Fuzzy-Fated Filters of 𝑅₀-Algebras

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 (∈,∈∨𝑞)-fuzzy-fated filter is introduced. Characterizations of an (∈,∈∨𝑞)-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 “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 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 “belongs to” relation (∈) and “quasicoincident with” relation (𝑞) between a fuzzy point and a fuzzy subgroup and introduced the concept of an (∈,∈∨𝑞)-fuzzy subgroup. In particular, an (∈,∈∨𝑞)-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 (∈,∈∨𝑞)-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 (∈,∈∨𝑞)-fuzzy-fated filters and investigate related properties. We establish a relation between an (∈,∈∨𝑞)-fuzzy filter and an (∈,∈∨𝑞)-fuzzy-fated filter and provide conditions for an (∈,∈∨𝑞)-fuzzy filter to be an (∈,∈∨𝑞)-fuzzy-fated filter. We deal with characterizations of an (∈,∈∨𝑞)-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 ¬ and a binary operation →, then (𝐿,∧,∨,¬,→) is called an 𝑅₀-algebra (see [2]) 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.

For any elements 𝑥,𝑦, and 𝑧 of an 𝑅₀-algebra 𝐿, we have the following properties (see [12]): (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)(∀𝑥∈𝐴)(∀𝑦∈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)(∀𝑥,𝑦∈𝐴)(𝑥⊙𝑦∈𝐴), (b4)(∀𝑦∈𝐿)(∀𝑥∈𝐴)(𝑥≤𝑦⇒𝑦∈𝐴).

Definition 2.1. A fuzzy subset 𝜇 of an 𝑅₀-algebra 𝐿 is called a fuzzy filter of 𝐿 if it satisfies: (c1)(∀𝑥,𝑦∈𝐿)(𝜇(𝑥⊙𝑦)≥min{𝜇(𝑥),𝜇(𝑦)}), (c2)𝜇 is order preserving, that is, (∀𝑥,𝑦∈𝐿)(𝑥≤𝑦⇒𝜇(𝑥)≤𝜇(𝑦)).

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

Theorem 2.2. A fuzzy subset 𝜇 of an 𝑅₀-algebra 𝐿 is a fuzzy filter of 𝐿 if and only if it satisfies the following: (c3)(∀𝑥∈𝐿)(𝜇(1)≥𝜇(𝑥)), (c4)(∀𝑥,𝑦∈𝐿)(𝜇(𝑦)≥min{𝜇(𝑥→𝑦),𝜇(𝑥)}).

For any fuzzy subset 𝜇 of 𝐿 and 𝑡∈(0,1], the set

𝑈(𝜇;𝑡)={𝑥∈𝐿∣𝜇(𝑥)≥𝑡} is called a level subset of 𝐿. It is well known that a fuzzy subset 𝜇 of 𝐿 is a fuzzy filter of 𝐿 if and only if the nonempty level subset 𝑈(𝜇;𝑡),𝑡∈(0,1], of 𝜇 is a filter of 𝐿.

A fuzzy subset 𝜇 of a set 𝐿 of the form 𝜇(𝑦)∶=𝑡∈(0,1]0if𝑦=𝑥,if𝑦≠𝑥,(2.1) is said to be a fuzzy point with support 𝑥 and value 𝑡 and is denoted by (𝑥,𝑡).

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(∀𝑥,𝑦∈𝐿)(∀𝑎∈𝐴)(𝑎⟶((𝑥⟶𝑦)⟶𝑥)∈𝐴⟹𝑥∈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: (∀𝑥,𝑦,𝑧∈𝐿)(𝑥⟶(𝑦⟶𝑧)∈𝐹,𝑥⟶𝑦∈𝐹⟹𝑥⟶𝑧∈𝐹).(3.2)

Lemma 3.2 (see [4]). A filter 𝐹 of 𝐿 is fated if and only if the following assertion is valid: (∀𝑥,𝑦∈𝐿)((𝑥⟶𝑦)⟶𝑥∈𝐹⟹𝑥∈𝐹).(3.3)

Definition 3.3. A fuzzy subset 𝜇 of 𝐿 is called a fuzzy-fated filter of 𝐿 if it satisfies the following assertion: (∀𝑡∈[0,1])(𝑈(𝜇;𝑡)∈𝐹𝐹(𝐿)∪{∅}).(3.4)

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

Example 3.4. Let 𝐿={0,𝑎,𝑏,𝑐,𝑑,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 subset 𝜇 of 𝐿 by 𝜇∶𝐿⟶[0,1],𝑥⟼0.70.2if𝑥∈{𝑐,𝑑,1},otherwise.(3.5) Then 𝑈(𝜇;𝑡)=⎧⎪⎨⎪⎩∅{𝑐,𝑑,1}if0.7<𝑡≤1,𝐿if0.2<𝑡≤0.7,if0≤𝑡≤0.2,(3.6) which is a fated filter of 𝐿. Therefore, 𝜇 is a fuzzy-fated filter of 𝐿.

Table 1 

Hasse diagram and Cayley tables.

We provide a characterization of a fuzzy-fated filter.

Theorem 3.5. For a fuzzy subset 𝜇 of 𝐿,𝜇∈ℱ_𝐹𝐹(𝐿) if and only if it satisfies the following conditions: (1)(∀𝑥∈𝐿)(𝜇(1)≥𝜇(𝑥)), (2)(∀𝑥,𝑎,𝑦∈𝐿)(𝜇(𝑥)≥min{𝜇(𝑎→((𝑥→𝑦)→𝑥)),𝜇(𝑎)}).

Proof. Suppose that 𝜇 is a fuzzy-fated filter of 𝐿. For any 𝑥∈𝐿, let 𝜇(𝑥)=𝑡, then 𝑥∈𝑈(𝜇;𝑡), that is, 𝑈(𝜇;𝑡)≠∅, and so 𝑈(𝜇;𝑡) is a fated filter of 𝐿. Thus, 1∈𝑈(𝜇;𝑡), and hence 𝜇(1)≥𝑡=𝜇(𝑥) for all 𝑥∈𝐿. For any 𝑥,𝑎,𝑦∈𝐿, let 𝑡_𝑎∶=min{𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥)),𝜇(𝑎)}.(3.7) Then 𝑎∈𝑈(𝜇;𝑡_𝑎) and 𝑎→((𝑥→𝑦)→𝑥)∈𝑈(𝜇;𝑡_𝑎). Since 𝑈(𝜇;𝑡_𝑎) is a fated filter of 𝐿, it follows from (3.1) that 𝑥∈𝑈(𝜇;𝑡_𝑎), so that 𝜇(𝑥)≥𝑡_𝑎=min{𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥)),𝜇(𝑎)},(3.8) for all 𝑥,𝑎,𝑦∈𝐿.
Conversely, let 𝜇 satisfy two conditions (1) and (2), and let 𝑡∈[0,1] be such that 𝑈(𝜇;𝑡)≠∅, then there exists 𝑥₀∈𝑈(𝜇;𝑡) such that 𝜇(𝑥₀)≥𝑡. Using (1), we have 𝜇(1)≥𝜇(𝑥₀)≥𝑡, and so 1∈𝑈(𝜇;𝑡). Let 𝑥,𝑎,𝑦∈𝐿 be such that 𝑎∈𝑈(𝜇;𝑡) and 𝑎→((𝑥→𝑦)→𝑥)∈𝑈(𝜇;𝑡), then 𝜇(𝑎)≥𝑡 and 𝜇(𝑎→((𝑥→𝑦)→𝑥))≥𝑡. It follows from (2) that 𝜇(𝑥)≥min{𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥)),𝜇(𝑎)}≥𝑡,(3.9) so that 𝑥∈𝑈(𝜇;𝑡). Hence, 𝑈(𝜇;𝑡) is a fated filter of 𝐿, and therefore 𝜇 is a fuzzy-fated filter of 𝐿.

Theorem 3.6. For any fuzzy subset 𝜇 of 𝐿, one has ℱ_𝐹𝐹(𝐿)⊆ℱ_𝐹(𝐿).

Proof. Let 𝜇 be a fuzzy-fated filter of 𝐿. Replacing 𝑎 and 𝑥 by 𝑥 and 𝑦, respectively, in Theorem 3.5(2) and using (R2), we have 𝜇(𝑦)≥==mmmiiinnn{{{𝜇𝜇𝜇(((𝑥𝑥𝑥⟶⟶⟶𝑦(()1(,𝑦⟶𝜇⟶(𝑥)𝑦}𝑦)),),⟶𝜇(𝑥)𝑦})),𝜇(𝑥)}(3.10) for all 𝑥,𝑦∈𝐿. Using Theorem 2.2, 𝜇 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,𝑎,𝑏,𝑐,1} be a set with Hasse diagram and Cayley tables which are given in Table 2, 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 subset 𝜇 of 𝐿 by 𝜇∶𝐿⟶[0,1],𝑥⟼0.60.2if𝑥∈{𝑐,1},otherwise.(3.11) Then 𝜇 is a fuzzy filter of 𝐿, but it is not a fuzzy-fated filter of 𝐿 since 𝜇(𝑏)=0.2≱0.6=min{𝜇(𝑐⟶((𝑏⟶𝑎)⟶𝑏)),𝜇(𝑐)}.(3.12)

Table 2 

Hasse diagram and Cayley tables.

Theorem 3.8. For any fuzzy filter 𝜇 of 𝐿, the following assertions are equivalent: (1)𝜇 is a fuzzy-fated filter of 𝐿,(2)𝜇 satisfies the following inequality:(∀𝑥,𝑦∈𝐿)(𝜇(𝑥)≥𝜇((𝑥⟶𝑦)⟶𝑥)).(3.13)

Proof. Assume that 𝜇 is a fuzzy-fated filter of 𝐿. Putting 𝑎=1 in Theorem 3.5(2) and using (R2) and Theorem 3.5(1), we have 𝜇(𝑥)≥==m𝜇m(iin(n𝑥{{⟶𝜇𝜇(((1𝑥𝑦⟶)⟶⟶(𝑦()𝑥𝑥⟶)⟶,𝑥𝑦)),⟶𝜇(1)𝑥})),𝜇(1)}(3.14) for all 𝑥,𝑦∈𝐿.
Conversely, suppose that 𝜇 satisfies (3.13), it follows from (c4) that 𝜇(𝑥)≥𝜇((𝑥⟶𝑦)⟶𝑥)≥min{𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥)),𝜇(𝑎)},(3.15) for all 𝑥,𝑎,𝑦∈𝐿, so from Theorem 3.5 that 𝜇 is a fuzzy-fated filter of 𝐿.

Theorem 3.9. Let 𝜇 be a fuzzy filter of 𝐿, then 𝜇∈ℱ_𝐹𝐹(𝐿) if and only if it satisfies 𝜇(𝑥⟶𝑧)≥min{𝜇(𝑥⟶(𝑦⟶𝑧)),𝜇(𝑥⟶𝑦)},(3.16) for all 𝑥,𝑦,𝑧∈𝐿.

Proof. Assume that 𝜇∈ℱ_𝐹𝐹(𝐿). If 𝑈(𝜇;𝑡)≠∅ for all 𝑡∈[0,1], then 𝑈(𝜇;𝑡)∈𝐹𝐹(𝐿). Suppose that 𝜇(𝑥⟶𝑧)<min{𝜇(𝑥⟶(𝑦⟶𝑧)),𝜇(𝑥⟶𝑦)},(3.17) for some 𝑥,𝑦,𝑧∈𝐿, then there exists 𝑡∈(0,1] such that 𝜇(𝑥⟶𝑧)<𝑡≤min{𝜇(𝑥⟶(𝑦⟶𝑧)),𝜇(𝑥⟶𝑦)}.(3.18) It follows that 𝑥→(𝑦→𝑧)∈𝑈(𝜇;𝑡) and 𝑥→𝑦∈𝑈(𝜇;𝑡), so from Lemma 3.1 that 𝑥→𝑧∈𝑈(𝜇;𝑡), that is, 𝜇(𝑥→𝑧)≥𝑡. This is a contradiction, and so 𝜇(𝑥→𝑧)≥min{𝜇(𝑥→(𝑦→𝑧)),𝜇(𝑥→𝑦)} for all 𝑥,𝑦,𝑧∈𝐿.
Conversely, let 𝜇 be a fuzzy filter of 𝐿 that satisfies (3.16). Let 𝑡∈[0,1] be such that 𝑈(𝜇;𝑡)≠∅, then 𝑈(𝜇;𝑡)∈𝐹(𝐿) by Theorem 3.5 in [5]. Assume that 𝑥→(𝑦→𝑧)∈𝑈(𝜇;𝑡) and 𝑥→𝑦∈𝑈(𝜇;𝑡) for all 𝑥,𝑦,𝑧∈𝐿, then 𝜇(𝑥→(𝑦→𝑧))≥𝑡 and 𝜇(𝑥→𝑦)≥𝑡. Using (3.16), we have 𝜇(𝑥⟶𝑧)≥min{𝜇(𝑥⟶(𝑦⟶𝑧)),𝜇(𝑥⟶𝑦)}≥𝑡,(3.19) and so 𝑥→𝑧∈𝑈(𝜇;𝑡). Therefore, 𝑈(𝜇;𝑡)∈𝐹𝐹(𝐿), and thus 𝜇 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 (𝑥,𝑡) and a fuzzy subset 𝜇 of 𝐿, Pu and Liu [8] introduced the symbol (𝑥,𝑡)𝛼𝜇, where 𝛼∈{∈,𝑞,∈∨𝑞}. We say that (i)(𝑥,𝑡)belong to 𝜇, denoted by (𝑥,𝑡)∈𝜇 if 𝜇(𝑥)≥𝑡,(ii)(𝑥,𝑡) is quasicoincident with 𝜇, denoted by (𝑥,𝑡)𝑞𝜇, if 𝜇(𝑥)+𝑡>1,(iii)(𝑥,𝑡)∈∨𝑞𝜇 if (𝑥,𝑡)∈𝜇 or (𝑥,𝑡)𝑞𝜇,(iv)(𝑥,𝑡)𝛼𝜇 if (𝑥,𝑡)𝛼𝜇 does not hold for 𝛼∈{∈,𝑞,∈∨𝑞}.

Definition 4.1 (see [11]). A fuzzy subset 𝜇 of 𝐿 is said to be an (∈,∈∨𝑞)-fuzzy filter of 𝐿 if it satisfies (1)(𝑥,𝑡)∈𝜇and(𝑦,𝑟)∈𝜇⇒(𝑥⊙𝑦,min{𝑡,𝑟})∈∨𝑞𝜇, (2)(𝑥,𝑡)∈𝜇and𝑥≤𝑦⇒(𝑦,𝑡)∈∨𝑞𝜇, for all 𝑥,𝑦∈𝐿 and 𝑡,𝑟∈(0,1].

Theorem 4.2 (see [11]). A fuzzy subset 𝜇 of 𝐿 is an (∈,∈∨𝑞)-fuzzy filter of 𝐿 if and only if the following conditions are valid: (1)(∀𝑥∈𝐿)(𝜇(1)≥min{𝜇(𝑥),0.5}), (2)(∀𝑥,𝑦∈𝐿)(𝜇(𝑦)≥min{𝜇(𝑥),𝜇(𝑥→𝑦),0.5}).

Definition 4.3. A fuzzy subset 𝜇 of 𝐿 is said to be an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿 if it satisfies (1)(𝑥,𝑡)∈𝜇⇒(1,𝑡)∈∨𝑞𝜇, (2)(𝑎→((𝑥→𝑦)→𝑥),𝑡)∈𝜇and(𝑎,𝑟)∈𝜇⇒(𝑥,min{𝑡,𝑟})∈∨𝑞𝜇, for all 𝑥,𝑎,𝑦∈𝐿 and 𝑡,𝑟∈(0,1].

If a fuzzy subset 𝜇 of 𝐿 satisfies (c3) and Definition 4.3(2), then we say that 𝜇 is a strong (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿.

Example 4.4. Consider an 𝑅₀-algebra 𝐿={0,𝑎,𝑏,𝑐,𝑑,1} which appeared in Example 3.4. Define a fuzzy subset 𝜇 of 𝐿 by 𝜇=00.30𝑎.30𝑏.30𝑐.70𝑑.601.8.(4.1) It is routine to verify that 𝜇 is a strong (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿. A fuzzy subset 𝜈 of 𝐿 defined by 𝜈=00.40𝑎.40𝑏.40𝑐.80𝑑.801.7(4.2) is an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿, but it is not a strong (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿.

Obviously, every strong (∈,∈∨𝑞)-fuzzy-fated filter is an (∈,∈∨𝑞)-fuzzy-fated filter, but not converse as seen in Example 4.4.

We provided characterizations of an (∈,∈∨𝑞)-fuzzy-fated filter.

Theorem 4.5. A fuzzy subset 𝜇 of 𝐿 is an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿 if and only if it satisfies the following inequalities: (1)(∀𝑥∈𝐿)(𝜇(1)≥min{𝜇(𝑥),0.5}), (2)(∀𝑥,𝑎,𝑦∈𝐿)(𝜇(𝑥)≥min{𝜇(𝑎→((𝑥→𝑦)→𝑥)),𝜇(𝑎),0.5}).

Proof. Let 𝜇 be an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿. Assume that there exists 𝑎∈𝐿 such that 𝜇(1)<min{𝜇(𝑎),0.5}, then 𝜇(1)<𝑡≤min{𝜇(𝑎),0.5} for some 𝑡∈(0,0.5], and so (𝑎,𝑡)∈𝜇. It follows from Definition 4.3(1) that (1,𝑡)∈∨𝑞𝜇, that is, (1,𝑡)∈𝜇 or (1,𝑡)𝑞𝜇, so that 𝜇(1)≥𝑡 or 𝜇(1)+𝑡>1. This is a contradiction. Hence, 𝜇(1)≥min{𝜇(𝑥),0.5} for all 𝑥∈𝐿. Suppose that there exist 𝑎,𝑏,𝑐∈𝐿 such that 𝜇(𝑏)<min{𝜇(𝑎⟶((𝑏⟶𝑐)⟶𝑏)),𝜇(𝑎),0.5},(4.3) then 𝜇(𝑏)<𝑡_𝑏≤min{𝜇(𝑎→((𝑏→𝑐)→𝑏)),𝜇(𝑎),0.5} for some 𝑡_𝑏∈(0,0.5]. Thus (𝑎→((𝑏→𝑐)→𝑏),𝑡_𝑏)∈𝜇 and (𝑎,𝑡_𝑏)∈𝜇. Using Definition 4.3(2), we have (𝑏,𝑡_𝑏)=(𝑏,min{𝑡_𝑏,𝑡_𝑏})∈∨𝑞𝜇, which implies that 𝜇(𝑏)≥𝑡_𝑏 or 𝜇(𝑏)+𝑡_𝑏>1. This is a contradiction, and therefore 𝜇(𝑥)≥min{𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥)),𝜇(𝑎),0.5},(4.4) for all 𝑥,𝑎,𝑦∈𝐿. Conversely, let 𝜇 be a fuzzy subset of 𝐿 that satisfies two conditions (1) and (2). Let 𝑥∈𝐿 and 𝑡∈(0,1] be such that (𝑥,𝑡)∈𝜇, then 𝜇(𝑥)≥𝑡, which implies from (1) that 𝜇(1)≥min{𝜇(𝑥),0.5}≥min{𝑡,0.5}. If 𝑡≤0.5, then 𝜇(1)≥𝑡, that is, (1,𝑡)∈𝜇. If 𝑡>0.5, then 𝜇(1)≥0.5 and so 𝜇(1)+𝑡>0.5+0.5=1, that is, (1,𝑡)𝑞𝜇. Hence, (1,𝑡)∈∨𝑞𝜇. Let 𝑥,𝑎,𝑦∈𝐿 and 𝑡,𝑟∈(0,1] be such that (𝑎→((𝑥→𝑦)→𝑥),𝑡)∈𝜇 and (𝑎,𝑟)∈𝜇, then 𝜇(𝑎→((𝑥→𝑦)→𝑥))≥𝑡 and 𝜇(𝑎)≥𝑟. It follows from (2) that 𝜇(𝑥)≥≥mmiinn{{𝜇𝑡,𝑟(𝑎,0⟶.5}.((𝑥⟶𝑦)⟶𝑥)),𝜇(𝑎),0.5}(4.5) If min{𝑡,𝑟}≤0.5, then 𝜇(𝑥)≥min{𝑡,𝑟}, which shows that (𝑥,min{𝑡,𝑟})∈𝜇. If min{𝑡,𝑟}>0.5, then 𝜇(𝑥)≥0.5, and thus 𝜇(𝑥)+min{𝑡,𝑟}>1, that is, (𝑥,min{𝑡,𝑟})𝑞𝜇. Hence, (𝑥,min{𝑡,𝑟})∈∨𝑞𝜇. Consequently, 𝜇 is an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿.

Corollary 4.6. Every strong (∈,∈∨𝑞)-fuzzy-fated filter 𝜇 of 𝐿 satisfies the following inequalities: (1)(∀𝑥∈𝐿)(𝜇(1)≥min{𝜇(𝑥),0.5}), (2)(∀𝑥,𝑎,𝑦∈𝐿)(𝜇(𝑥)≥min{𝜇(𝑎→((𝑥→𝑦)→𝑥)),𝜇(𝑎),0.5}).

Theorem 4.7. A fuzzy subset 𝜇 of 𝐿 is an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿 if and only if it satisfies the following assertion: (∀𝑡∈(0,0.5])(𝑈(𝜇;𝑡)∈𝐹𝐹(𝐿)∪{∅}).(4.6)

Proof. Assume that 𝜇 is an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿. Let 𝑡∈(0,0.5] be such that 𝑈(𝜇;𝑡)≠∅, then there exists 𝑥∈𝑈(𝜇;𝑡), and so 𝜇(𝑥)≥𝑡. Using Theorem 4.5(1), we get 𝜇(1)≥min{𝜇(𝑥),0.5}≥min{𝑡,0.5}=𝑡,(4.7) that is, 1∈𝑈(𝜇;𝑡). Assume that 𝑎→((𝑥→𝑦)→𝑥)∈𝑈(𝜇;𝑡) for all 𝑥,𝑦∈𝐿 and 𝑎∈𝑈(𝜇;𝑡), then 𝜇(𝑎→((𝑥→𝑦)→𝑥))≥𝑡 and 𝜇(𝑎)≥𝑡. It follows from Theorem 4.5 (2) that 𝜇(𝑥)≥≥mmiinn{{𝜇𝑡,0(𝑎.5⟶}=𝑡(,(𝑥⟶𝑦)⟶𝑥)),𝜇(𝑎),0.5}(4.8) so that 𝑥∈𝑈(𝜇;𝑡). Therefore, 𝑈(𝜇;𝑡) is a fated filter of 𝐿.
Conversely, let 𝜇 be a fuzzy subset of 𝐿 satisfying the assertion (4.6). Assume that 𝜇(1)<min{𝜇(𝑎),0.5} for some 𝑎∈𝐿. Putting 𝑡_𝑎∶=min{𝜇(𝑎),0.5}, we have 𝑎∈𝑈(𝜇;𝑡_𝑎) and so 𝑈(𝜇;𝑡_𝑎)≠∅. Hence, 𝑈(𝜇;𝑡_𝑎) is a fated filter of 𝐿 by (4.6), which implies that 1∈𝑈(𝜇;𝑡_𝑎). Thus, 𝜇(1)≥𝑡_𝑎, which is a contradiction. Therefore, 𝜇(1)≥min{𝜇(𝑥),0.5} for all 𝑥∈𝐿. Suppose that 𝜇(𝑥)<min{𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥)),𝜇(𝑎),0.5},(4.9) for some 𝑥,𝑎,𝑦∈𝐿. Taking 𝑡_𝑥∶=min{𝜇(𝑎→((𝑥→𝑦)→𝑥)),𝜇(𝑎),0.5}, we get 𝑎→((𝑥→𝑦)→𝑥)∈𝑈(𝜇;𝑡_𝑥) and 𝑎∈𝑈(𝜇;𝑡_𝑥). It follows from (3.1) that 𝑥∈𝑈(𝜇;𝑡_𝑥), that is, 𝜇(𝑥)≥𝑡_𝑥. This is a contradiction. Hence, 𝜇(𝑥)≥min{𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥)),𝜇(𝑎),0.5},(4.10) for all 𝑥,𝑎,𝑦∈𝐿. Using Theorem 4.5, we conclude that 𝜇 is an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿.

Proposition 4.8. Every (∈,∈∨𝑞)-fuzzy-fated filter 𝜇 of 𝐿 satisfies the following inequalities: (1)𝜇(𝑥→𝑧)≥min{𝜇(𝑥→(𝑦→𝑧)),𝜇(𝑥→𝑦),0.5}, (2)𝜇(𝑥)≥min{𝜇((𝑥→𝑦)→𝑥),0.5}, for all 𝑥,𝑦,𝑧∈𝐿.

Proof. (1) Suppose that there exist 𝑎,𝑏,𝑐∈𝐿 such that 𝜇(𝑎⟶𝑐)<min{𝜇(𝑎⟶(𝑏⟶𝑐)),𝜇(𝑎⟶𝑏),0.5}.(4.11) Taking 𝑡∶=min{𝜇(𝑎→(𝑏→𝑐)),𝜇(𝑎→𝑏),0.5} implies that 𝑡∈(0,0.5],𝑎→(𝑏→𝑐)∈𝑈(𝜇;𝑡) and 𝑎→𝑏∈𝑈(𝜇;𝑡). Since 𝑈(𝜇;𝑡)∈𝐹𝐹(𝐿) by Theorem 4.7, it follows from Lemma 3.1 that 𝑎→𝑐∈𝑈(𝜇;𝑡), that is, 𝜇(𝑎→𝑐)≥𝑡. This is a contradiction, and therefore 𝜇 satisfies (1).
(2) If 𝜇 is an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿, then 𝑈(𝜇;𝑡)∈𝐹𝐹(𝐿)∪{∅} for all 𝑡∈(0,0.5] by Theorem 4.7. Hence, 𝑈(𝜇;𝑡)∈𝐹(𝐿)∪{∅} for all 𝑡∈(0,0.5]. Suppose that 𝜇(𝑥)<𝑡≤min{𝜇((𝑥⟶𝑦)⟶𝑥),0.5},(4.12) for some 𝑥,𝑦∈𝐿 and 𝑡∈(0,0.5] then (𝑥→𝑦)→𝑥∈𝑈(𝜇;𝑡), which implies from Lemma 3.2 that 𝑥∈𝑈(𝜇;𝑡), that is, 𝜇(𝑥)≥𝑡. This is a contradiction. Hence, 𝜇(𝑥)≥min{𝜇((𝑥→𝑦)→𝑥),0.5} for all 𝑥,𝑦∈𝐿.

Theorem 4.9. If 𝐹 is a fated filter of 𝐿, then a fuzzy subset 𝜇 of 𝐿 defined by 𝜇∶𝐿⟶[0,1],𝑥⟼𝑡¹𝑡₂ifx∈F,otherwise,(4.13) where 𝑡₁∈[0.5,1] and 𝑡₂∈(0,0.5) is an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿.

Proof. Note that 𝑈(𝜇;𝑟)=𝐹𝐿if𝑟∈𝑡₂,0.5,if𝑟∈0,𝑡₂,(4.14) which is a fated filter of 𝐿. It follows from Theorem 4.7 that 𝜇 is an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿.

Theorem 4.10. Let 𝜇 be an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿. If 𝜇(1)<0.5, then 𝜇 is a fuzzy-fated filter of 𝐿.

Proof. It is straightforward.
For any fuzzy subset 𝜇 of 𝐿 and any 𝑡∈(0,1], we consider two subsets: 𝑄(𝜇;𝑡)∶={𝑥∈𝐿∣(𝑥,𝑡)𝑞𝜇},[𝜇]_𝑡∶={𝑥∈𝐿∣(𝑥,𝑡)∈∨𝑞𝜇}.(4.15) It is clear that [𝜇]_𝑡=𝑈(𝜇;𝑡)∪𝑄(𝜇;𝑡).

Theorem 4.11. If 𝜇 is an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿, then (∀𝑡∈(0.5,1])(𝑄(𝜇;𝑡)∈𝐹𝐹(𝐿)∪{∅}).(4.16)

Proof. Assume that 𝑄(𝜇;𝑡)≠∅ for all 𝑡∈(0.5,1], then there exists 𝑥∈𝑄(𝜇;𝑡), and so 𝜇(𝑥)+𝑡>1. Using Theorem 4.5(1), we have 𝜇(1)≥=m0in.5{𝜇(𝑥),0.5}𝜇(𝑥)if𝜇(𝑥)≥0.5,>1−𝑡,if𝜇(𝑥)<0.5(4.17) which implies that 1∈𝑄(𝜇;𝑡). Assume that 𝑎→((𝑥→𝑦)→𝑥)∈𝑄(𝜇;𝑡) and 𝑎∈𝑄(𝜇;𝑡) for all 𝑥,𝑎,𝑦∈𝐿, then (𝑎→((𝑥→𝑦)→𝑥),𝑡)𝑞𝜇 and (𝑎,𝑡)𝑞𝜇, that is, 𝜇(𝑎→((𝑥→𝑦)→𝑥))>1−𝑡 and 𝜇(𝑎)>1−𝑡. Using Theorem 4.5(2), we get 𝜇(𝑥)≥min{𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥)),𝜇(𝑎),0.5}.(4.18) Thus, if min{𝜇(𝑎→((𝑥→𝑦)→𝑥)),𝜇(𝑎)}<0.5, then 𝜇(𝑥)≥min{𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥)),𝜇(𝑎)}>1−𝑡.(4.19) If min{𝜇(𝑎→((𝑥→𝑦)→𝑥)),𝜇(𝑎)}≥0.5, then 𝜇(𝑥)≥0.5>1−𝑡. It follows that (𝑥,𝑡)𝑞𝜇 so that 𝑥∈𝑄(𝜇;𝑡). Therefore, 𝑄(𝜇;𝑡) is a fated filter of 𝐿.

Corollary 4.12. If 𝜇 is a strong (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿, then (∀𝑡∈(0.5,1])(𝑄(𝜇;𝑡)∈𝐹𝐹(𝐿)∪{∅}).(4.20)

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

Example 4.13. Consider the (∈,∈∨𝑞)-fuzzy-fated filter 𝜈 of 𝐿 which is given in Example 4.4, then 𝑄(𝜈;𝑡)={𝑐,𝑑,1}𝐿if𝑡∈(0.5,0.6],if𝑡∈(0.6,1](4.21) is a fated filter of 𝐿. But 𝜈 is not a strong (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿.

Theorem 4.14. For a fuzzy subset 𝜇 of 𝐿, the following assertions are equivalent: (1)𝜇 is an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿,(2)(∀𝑡∈(0,1])([𝜇]_𝑡∈𝐹𝐹(𝐿)∪{∅}).

Proof. Assume that 𝜇 is an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿, and let 𝑡∈(0,1] be such that [𝜇]_𝑡≠∅, then there exists 𝑥∈[𝜇]_𝑡=𝑈(𝜇;𝑡)∪𝑄(𝜇;𝑡), and so 𝑥∈𝑈(𝜇;𝑡) or 𝑥∈𝑄(𝜇;𝑡). If 𝑥∈𝑈(𝜇;𝑡), then 𝜇(𝑥)≥𝑡. It follows from Theorem 4.5(1) that 𝜇(1)≥=m𝑡in{𝜇(𝑥),0.5}≥min{𝑡,0.5}0.5>1−𝑡if𝑡≤0.5,if𝑡>0.5,(4.22) so that 1∈𝑈(𝜇;𝑡)∪𝑄(𝜇;𝑡)=[𝜇]_𝑡. If 𝑥∈𝑄(𝜇;𝑡), then (𝑥,𝑡)𝑞𝜇, that is, 𝜇(𝑥)+𝑡>1. Thus, 𝜇(1)≥=m1in−{𝑡𝜇(𝑥),0.5}≥min{1−𝑡,0.5}0.5≥𝑡if𝑡>0.5,if𝑡≤0.5,(4.23) and so 1∈𝑄(𝜇;𝑡)∪𝑈(𝜇;𝑡)=[𝜇]_𝑡. Let 𝑥,𝑎,𝑦∈𝐿 be such that 𝑎∈[𝜇]_𝑡 and 𝑎→((𝑥→𝑦)→𝑥)∈[𝜇]_𝑡, then 𝜇(𝑎)≥𝑡 or 𝜇(𝑎)+𝑡>1, and 𝜇(𝑎→((𝑥→𝑦)→𝑥))≥𝑡 or 𝜇(𝑎→((𝑥→𝑦)→𝑥))+𝑡>1. We can consider four cases: 𝜇(𝑎)≥𝑡𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥))≥𝑡,(4.24)𝜇(𝑎)≥𝑡𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥))+𝑡>1,(4.25)𝜇(𝑎)+𝑡>1𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥))≥𝑡,(4.26)𝜇(𝑎)+𝑡>1𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥))+𝑡>1.(4.27) For the first case, Theorem 4.5(2) implies that 𝜇(𝑥)≥≥mmiinn{{𝜇𝑡,0(𝑎.5⟶}=((𝑥0.5⟶𝑦)⟶𝑥)),𝜇(𝑎),0.5}𝑡if𝑡>0.5,if𝑡≤0.5,(4.28) so that 𝑥∈𝑈(𝜇;𝑡) or 𝜇(𝑥)+𝑡>0.5+0.5=1, that is, 𝑥∈𝑄(𝜇;𝑡). Hence, 𝑥∈[𝜇]_𝑡. Case (4.25) implies that 𝜇(𝑥)≥≥mmiinn{{𝜇1−(𝑎𝑡⟶,𝑡,0.5((}𝑥=⟶1𝑦−)⟶𝑡𝑥)),𝜇(𝑎),0.5}𝑡if𝑡>0.5,if𝑡≤0.5.(4.29) Thus, 𝑥∈𝑄(𝜇;𝑡)∪𝑈(𝜇;𝑡)=[𝜇]_𝑡. Similarly, 𝑥∈[𝜇]_𝑡 for the case (4.26). The final case implies that 𝜇(𝑥)≥≥mmiinn{{𝜇1−(𝑎𝑡⟶,0.5}((=𝑥⟶1−𝑦)𝑡⟶𝑥)),𝜇(𝑎),0.5}0.5if𝑡>0.5,if𝑡≤0.5,(4.30) so that 𝑥∈𝑄(𝜇;𝑡)∪𝑈(𝜇;𝑡)=[𝜇]_𝑡. Consequently, [𝜇]_𝑡 is a fuzzy-fated filter of 𝐿.
Conversely, let 𝜇 be a fuzzy subset of 𝐿 such that [𝜇]_𝑡 is a fated filter of 𝐿 whenever it is nonempty for all 𝑡∈(0,1]. If there exists 𝑎∈𝐿 such that 𝜇(1)<min{𝜇(𝑎),0.5}, then 𝜇(1)<𝑡_𝑎≤min{𝜇(𝑎),0.5} for some 𝑡_𝑎∈(0,0.5]. It follows that 𝑎∈𝑈(𝜇;𝑡_𝑎) but 1∉𝑈(𝜇;𝑡_𝑎). Also, 𝜇(1)+𝑡_𝑎<2𝑡_𝑎≤1 and so 1∉𝑄(𝜇;𝑡_𝑎). Hence, 1∉𝑈(𝜇;𝑡_𝑎)∪𝑄(𝜇;𝑡_𝑎)=[𝜇]_𝑡𝑎, which is a contradiction. Therefore, 𝜇(1)≥min{𝜇(𝑥),0.5} for all 𝑥∈𝐿. Suppose that 𝜇(𝑥)<min{𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥)),𝜇(𝑎),0.5},(4.31) for some 𝑥,𝑎,𝑦∈𝐿. Taking 𝑡∶=min{𝜇(𝑎→((𝑥→𝑦)→𝑥)),𝜇(𝑎),0.5} implies that 𝑡∈(0,0.5],𝑎∈𝑈(𝜇;𝑡)⊆[𝜇]_𝑡, and 𝑎→((𝑥→𝑦)→𝑥)∈𝑈(𝜇;𝑡)⊆[𝜇]_𝑡. Since [𝜇]_𝑡∈𝐹𝐹(𝐿), it follows that 𝑥∈[𝜇]_𝑡=𝑈(𝜇;𝑡)∪𝑄(𝜇;𝑡). But (4.31) induces 𝑥∉𝑈(𝜇;𝑡) and 𝜇(𝑥)+𝑡<2𝑡≤1, that is, 𝑥∉𝑄(𝜇;𝑡). This is a contradiction, and thus 𝜇(𝑥)≥min{𝜇(𝑎→((𝑥→𝑦)→𝑥)),𝜇(𝑎),0.5} for all 𝑥,𝑎,𝑦∈𝐿. Using Theorem 4.5, we conclude that 𝜇 is an (∈,∈∨𝑞)-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 ∧,∨,¬,→ 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[𝑥∈𝜇]=𝜇(𝑥),(5.1)[Φ∧Ψ]=min{[Φ],[Ψ]},(5.2)[Φ⟶Ψ]=min{1,1−[Φ]+[Ψ]},(5.3)[∀𝑥Φ(𝑥)]=inf_𝑥∈𝑈[Φ(𝑥)],(5.4)⊨Φifandonlyif[Φ]=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 (𝐼_GR): 𝐼_GR(𝑎,𝑏)=10if𝑎≤𝑏,otherwise.(5.6)(b)Gödel implication operator (𝐼_G): 𝐼_𝐺(𝑎,𝑏)=1𝑏if𝑎≤𝑏,otherwise.(5.7)(c)The contraposition of Gödel implication operator (𝐼_cG): 𝐼_cG(𝑎,𝑏)=11−𝑎if𝑎≤𝑏,otherwise.(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 𝜇 of 𝐿 is called a fuzzifying fated filter of 𝐿 if it satisfies the following conditions: (1)for all 𝑥∈𝐿, we have ⊨[𝑥∈𝜇]⟶[1∈𝜇],(5.9)(2)for all 𝑥,𝑎,𝑦∈𝐿, we get ⊨[𝑎∈𝜇]∧[𝑎⟶((𝑥→𝑦)⟶𝑥)∈𝜇]⟶[𝑥∈𝜇].(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 𝑡-tautology is introduced, that is,⊨_𝑡Φifandonlyif[Φ]≥𝑡forallvaluations.(5.11)

Definition 5.2. Let 𝜇 be a fuzzy subset of 𝐿 and 𝑡∈(0,1], then 𝜇 is called a 𝑡-implication-based fuzzy-fated filter of 𝐿 if it satisfies the following conditions: (1)for all 𝑥∈𝐿, we have ⊨_𝑡[𝑥∈𝜇]⟶[1∈𝜇],(5.12)(2)for all 𝑥,𝑎,𝑦∈𝐿, we get ⊨_𝑡[𝑎∈𝜇]∧[𝑎→((𝑥⟶𝑦)⟶𝑥)∈𝜇]⟶[𝑥∈𝜇].(5.13)
Let 𝐼 be an implication operator. Clearly, 𝜇 is a 𝑡-implication-based fuzzy-fated filter of 𝐿 if and only if it satisfies (1)(∀𝑥∈𝐿)(𝐼(𝜇(𝑥),𝜇(1))≥𝑡), (2)(∀𝑥,𝑦∈𝐿)(𝐼(min{𝜇(𝑎),𝜇(𝑎→((𝑥→𝑦)→𝑥))},𝜇(𝑥))≥𝑡).

Theorem 5.3. For any fuzzy subset 𝜇 of 𝐿, one has the following: (1)if 𝐼=𝐼_GR, then 𝜇 is a 0.5-implication-based fuzzy-fated filter of 𝐿 if and only if 𝜇 is a fuzzy-fated filter of 𝐿,(2)if 𝐼=𝐼_G, then 𝜇 is a 0.5-implication-based fuzzy-fated filter of 𝐿 if and only if 𝜇 is an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿,(3)if 𝐼=𝐼_cG, then 𝜇 is a 0.5-implication-based fuzzy-fated filter of 𝐿 if and only if 𝜇 satisfies the following conditions:(3.1)max{𝜇(1),0.5}≥min{𝜇(𝑥),1}, (3.2)max{𝜇(𝑥),0.5}≥min{𝜇(𝑎),𝜇(𝑎→((𝑥→𝑦)→𝑥)),1}
for all 𝑥,𝑎,𝑦∈𝐿.

Proof. (1) It is Straightforward.
(2) Assume that 𝜇 is a 0.5-implication-based fuzzy-fated filter of 𝐿, then(i)(∀𝑥∈𝐿)(𝐼_G(𝜇(𝑥),𝜇(1))≥0.5),(ii)(∀𝑥,𝑦∈𝐿)(𝐼_G(min{𝜇(𝑎),𝜇(𝑎→((𝑥→𝑦)→𝑥))},𝜇(𝑥))≥0.5). From (i), we have 𝜇(1)≥𝜇(𝑥) or 𝜇(𝑥)≥𝜇(1)≥0.5, and so 𝜇(1)≥min{𝜇(𝑥),0.5} for all 𝑥∈𝐿. The second case implies that 𝜇(𝑥)≥min{𝜇(𝑎),𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥))}(5.14) or min{𝜇(𝑎),𝜇(𝑎→((𝑥→𝑦)→𝑥))}>𝜇(𝑥)≥0.5. It follows that 𝜇(𝑥)≥min{𝜇(𝑎),𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥)),0.5},(5.15) for all 𝑥,𝑎,𝑦∈𝐿. Using Theorem 4.5, we know that 𝜇 is an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿.
Conversely, suppose that 𝜇 is an (∈,∈∨𝑞)-fuzzy-fated filter of 𝐿. From Theorem 4.5(1), if min{𝜇(𝑥),0.5}=𝜇(𝑥), then 𝐼_G(𝜇(𝑥),𝜇(1))=1≥0.5. Otherwise, 𝐼_G(𝜇(𝑥),𝜇(1))≥0.5. From Theorem 4.5(2), if min{𝜇(𝑎),𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥)),0.5}=min{𝜇(𝑎),𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥))},(5.16) then 𝜇(𝑥)≥min{𝜇(𝑎),𝜇(𝑎→((𝑥→𝑦)→𝑥))}, and so 𝐼_G(min{𝜇(𝑎),𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥))},𝜇(𝑥))=1≥0.5.(5.17) If min{𝜇(𝑎),𝜇(𝑎→((𝑥→𝑦)→𝑥)),0.5}=0.5, then 𝜇(𝑥)≥0.5, and thus 𝐼_G(min{𝜇(𝑎),𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥))},𝜇(𝑥))≥0.5.(5.18) Consequently, 𝜇 is a 0.5-implication-based fuzzy-fated filter of 𝐿.
(3) Suppose that 𝜇 satisfies (3.1) and (3.2). In (3.1), if 𝜇(𝑥)=1, then max{𝜇(1),0.5}=1, and hence 𝐼_cG(𝜇(𝑥),𝜇(1))=1≥0.5. If 𝜇(𝑥)<1, then max{𝜇(1),0.5}≥𝜇(𝑥).(5.19) If max{𝜇(1),0.5}=𝜇(1) in (5.19), then 𝜇(1)≥𝜇(𝑥). Hence, 𝐼_cG(𝜇(𝑥),𝜇(1))=1≥0.5.(5.20) If max{𝜇(1),0.5}=0.5 in (5.19), then 𝜇(𝑥)≤0.5 which implies that 𝐼_cG(𝜇(𝑥),𝜇(1))=1≥0.51−𝜇(𝑥)≥0.5if𝜇(1)≥𝜇(𝑥),otherwise.(5.21) In (3.2), if min{𝜇(𝑎),𝜇(𝑎→((𝑥→𝑦)→𝑥)),1}=1, then max{𝜇(𝑥),0.5}=1,(5.22) and so 𝜇(𝑥)=1≥min{𝜇(𝑎),𝜇(𝑎→((𝑥→𝑦)→𝑥))}. Therefore, 𝐼_cG(min{𝜇(𝑎),𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥))},𝜇(𝑥))=1≥0.5.(5.23) If min{𝜇(𝑎),𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥)),1}=min{𝜇(𝑎),𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥))},(5.24) then max{𝜇(𝑥),0.5}≥min{𝜇(𝑎),𝜇(𝑎⟶((𝑥→𝑦)⟶𝑥))}.(5.25) Thus, if max{𝜇(𝑥),0.5}=0.5 in (5.25), then 𝜇(𝑥)≤0.5 and min{𝜇(𝑎),𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥))}≤0.5.(5.26) Therefore, 𝐼_cG(min{𝜇(𝑎),𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥))},𝜇(𝑥))=1≥0.5,(5.27) whenever 𝜇(𝑥)≥min{𝜇(𝑎),𝜇(𝑎→((𝑥→𝑦)→𝑥))}, 𝐼_cG(mi=n1{𝜇−(m𝑎)i,n𝜇{(𝜇𝑎(⟶𝑎),𝜇(((𝑎𝑥⟶⟶𝑦(()𝑥⟶⟶𝑥𝑦)))}⟶,𝜇(𝑥𝑥))))}≥0.5,(5.28) whenever 𝜇(𝑥)<min{𝜇(𝑎),𝜇(𝑎→((𝑥→𝑦)→𝑥))}. Now, if max{𝜇(𝑥),0.5}=𝜇(𝑥)(5.29) in (5.25), then 𝜇(𝑥)≥min{𝜇(𝑎),𝜇(𝑎→((𝑥→𝑦)→𝑥))}, and so 𝐼_cG(min{𝜇(𝑎),𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥))},𝜇(𝑥))=1≥0.5.(5.30)
Consequently, 𝜇 is a 0.5-implication-based fuzzy-fated filter of 𝐿.
Conversely, assume that 𝜇 is a 0.5-implication-based fuzzy-fated filter of 𝐿, then (iii)𝐼_cG(𝜇(𝑥),𝜇(1))≥0.5, (iv)𝐼_cG(min{𝜇(𝑎),𝜇(𝑎→((𝑥→𝑦)→𝑥))},𝜇(𝑥))≥0.5, for all 𝑥,𝑎,𝑦∈𝐿. The case (iii) implies that 𝐼_cG(𝜇(𝑥),𝜇(1))=1, that is, 𝜇(𝑥)≤𝜇(1), or 1−𝜇(𝑥)≥0.5 and so 𝜇(𝑥)≤0.5. It follows that max{𝜇(1),0.5}≥𝜇(𝑥)=min{𝜇(𝑥),1}.(5.31) From (iv), we have 𝐼_cG(min{𝜇(𝑎),𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥))},𝜇(𝑥))=1,(5.32) that is, min{𝜇(𝑎),𝜇(𝑎→((𝑥→𝑦)→𝑥))}≤𝜇(𝑥), or 1−min{𝜇(𝑎),𝜇(𝑎⟶((𝑥⟶𝑦)⟶𝑥))}≥0.5.(5.33) Hence, max{𝜇(𝑥),0.5}≥=mmiinn{{𝜇𝜇((𝑎𝑎)),,𝜇𝜇((𝑎𝑎⟶⟶((((𝑥𝑥⟶⟶𝑦𝑦))⟶⟶𝑥𝑥)))),}1},(5.34) for all 𝑥,𝑦,𝑧∈𝐿. This completes the proof.

6. Conclusion

Using the “belongs to” relation (∈) and quasicoincidence with relation (𝑞) between fuzzy points and fuzzy sets, we introduced the notion of an (∈,∈∨𝑞)-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 (𝐼_GR), Gödel implication operator (𝐼_G), and the contraposition of Gödel implication operator (𝐼_cG) to study 𝑡-implication-based fuzzy-fated filters.

There are also other situations concerning the relations between this kind of results, another type of structures (e.g., (∈,∈∨𝑞)-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 “belongs to” relation (∈) and intuitionistic quasicoincidence with relation (𝑞) 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.