ(πœ‡;𝑑), which implies from Lemma 3.2 that π‘₯βˆˆπ‘ˆ(πœ‡;𝑑), that is, πœ‡(π‘₯)β‰₯𝑑. This is a contradiction. Hence, πœ‡(π‘₯)β‰₯min{πœ‡((π‘₯→𝑦)β†’π‘₯),0.5} for all π‘₯,π‘¦βˆˆπΏ.

Theorem 4.9. If 𝐹 is a fated filter of 𝐿, then a fuzzy subset πœ‡ of 𝐿 defined by []ξ‚»π‘‘πœ‡βˆΆπΏβŸΆ0,1,π‘₯⟼1ifx∈F,𝑑2otherwise,(4.13) where 𝑑1∈[0.5,1] and 𝑑2∈(0,0.5) is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿.

Proof. Note that ξ‚»πΉπ‘ˆ(πœ‡;π‘Ÿ)=ifξ€·π‘‘π‘Ÿβˆˆ2ξ€»,𝐿,0.5ifξ€·π‘Ÿβˆˆ0,𝑑2ξ€»,(4.14) which is a fated filter of 𝐿. It follows from Theorem 4.7 that πœ‡ is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿.

Theorem 4.10. Let πœ‡ be an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿. If πœ‡(1)<0.5, then πœ‡ is a fuzzy-fated filter of 𝐿.

Proof. It is straightforward.
For any fuzzy subset πœ‡ of 𝐿 and any π‘‘βˆˆ(0,1], we consider two subsets: [πœ‡]𝑄(πœ‡;𝑑)∢={π‘₯∈𝐿∣(π‘₯,𝑑)π‘žπœ‡},π‘‘βˆΆ={π‘₯∈𝐿∣(π‘₯,𝑑)βˆˆβˆ¨π‘žπœ‡}.(4.15) It is clear that [πœ‡]𝑑=π‘ˆ(πœ‡;𝑑)βˆͺ𝑄(πœ‡;𝑑).

Theorem 4.11. If πœ‡ is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿, then ](βˆ€π‘‘βˆˆ(0.5,1)(𝑄(πœ‡;𝑑)∈𝐹𝐹(𝐿)βˆͺ{βˆ…}).(4.16)

Proof. Assume that 𝑄(πœ‡;𝑑)β‰ βˆ… for all π‘‘βˆˆ(0.5,1], then there exists π‘₯βˆˆπ‘„(πœ‡;𝑑), and so πœ‡(π‘₯)+𝑑>1. Using Theorem 4.5(1), we have =ξ‚»πœ‡(1)β‰₯min{πœ‡(π‘₯),0.5}0.5ifπœ‡(π‘₯)β‰₯0.5,πœ‡(π‘₯)ifπœ‡(π‘₯)<0.5>1βˆ’π‘‘,(4.17) which implies that 1βˆˆπ‘„(πœ‡;𝑑). Assume that π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)βˆˆπ‘„(πœ‡;𝑑) and π‘Žβˆˆπ‘„(πœ‡;𝑑) for all π‘₯,π‘Ž,π‘¦βˆˆπΏ, then (π‘Žβ†’((π‘₯→𝑦)β†’π‘₯),𝑑)π‘žπœ‡ and (π‘Ž,𝑑)π‘žπœ‡, that is, πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯))>1βˆ’π‘‘ and πœ‡(π‘Ž)>1βˆ’π‘‘. Using Theorem 4.5(2), we get πœ‡(π‘₯)β‰₯min{πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),πœ‡(π‘Ž),0.5}.(4.18) Thus, if min{πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)),πœ‡(π‘Ž)}<0.5, then πœ‡(π‘₯)β‰₯min{πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),πœ‡(π‘Ž)}>1βˆ’π‘‘.(4.19) If min{πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)),πœ‡(π‘Ž)}β‰₯0.5, then πœ‡(π‘₯)β‰₯0.5>1βˆ’π‘‘. It follows that (π‘₯,𝑑)π‘žπœ‡ so that π‘₯βˆˆπ‘„(πœ‡;𝑑). Therefore, 𝑄(πœ‡;𝑑) is a fated filter of 𝐿.

Corollary 4.12. If πœ‡ is a strong (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿, then ](βˆ€π‘‘βˆˆ(0.5,1)(𝑄(πœ‡;𝑑)∈𝐹𝐹(𝐿)βˆͺ{βˆ…}).(4.20)

The converse of Corollary 4.12 is not true as shown by the following example.

Example 4.13. Consider the (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter 𝜈 of 𝐿 which is given in Example 4.4, then 𝑄(𝜈;𝑑)={𝑐,𝑑,1}if],πΏπ‘‘βˆˆ(0.5,0.6if]π‘‘βˆˆ(0.6,1(4.21) is a fated filter of 𝐿. But 𝜈 is not a strong (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿.

Theorem 4.14. For a fuzzy subset πœ‡ of 𝐿, the following assertions are equivalent: (1)πœ‡ is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿,(2)(βˆ€π‘‘βˆˆ(0,1])([πœ‡]π‘‘βˆˆπΉπΉ(𝐿)βˆͺ{βˆ…}).

Proof. Assume that πœ‡ is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿, and let π‘‘βˆˆ(0,1] be such that [πœ‡]π‘‘β‰ βˆ…, then there exists π‘₯∈[πœ‡]𝑑=π‘ˆ(πœ‡;𝑑)βˆͺ𝑄(πœ‡;𝑑), and so π‘₯βˆˆπ‘ˆ(πœ‡;𝑑) or π‘₯βˆˆπ‘„(πœ‡;𝑑). If π‘₯βˆˆπ‘ˆ(πœ‡;𝑑), then πœ‡(π‘₯)β‰₯𝑑. It follows from Theorem 4.5(1) that =ξ‚»π‘‘πœ‡(1)β‰₯min{πœ‡(π‘₯),0.5}β‰₯min{𝑑,0.5}if𝑑≀0.5,0.5>1βˆ’π‘‘if𝑑>0.5,(4.22) so that 1βˆˆπ‘ˆ(πœ‡;𝑑)βˆͺ𝑄(πœ‡;𝑑)=[πœ‡]𝑑. If π‘₯βˆˆπ‘„(πœ‡;𝑑), then (π‘₯,𝑑)π‘žπœ‡, that is, πœ‡(π‘₯)+𝑑>1. Thus, =ξ‚»πœ‡(1)β‰₯min{πœ‡(π‘₯),0.5}β‰₯min{1βˆ’π‘‘,0.5}1βˆ’π‘‘if𝑑>0.5,0.5β‰₯𝑑if𝑑≀0.5,(4.23) and so 1βˆˆπ‘„(πœ‡;𝑑)βˆͺπ‘ˆ(πœ‡;𝑑)=[πœ‡]𝑑. Let π‘₯,π‘Ž,π‘¦βˆˆπΏ be such that π‘Žβˆˆ[πœ‡]𝑑 and π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)∈[πœ‡]𝑑, then πœ‡(π‘Ž)β‰₯𝑑 or πœ‡(π‘Ž)+𝑑>1, and πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯))β‰₯𝑑 or πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯))+𝑑>1. We can consider four cases: πœ‡(π‘Ž)β‰₯π‘‘πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))β‰₯𝑑,(4.24)πœ‡(π‘Ž)β‰₯π‘‘πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))+𝑑>1,(4.25)πœ‡(π‘Ž)+𝑑>1πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))β‰₯𝑑,(4.26)πœ‡(π‘Ž)+𝑑>1πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))+𝑑>1.(4.27) For the first case, Theorem 4.5(2) implies that ξ‚»πœ‡(π‘₯)β‰₯min{πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),πœ‡(π‘Ž),0.5}β‰₯min{𝑑,0.5}=0.5if𝑑𝑑>0.5,if𝑑≀0.5,(4.28) so that π‘₯βˆˆπ‘ˆ(πœ‡;𝑑) or πœ‡(π‘₯)+𝑑>0.5+0.5=1, that is, π‘₯βˆˆπ‘„(πœ‡;𝑑). Hence, π‘₯∈[πœ‡]𝑑. Case (4.25) implies that ξ‚»πœ‡(π‘₯)β‰₯min{πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),πœ‡(π‘Ž),0.5}β‰₯min{1βˆ’π‘‘,𝑑,0.5}=1βˆ’π‘‘if𝑑𝑑>0.5,if𝑑≀0.5.(4.29) Thus, π‘₯βˆˆπ‘„(πœ‡;𝑑)βˆͺπ‘ˆ(πœ‡;𝑑)=[πœ‡]𝑑. Similarly, π‘₯∈[πœ‡]𝑑 for the case (4.26). The final case implies that ξ‚»πœ‡(π‘₯)β‰₯min{πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),πœ‡(π‘Ž),0.5}β‰₯min{1βˆ’π‘‘,0.5}=1βˆ’π‘‘if𝑑>0.5,0.5if𝑑≀0.5,(4.30) so that π‘₯βˆˆπ‘„(πœ‡;𝑑)βˆͺπ‘ˆ(πœ‡;𝑑)=[πœ‡]𝑑. Consequently, [πœ‡]𝑑 is a fuzzy-fated filter of 𝐿.
Conversely, let πœ‡ be a fuzzy subset of 𝐿 such that [πœ‡]𝑑 is a fated filter of 𝐿 whenever it is nonempty for all π‘‘βˆˆ(0,1]. If there exists π‘ŽβˆˆπΏ such that πœ‡(1)<min{πœ‡(π‘Ž),0.5}, then πœ‡(1)<π‘‘π‘Žβ‰€min{πœ‡(π‘Ž),0.5} for some π‘‘π‘Žβˆˆ(0,0.5]. It follows that π‘Žβˆˆπ‘ˆ(πœ‡;π‘‘π‘Ž) but 1βˆ‰π‘ˆ(πœ‡;π‘‘π‘Ž). Also, πœ‡(1)+π‘‘π‘Ž<2π‘‘π‘Žβ‰€1 and so 1βˆ‰π‘„(πœ‡;π‘‘π‘Ž). Hence, 1βˆ‰π‘ˆ(πœ‡;π‘‘π‘Ž)βˆͺ𝑄(πœ‡;π‘‘π‘Ž)=[πœ‡]π‘‘π‘Ž, which is a contradiction. Therefore, πœ‡(1)β‰₯min{πœ‡(π‘₯),0.5} for all π‘₯∈𝐿. Suppose that ,πœ‡(π‘₯)<min{πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),πœ‡(π‘Ž),0.5}(4.31) for some π‘₯,π‘Ž,π‘¦βˆˆπΏ. Taking π‘‘βˆΆ=min{πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)),πœ‡(π‘Ž),0.5} implies that π‘‘βˆˆ(0,0.5],π‘Žβˆˆπ‘ˆ(πœ‡;𝑑)βŠ†[πœ‡]𝑑, and π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)βˆˆπ‘ˆ(πœ‡;𝑑)βŠ†[πœ‡]𝑑. Since [πœ‡]π‘‘βˆˆπΉπΉ(𝐿), it follows that π‘₯∈[πœ‡]𝑑=π‘ˆ(πœ‡;𝑑)βˆͺ𝑄(πœ‡;𝑑). But (4.31) induces π‘₯βˆ‰π‘ˆ(πœ‡;𝑑) and πœ‡(π‘₯)+𝑑<2𝑑≀1, that is, π‘₯βˆ‰π‘„(πœ‡;𝑑). This is a contradiction, and thus πœ‡(π‘₯)β‰₯min{πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)),πœ‡(π‘Ž),0.5} for all π‘₯,π‘Ž,π‘¦βˆˆπΏ. Using Theorem 4.5, we conclude that πœ‡ is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿.

5. Implication-Based Fuzzy-Fated 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 ∧,∨,Β¬,β†’ 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 Ξ¦ is denoted by [Ξ¦]. For a universe π‘ˆ of discourse, we display the fuzzy logical and corresponding set-theoretical notations used in this paper[]π‘₯βˆˆπœ‡=πœ‡(π‘₯),(5.1)[][Ξ¦],[Ξ¨]},Φ∧Ψ=min{(5.2)[][Ξ¦]+[Ξ¨]Φ⟢Ψ=min{1,1βˆ’},(5.3)[]βˆ€π‘₯Ξ¦(π‘₯)=infπ‘₯βˆˆπ‘ˆ[],Ξ¦(π‘₯)(5.4)⊨Φifandonlyif[Ξ¦]=1forallvaluations.(5.5) The truth valuation rules given in (5.3) 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): 𝐼GRξ‚»1(π‘Ž,𝑏)=if0π‘Žβ‰€π‘,otherwise.(5.6)(b)GΓΆdel implication operator (𝐼G): 𝐼𝐺1(π‘Ž,𝑏)=ifπ‘π‘Žβ‰€π‘,otherwise.(5.7)(c)The contraposition of GΓΆdel implication operator (𝐼cG): 𝐼cGξ‚»1(π‘Ž,𝑏)=ifπ‘Žβ‰€π‘,1βˆ’π‘Žotherwise.(5.8)

Ying [13] introduced the concept of fuzzifying topology. We can expand his/her idea to 𝑅0-algebras, and we define a fuzzifying fated filter as follows.

Definition 5.1. A fuzzy subset πœ‡ of 𝐿 is called a fuzzifying fated filter of 𝐿 if it satisfies the following conditions: (1)for all π‘₯∈𝐿, we have ⊨[]⟢[],π‘₯βˆˆπœ‡1βˆˆπœ‡(5.9)(2)for all π‘₯,π‘Ž,π‘¦βˆˆπΏ, we get ⊨[]∧[]⟢[].π‘Žβˆˆπœ‡π‘ŽβŸΆ((π‘₯→𝑦)⟢π‘₯)βˆˆπœ‡π‘₯βˆˆπœ‡(5.10)

Obviously, conditions (5.9) and (5.10) are equivalent to Theorem 3.5(1) and Theorem 3.5(2), respectively. Therefore, a fuzzifying fated filter is an ordinary fuzzy-fated filter.

In [14], the concept of 𝑑-tautology is introduced, that is,βŠ¨π‘‘Ξ¦ifandonlyif[Ξ¦]β‰₯𝑑forallvaluations.(5.11)

Definition 5.2. Let πœ‡ be a fuzzy subset of 𝐿 and π‘‘βˆˆ(0,1], then πœ‡ is called a 𝑑-implication-based fuzzy-fated filter of 𝐿 if it satisfies the following conditions: (1)for all π‘₯∈𝐿, we have βŠ¨π‘‘[]⟢[],π‘₯βˆˆπœ‡1βˆˆπœ‡(5.12)(2)for all π‘₯,π‘Ž,π‘¦βˆˆπΏ, we get βŠ¨π‘‘[]∧[]⟢[].π‘Žβˆˆπœ‡π‘Žβ†’((π‘₯βŸΆπ‘¦)⟢π‘₯)βˆˆπœ‡π‘₯βˆˆπœ‡(5.13)
Let 𝐼 be an implication operator. Clearly, πœ‡ is a 𝑑-implication-based fuzzy-fated filter of 𝐿 if and only if it satisfies (1)(βˆ€π‘₯∈𝐿)(𝐼(πœ‡(π‘₯),πœ‡(1))β‰₯𝑑), (2)(βˆ€π‘₯,π‘¦βˆˆπΏ)(𝐼(min{πœ‡(π‘Ž),πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯))},πœ‡(π‘₯))β‰₯𝑑).

Theorem 5.3. For any fuzzy subset πœ‡ of 𝐿, one has the following: (1)if 𝐼=𝐼GR, then πœ‡ is a 0.5-implication-based fuzzy-fated filter of 𝐿 if and only if πœ‡ is a fuzzy-fated filter of 𝐿,(2)if 𝐼=𝐼G, then πœ‡ is a 0.5-implication-based fuzzy-fated filter of 𝐿 if and only if πœ‡ is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿,(3)if 𝐼=𝐼cG, then πœ‡ is a 0.5-implication-based fuzzy-fated filter of 𝐿 if and only if πœ‡ satisfies the following conditions:(3.1)max{πœ‡(1),0.5}β‰₯min{πœ‡(π‘₯),1}, (3.2)max{πœ‡(π‘₯),0.5}β‰₯min{πœ‡(π‘Ž),πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)),1}
for all π‘₯,π‘Ž,π‘¦βˆˆπΏ.

Proof. (1) It is Straightforward.
(2) Assume that πœ‡ is a 0.5-implication-based fuzzy-fated filter of 𝐿, then(i)(βˆ€π‘₯∈𝐿)(𝐼G(πœ‡(π‘₯),πœ‡(1))β‰₯0.5),(ii)(βˆ€π‘₯,π‘¦βˆˆπΏ)(𝐼G(min{πœ‡(π‘Ž),πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯))},πœ‡(π‘₯))β‰₯0.5). From (i), we have πœ‡(1)β‰₯πœ‡(π‘₯) or πœ‡(π‘₯)β‰₯πœ‡(1)β‰₯0.5, and so πœ‡(1)β‰₯min{πœ‡(π‘₯),0.5} for all π‘₯∈𝐿. The second case implies that πœ‡(π‘₯)β‰₯min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))}(5.14) or min{πœ‡(π‘Ž),πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯))}>πœ‡(π‘₯)β‰₯0.5. It follows that πœ‡(π‘₯)β‰₯min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),0.5},(5.15) for all π‘₯,π‘Ž,π‘¦βˆˆπΏ. Using Theorem 4.5, we know that πœ‡ is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿.
Conversely, suppose that πœ‡ is an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter of 𝐿. From Theorem 4.5(1), if min{πœ‡(π‘₯),0.5}=πœ‡(π‘₯), then 𝐼G(πœ‡(π‘₯),πœ‡(1))=1β‰₯0.5. Otherwise, 𝐼G(πœ‡(π‘₯),πœ‡(1))β‰₯0.5. From Theorem 4.5(2), if ,min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),0.5}=min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))}(5.16) then πœ‡(π‘₯)β‰₯min{πœ‡(π‘Ž),πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯))}, and so 𝐼G(min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))},πœ‡(π‘₯))=1β‰₯0.5.(5.17) If min{πœ‡(π‘Ž),πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)),0.5}=0.5, then πœ‡(π‘₯)β‰₯0.5, and thus 𝐼G(min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))},πœ‡(π‘₯))β‰₯0.5.(5.18) Consequently, πœ‡ is a 0.5-implication-based fuzzy-fated filter of 𝐿.
(3) Suppose that πœ‡ satisfies (3.1) and (3.2). In (3.1), if πœ‡(π‘₯)=1, then max{πœ‡(1),0.5}=1, and hence 𝐼cG(πœ‡(π‘₯),πœ‡(1))=1β‰₯0.5. If πœ‡(π‘₯)<1, then max{πœ‡(1),0.5}β‰₯πœ‡(π‘₯).(5.19) If max{πœ‡(1),0.5}=πœ‡(1) in (5.19), then πœ‡(1)β‰₯πœ‡(π‘₯). Hence, 𝐼cG(πœ‡(π‘₯),πœ‡(1))=1β‰₯0.5.(5.20) If max{πœ‡(1),0.5}=0.5 in (5.19), then πœ‡(π‘₯)≀0.5 which implies that 𝐼cGξ‚»(πœ‡(π‘₯),πœ‡(1))=1β‰₯0.5ifπœ‡(1)β‰₯πœ‡(π‘₯),1βˆ’πœ‡(π‘₯)β‰₯0.5otherwise.(5.21) In (3.2), if min{πœ‡(π‘Ž),πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯)),1}=1, then max{πœ‡(π‘₯),0.5}=1,(5.22) and so πœ‡(π‘₯)=1β‰₯min{πœ‡(π‘Ž),πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯))}. Therefore, 𝐼cG(min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))},πœ‡(π‘₯))=1β‰₯0.5.(5.23) If min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),1}=min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))},(5.24) then max{πœ‡(π‘₯),0.5}β‰₯min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯→𝑦)⟢π‘₯))}.(5.25) Thus, if max{πœ‡(π‘₯),0.5}=0.5 in (5.25), then πœ‡(π‘₯)≀0.5 and min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))}≀0.5.(5.26) Therefore, 𝐼cG,(min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))},πœ‡(π‘₯))=1β‰₯0.5(5.27) whenever πœ‡(π‘₯)β‰₯min{πœ‡(π‘Ž),πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯))}, 𝐼cG(min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))},πœ‡(π‘₯))=1βˆ’min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))}β‰₯0.5,(5.28) whenever πœ‡(π‘₯)<min{πœ‡(π‘Ž),πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯))}. Now, if max{πœ‡(π‘₯),0.5}=πœ‡(π‘₯)(5.29) in (5.25), then πœ‡(π‘₯)β‰₯min{πœ‡(π‘Ž),πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯))}, and so 𝐼cG(min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))},πœ‡(π‘₯))=1β‰₯0.5.(5.30)
Consequently, πœ‡ is a 0.5-implication-based fuzzy-fated filter of 𝐿.
Conversely, assume that πœ‡ is a 0.5-implication-based fuzzy-fated filter of 𝐿, then (iii)𝐼cG(πœ‡(π‘₯),πœ‡(1))β‰₯0.5, (iv)𝐼cG(min{πœ‡(π‘Ž),πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯))},πœ‡(π‘₯))β‰₯0.5, for all π‘₯,π‘Ž,π‘¦βˆˆπΏ. The case (iii) implies that 𝐼cG(πœ‡(π‘₯),πœ‡(1))=1, that is, πœ‡(π‘₯)β‰€πœ‡(1), or 1βˆ’πœ‡(π‘₯)β‰₯0.5 and so πœ‡(π‘₯)≀0.5. It follows that max{πœ‡(1),0.5}β‰₯πœ‡(π‘₯)=min{πœ‡(π‘₯),1}.(5.31) From (iv), we have 𝐼cG(min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))},πœ‡(π‘₯))=1,(5.32) that is, min{πœ‡(π‘Ž),πœ‡(π‘Žβ†’((π‘₯→𝑦)β†’π‘₯))}β‰€πœ‡(π‘₯), or 1βˆ’min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))}β‰₯0.5.(5.33) Hence, max{πœ‡(π‘₯),0.5}β‰₯min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯))}=min{πœ‡(π‘Ž),πœ‡(π‘ŽβŸΆ((π‘₯βŸΆπ‘¦)⟢π‘₯)),1},(5.34) for all π‘₯,𝑦,π‘§βˆˆπΏ. This completes the proof.

6. Conclusion

Using the β€œbelongs to” relation (∈) and quasicoincidence with relation (π‘ž) between fuzzy points and fuzzy sets, we introduced the notion of an (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter, this is a generalization of a fuzzy implicative filter. In fuzzy logic, one can see that various implication operators have been defined. We used Gaines-Rescher implication operator (𝐼GR), GΓΆdel implication operator (𝐼G), and the contraposition of GΓΆdel implication operator (𝐼cG) to study 𝑑-implication-based fuzzy-fated filters.

There are also other situations concerning the relations between this kind of results, another type of structures (e.g., (∈,βˆˆβˆ¨π‘ž)-fuzzy-fated filter), and (fuzzy) soft and rough set theory. How to deal with these situations will be one of our future topics. We will also try to study the intuitionistic fuzzy version of several type of filters in 𝑅0-algebras related to the intuitionistic β€œbelongs to” relation (∈) and intuitionistic quasicoincidence with relation (π‘ž) between intuitionistic fuzzy points and intuitionistic fuzzy sets.

Acknowledgments

The authors wish to thank the anonymous reviewers for their valuable comments and suggestions for improving this paper.

References

  1. L. A. Zadeh, β€œToward a generalized theory of uncertainty (GTU)—an outline,” Information Sciences, vol. 172, no. 1-2, pp. 1–40, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  2. G. J. Wang, Non-Classical Mathematical Logic and Approximate Reasoning, Science Press, Beijing, China, 2000. View at Zentralblatt MATH
  3. G.-J. Wang, β€œOn the logic foundation of fuzzy reasoning,” Information Sciences, vol. 117, no. 1-2, pp. 47–88, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  4. Y. B. Jun and L. Lianzhen, β€œFilters of R0-algebras,” International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 93249, 9 pages, 2006. View at Google Scholar Β· View at Zentralblatt MATH
  5. L. Lianzhen and L. Kaitai, β€œFuzzy implicative and Boolean filters of R0-algebras,” Information Sciences, vol. 171, no. 1–3, pp. 61–71, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  6. L. A. Zadeh, β€œFuzzy sets,” Information and Control, vol. 8, pp. 338–353, 1965. View at Google Scholar Β· View at Zentralblatt MATH
  7. V. Murali, β€œFuzzy points of equivalent fuzzy subsets,” Information Sciences, vol. 158, pp. 277–288, 2004. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  8. P. M. Pu and Y. M. Liu, β€œFuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergence,” Journal of Mathematical Analysis and Applications, vol. 76, no. 2, pp. 571–599, 1980. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  9. S. K. Bhakat and P. Das, β€œOn the definition of a fuzzy subgroup,” Fuzzy Sets and Systems, vol. 51, no. 2, pp. 235–241, 1992. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  10. S. K. Bhakat and P. Das, β€œ(,q)-fuzzy subgroup,” Fuzzy Sets and Systems, vol. 80, no. 3, pp. 359–368, 1996. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  11. X. Ma, J. Zhan, and Y. B. Jun, β€œOn (,q)-fuzzy filters of R0-algebras,” Mathematical Logic Quarterly, vol. 55, no. 5, pp. 493–508, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  12. D. W. Pei and G. J. Wang, β€œThe completeness and application of formal systems 𝔏,” Science in China E, vol. 32, no. 1, pp. 56–64, 2002. View at Google Scholar Β· View at Zentralblatt MATH
  13. M. S. Ying, β€œA new approach for fuzzy topology (I),” Fuzzy Sets and Systems., vol. 39, no. 3, pp. 303–321, 1991. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  14. M. S. Ying, β€œOn standard models of fuzzy modal logics,” Fuzzy Sets and Systems, vol. 26, no. 3, pp. 357–363, 1988. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet