# Fuzzyn-Fold Filters of Pseudoresiduated Lattices

Authors: Albert Kadji; Celestin Lele; Marcel Tonga
Journal: International Journal of Mathematics and Mathematical Sciences
 (2015)

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## Abstract

Given a pseudoresiduated latticeM and a lattice L, we introduce and characterize the fuzzy versions of different n-fold implicative (resp., obstinate, Boolean, normal, and extended involutive) filters of M. Moreover, we study some relationships between these different types of fuzzy n-fold filters.


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## Body

## 1. Introduction

Uncertainty is present in almost every sphere of our daily life. Traditional mathematical tools are not sufficient to handle all practical problems in fields such as medical science, social science, engineering, and economics, involving uncertainty of various types. Zadeh, in 1965, came up with his remarkable theory of fuzzy sets for dealing with these types of uncertainties where conventional tools fail. His theory brought a grand paradigmatic change in mathematics. Later on, fuzzy logic became popular.Fuzzy logic has become a useful tool for computer science to deal with uncertain information. Some domains of research and application of fuzzy logic are automation (in manufacture), instrumentation (captors, instruments of measure, and recognition of voice), judgment and conception (consultation, decision making, and timetable for traffic). Recently, an algorithm based on fuzzy logic is used for the detection of the cancer tumors [1].The origin of residuated lattices lies in mathematical logic without contraction. Residuated lattices have been investigated by Krull [2], Dilworth [3], Balbes and Dwinger [4], and Pavelka [5]. Many algebraic structures have been proposed as the semantic aspects of logical systems. In 1958, Chang introduced the notion of MV-algebra formany-valued logic, and his completeness theorem (1959) stated the closed real unit interval [0,1] as a standard model of this logic. Later on, BL-algebra was introduced by Hàjek [6] as a commutative generalization of MV-algebra. Filters correspond to sets of formulae closed with respect to Modus Ponens. So filters are important tools in studying the above algebras and the completeness of nonclassical logic. From the logical point of view, various filters correspond to various sets of provable formulae.At present, there are two ways to generalize the existing types of filters: one is folding theory and the other is fuzzy set theory.In the folding approach, Haveshki and Eslami introduced the notion ofn-fold implicative (resp., n-fold positive implicative) BL-algebra [7]. Then, Turunen et al. in [8, 9] proved that n-fold implicative BL-algebras are Gödel algebras, and n-fold positive implicative BL-algebras are MV-algebras.In the fuzzy approach, a lot of work has been done on fuzzy filters which are particular cases of fuzzy sets introduced by Zadeh [10] (a fuzzy subset of a set A is a map f:A→I, where I≔[0;1] is the closed unit interval of real numbers); most of the authors (see, e.g., [11–22]) use the above original definition of a fuzzy subset.In the present work, we replace the closed real unit interval by an arbitrary lattice. So, a fuzzy subset ofA will be a map μ:A→L, where L is the underlying set of a lattice. It follows from our characterizations of fuzzy filters that most of the known results with I=[0;1] are easily proved and extended to more general cases. We also introduce some new notions and prove several new results in pseudoresiduated lattices.The paper is organized as follows. In Section2, some basic concepts and properties are recalled. In Section 3, we study the notions of L-fuzzy filter. Section 4 is devoted to the notions of L-fuzzy n-fold obstinate filter and L-fuzzy n-fold Boolean filter. In Section 5, we study the notion of L-fuzzy n-fold implicative filter and show that any L-fuzzy n-fold Boolean filter is L-fuzzy n-fold implicative. In Section 6, the notions of L-fuzzy n-fold normal (resp., extended involutive, fantastic, and involutive) filter are given and we obtain that an L-fuzzy filter is L-fuzzy n-fold Boolean if and only if it is L-fuzzy n-fold fantastic and L-fuzzy n-fold implicative.


## 2. Preliminaries

In this section, we recall some definitions and results which will be used in the sequel.A pseudoresiduated lattice is a nonempty setL with five binary operations ∧, ∨, ⊗, →, and ⇝ and two constants 0,1 satisfying the following:L-1:
L(L)≔(L,∧,∨,0,1) is a bounded lattice.L-2:
(L,⊗,1) is a monoid.L-3:
x⊗y≤z iff x≤y→z if and only if y≤x⇝z (pseudoresiduation).Apseudo-RL monoid is a pseudoresiduated lattice L which satisfies the following condition:L-4:
y⊗(y⇝x)=x∧y=(x→y)⊗x (pseudodivisibility).Apseudo-MTL algebra is a pseudoresiduated lattice L which satisfies the following condition:L-5:
(x→y)∨(y→x)=1=(x⇝y)∨(y⇝x) (pseudoprelinearity).Apseudo-BL-algebra is a pseudo-MTL-algebra L which satisfies the pseudodivisibility.Apseudo-MV-algebra is a pseudo-BL-algebra L which satisfies the following condition:L-6:
x~¯=x=x¯~, where x¯=x→0 and x~=x⇝0.Alternatively, apseudo-MV-algebra can be defined as a pseudoresiduated lattice L which satisfies the following condition:L-7:
(x⇝y)→y=(y⇝x)→x and (x→y)⇝y=(y→x)⇝x.AHeyting algebra is a pseudoresiduated lattice L which satisfies the following condition:L-8:
x∧y=x⊗y.ABoolean algebra is a pseudoresiduated lattice L which satisfies the following condition:L-9:
x∨x¯=1.Recall that, for anyx∈L, x0≔1, xn+1≔xn⊗x for all n≥0.Proposition 1 (see [23]).
For allx,y,z∈L, the following properties hold: (1)
x≤y if and only if x→y=1=x⇝y.(2)
x→(y→z)=(x⊗y)→z; (x⊗y)⇝z=y⇝(x⇝z); x→(y⇝z)=y⇝(x→z).(3)
(y→z)⊗(x→y)≤x→z; (x⇝y)⊗(y⇝z)≤x⇝z.(4)
Ifx≤y then: y→z≤x→z, y⇝z≤x⇝z, z⇝x≤z⇝y, x⊗z≤y⊗z, y¯≤x¯, y~≤x~, and z→x≤z→y.(5)
1→x=x=1⇝x; x→x=1=x⇝x; x→1=1=x⇝1; y⇝x≥x≤y→x.(6)
x¯⊗x=0=x⊗x~; x~¯~=x~; x¯~¯=x¯; x~¯≥x≤x¯~.(7)
x⊗(x⇝y)≤y≤x⇝(x⊗y); (x→y)⊗x≤x≤y→(x⊗y).(8)
1~=0=1¯; 0~=1=0¯.(9)
x⊗(y∨z)=(x⊗y)∨(x⊗z); (y∨z)⊗x=(y⊗x)∨(z⊗x).(10)
(x∨y)→z=(x→z)∧(y→z); (x∨y)⇝z=(x⇝z)∧(y⇝z).(11)
x∨y≤((x→y)⇝y)∧((y→x)⇝x); x∨y≤((x⇝y)→y)∧((y⇝x)→x).(12)
x→y≤(y→z)⇝(x→z); x⇝y≤(y⇝z)→(x⇝z).(13)
y→x≤(z→y)→(z→x); y⇝x≤(z⇝y)⇝(z⇝x).(14)
x⊗(x⇝y)≤x∧y; (x→y)⊗x≤x∧y; x⊗y≤x∧y.In the following, unless mentioned otherwise, any pseudoresiduated latticeM=(M,∧,∨,⊗,→,⇝,0,1) will often be referred to by its support set M and will be called a residuated lattice for short.Definition 2.
LetL1 and L2 be two residuated lattices. Then, a map f:L1→L2 is called a residuated lattice homomorphism if f preserves all the operations, that is, if the following conditions are satisfied: (i)
f is a morphism of bounded lattices.(ii)
For allx,y∈L1, f(x⊗y)=f(x)⊗f(y).(iii)
For allx,y∈L1, f(x→y)=f(x)→f(y).(iv)
For allx,y∈L1, f(x⇝y)=f(x)⇝f(y).
Iff is bijective, the homomorphism f is called a residuated lattice isomorphism. In this case, we write L1≅L2.Recall that a nonempty subsetF of M is called afilter if it satisfies the following two conditions:(F1)
For everyx,y∈F, x⊗y∈F.(F2)
For everyx,y∈M, if x≤y and x∈F, then y∈F.A filterF is said to be proper if {1}⊆F⊊M.A filterF is said to be commutative if, for all x,y∈M, x⇝y∈F iff x→y∈F.Remark 3.
LetF be a commutative filter of M. The binary relation ≡F, defined on M by x≡Fy if and only if x→y and y→x∈F, is congruence on M. The quotient structure M/F is also a pseudoresiduated lattice, where, for all x,y∈M,(1)xF∧yF=x∧yF;xF∨yF=x∨yF;xF⊗yF=x⊗yF;xF⟶yF=x⟶yF,xF⇝yF=x⇝yF.Note that M/{1}≅M.The following examples will be frequently used in the text.Example 4.
LetL={0,a,b,c,d,1} be a lattice such that 0<a,b<c<d<1, where a and b are incomparable. Define the operations ⊗, ⇝, and → by the three tables in (2). Then L is a residuated lattice which is not a pseudo-MTL-algebra, since (a⇝b)∨(b⇝a)=c≠1. Consider(2)
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1F={1} and F1={1,d} are the only proper filters of L.Example 5.
LetM={0,a,b,c,d,e,1} be a lattice such that 0<a<b,  c<d<e<1, where b and c are incomparable. Define the operations ⊗, ⇝, and → by the three tables in (3).   M is a residuated lattice which is not a pseudo-MTL-algebra, since (b⇝c)∨(c⇝b)=e≠1. Consider(3)
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1F1={1}; F2={1,e}; F3={1,e,d,c}; F4={1,e,d,c,b,a} are the proper filters of M.
F
1 and F3 are commutative filters, but F2 is not, since b⇝a=e∈F2 and b→a=d∉F2.We recall that a proper filterF is said to be prime if, for all x,y∈M, x∨y∈F implies x∈F or y∈F. So all filters of Example 5 are prime filters.Proposition 6 (see [12]).
The following conditions are equivalent for a filterF:(i)
F is a prime filter.(ii)
For any filterF1,F2 of M, F1∩F2=F implies F1=F or F2=F.Note. Similar to the case of commutative residuated lattices [24], we establish in [25] that(i)
a filterF is maximal if and only if, for any x∈M, x∉F implies that there are f∈F and natural numbers m,n≥1 such that (f⊗xm)n=0;(ii)
any proper filterF can be extended to a prime filter P, and any proper filter F can be extended to a maximal filter G. So, any maximal filter is prime.


## 3. Fuzzy Filter

### 3.1. Definitions and First Properties

Definition 7.
LetM be a residuated lattice and let L be a lattice.
(i) A functionμ:M→L is called an L-fuzzy subset of M.
For allα∈L, the set μα≔{x∈M:μ(x)≥α} is called the α-cut of μ.
(ii) AnL-fuzzy subset μ of M is called an L-fuzzy filter of M if, for all α∈L,  μα is either empty or a filter of M.For anyα∈L, let cα denote the constant function M→L with value α. Clearly, cα is an L-fuzzy filter of M (called a constant fuzzy filter).AnL-fuzzy filter is said to be proper if it is nonconstant.From now on, anL-fuzzy subset (resp., filter) will be called a fuzzy subset (resp., filter).Example 8.
LetG be a proper filter of M and α,β∈L such that α<β. Define the map (G)αβ:M→L by (G)αβ(x)=β, if x∈G, and α, if x∉G.
It is easy to see that(G)αβ is a nonconstant fuzzy filter of M.The following result was established in [12].Theorem 9.
A fuzzy subsetμ of M is fuzzy filter of M if and only if, for all x,y∈M,(i)
μ(x⊗y)=μ(x)∧μ(y),(ii)
x≤y implies μ(x)≤μ(y).Proposition 10.
The following assertions are equivalent for any fuzzy subsetη:M→L and any x,y∈M:(i)
η is a fuzzy filter.(ii)
η(x→y)∧η(x)≤η(y) for all x,y∈M, and η(x)≤η(1).(iii)
η(x⇝y)∧η(x)≤η(y) for all x,y∈M, and η(x)≤η(1).Proof.
(
i
)
⇒
(
i
i
) Since x≤1, we have η(x)≤η(1).
In addition, since(x→y)⊗x≤y, it follows that η(x→y)∧η(x)=η((x→y)⊗x)≤η(y).
(
i
i
)
⇒
(
i
) Let x,y∈M. If x≤y, then η((x∧y)→y)∧η(x∧y)≤η(y); that is, η(x)≤η(y). Now, since y≤x→x⊗y, it follows that η(x)∧η(y)≤η(x)∧η(x→x⊗y)≤η(x⊗y).
(
i
)
⇔
(
i
i
i
) It is similar to (i)⇔(ii).LetFuFil(M) denote the set of all fuzzy filters of M.Proposition 11.
For allx,y,z∈M and η, a fuzzy filter, the following assertions hold:(1)
η(x→y)=η(1)⇒η(x)≤η(y), and η(x⇝y)=η(1)⇒η(x)≤η(y).(2)
η(y⇝x)≥η(x)≤η(y→x).(3)
η((x⊗y)∨(x⊗z))=η(y∨z)∧η(x)=η((y⊗x)∨(z⊗x)).(4)
η(x⇝y)≤η(z⊗x⇝z⊗y), and η(x→y)≤η(x⊗z→y⊗z).(5)
η(x→(z∧y))=η(x→z)∧η(x→y), and η(x⇝(z∧y))=η(x⇝z)∧η(x⇝y).Proof.
Letη be a fuzzy filter, and x,y∈M:(1)
By Proposition10, (η(x→y)=η(1))⇒(η(x)=η(1)∧η(x)≤η(y)). Similarly, we show that η(x⇝y)=η(1)⇒η(x)≤η(y).(2)
This follows from the fact thatx≤y→x and x≤y⇝x.(3)
η((x⊗y)∨(x⊗z))=η(x⊗(y∨z))=η(x)∧η(y∨z)=η(y∨z)∧η(x)=η((y⊗x)∨(z⊗x)).(4)
Sincex⊗(x⇝y)≤y, we have z⊗x⊗(x⇝y)≤z⊗y; so x⇝y≤(z⊗x)⇝(z⊗y).(5)
Note thatx→(y∧z)=(x→y)∧(x→z).Definition 12.
A fuzzy filterμ is said to be a commutative fuzzy filter if, for all x,y∈M,  μ(x⇝y)=μ(1) if and only if μ(x→y)=μ(1).It is easy to show the following result.Lemma 13.
μ is a commutative fuzzy filter of M if and only if μμ(1) is a commutative filter of M.
In particular, ifG is a filter of M and α<β in L, then (G)αβ is a commutative fuzzy filter of M if and only if G is a commutative filter of M.A classF of filters of M is said to be closed under extension if, for all filters G,H of M such that G⊆H, G∈F implies H∈F.A classC of fuzzy filters is said to be closed under extension if, for all η,ϑ∈FuFil(M) such that η(1)=θ(1) and η≤θ,  η∈C implies θ∈C.


### 3.2.  Fuzzy Prime Filter

Definition 14.
A nonconstant fuzzy filterμ:M→L is called a fuzzy prime filter if, for every α∈L, μα is a prime filter whenever it is proper.Example 15.
LetL be a lattice:(i)
LetM be the residuated lattice of Example 5; then any fuzzy filter μ:M→L is fuzzy prime.(ii)
LetG be a filter of M, and α<β in L; then (G)αβ is a fuzzy prime filter if and only if G is a prime filter of M.In fact, we have the following result.Theorem 16.
A nonconstant fuzzy filterμ:M→L is fuzzy prime if and only if Im⁡(μ) is a chain, and μ(x∨y)=max⁡{μ(x),μ(y)} for all x,y∈M.Proof.
Letμ be a nonconstant fuzzy filter of M.
Assume thatμ is fuzzy prime; then x∨y∈μμ(x∨y), so μμ(x∨y) is M or a prime filter; thus x∈μμ(x∨y) or y∈μμ(x∨y); this implies μ(x)≥μ(x∨y) or μ(y)≥μ(x∨y); so μ(x)≥μ(y) or μ(x)≤μ(y).
Conversely, assume that the conclusion holds. Letα∈L such that μα is a proper filter, and x,y∈M such that x∨y∈μα. That is, μ(x∨y)≥α; since μ(x∨y)∈{μ(x),μ(y)}, we obtain x∈μα or y∈μα. So μα is a prime filter of M.


## 3.1. Definitions and First Properties

Definition 7.
LetM be a residuated lattice and let L be a lattice.
(i) A functionμ:M→L is called an L-fuzzy subset of M.
For allα∈L, the set μα≔{x∈M:μ(x)≥α} is called the α-cut of μ.
(ii) AnL-fuzzy subset μ of M is called an L-fuzzy filter of M if, for all α∈L,  μα is either empty or a filter of M.For anyα∈L, let cα denote the constant function M→L with value α. Clearly, cα is an L-fuzzy filter of M (called a constant fuzzy filter).AnL-fuzzy filter is said to be proper if it is nonconstant.From now on, anL-fuzzy subset (resp., filter) will be called a fuzzy subset (resp., filter).Example 8.
LetG be a proper filter of M and α,β∈L such that α<β. Define the map (G)αβ:M→L by (G)αβ(x)=β, if x∈G, and α, if x∉G.
It is easy to see that(G)αβ is a nonconstant fuzzy filter of M.The following result was established in [12].Theorem 9.
A fuzzy subsetμ of M is fuzzy filter of M if and only if, for all x,y∈M,(i)
μ(x⊗y)=μ(x)∧μ(y),(ii)
x≤y implies μ(x)≤μ(y).Proposition 10.
The following assertions are equivalent for any fuzzy subsetη:M→L and any x,y∈M:(i)
η is a fuzzy filter.(ii)
η(x→y)∧η(x)≤η(y) for all x,y∈M, and η(x)≤η(1).(iii)
η(x⇝y)∧η(x)≤η(y) for all x,y∈M, and η(x)≤η(1).Proof.
(
i
)
⇒
(
i
i
) Since x≤1, we have η(x)≤η(1).
In addition, since(x→y)⊗x≤y, it follows that η(x→y)∧η(x)=η((x→y)⊗x)≤η(y).
(
i
i
)
⇒
(
i
) Let x,y∈M. If x≤y, then η((x∧y)→y)∧η(x∧y)≤η(y); that is, η(x)≤η(y). Now, since y≤x→x⊗y, it follows that η(x)∧η(y)≤η(x)∧η(x→x⊗y)≤η(x⊗y).
(
i
)
⇔
(
i
i
i
) It is similar to (i)⇔(ii).LetFuFil(M) denote the set of all fuzzy filters of M.Proposition 11.
For allx,y,z∈M and η, a fuzzy filter, the following assertions hold:(1)
η(x→y)=η(1)⇒η(x)≤η(y), and η(x⇝y)=η(1)⇒η(x)≤η(y).(2)
η(y⇝x)≥η(x)≤η(y→x).(3)
η((x⊗y)∨(x⊗z))=η(y∨z)∧η(x)=η((y⊗x)∨(z⊗x)).(4)
η(x⇝y)≤η(z⊗x⇝z⊗y), and η(x→y)≤η(x⊗z→y⊗z).(5)
η(x→(z∧y))=η(x→z)∧η(x→y), and η(x⇝(z∧y))=η(x⇝z)∧η(x⇝y).Proof.
Letη be a fuzzy filter, and x,y∈M:(1)
By Proposition10, (η(x→y)=η(1))⇒(η(x)=η(1)∧η(x)≤η(y)). Similarly, we show that η(x⇝y)=η(1)⇒η(x)≤η(y).(2)
This follows from the fact thatx≤y→x and x≤y⇝x.(3)
η((x⊗y)∨(x⊗z))=η(x⊗(y∨z))=η(x)∧η(y∨z)=η(y∨z)∧η(x)=η((y⊗x)∨(z⊗x)).(4)
Sincex⊗(x⇝y)≤y, we have z⊗x⊗(x⇝y)≤z⊗y; so x⇝y≤(z⊗x)⇝(z⊗y).(5)
Note thatx→(y∧z)=(x→y)∧(x→z).Definition 12.
A fuzzy filterμ is said to be a commutative fuzzy filter if, for all x,y∈M,  μ(x⇝y)=μ(1) if and only if μ(x→y)=μ(1).It is easy to show the following result.Lemma 13.
μ is a commutative fuzzy filter of M if and only if μμ(1) is a commutative filter of M.
In particular, ifG is a filter of M and α<β in L, then (G)αβ is a commutative fuzzy filter of M if and only if G is a commutative filter of M.A classF of filters of M is said to be closed under extension if, for all filters G,H of M such that G⊆H, G∈F implies H∈F.A classC of fuzzy filters is said to be closed under extension if, for all η,ϑ∈FuFil(M) such that η(1)=θ(1) and η≤θ,  η∈C implies θ∈C.


## 3.2.  Fuzzy Prime Filter

Definition 14.
A nonconstant fuzzy filterμ:M→L is called a fuzzy prime filter if, for every α∈L, μα is a prime filter whenever it is proper.Example 15.
LetL be a lattice:(i)
LetM be the residuated lattice of Example 5; then any fuzzy filter μ:M→L is fuzzy prime.(ii)
LetG be a filter of M, and α<β in L; then (G)αβ is a fuzzy prime filter if and only if G is a prime filter of M.In fact, we have the following result.Theorem 16.
A nonconstant fuzzy filterμ:M→L is fuzzy prime if and only if Im⁡(μ) is a chain, and μ(x∨y)=max⁡{μ(x),μ(y)} for all x,y∈M.Proof.
Letμ be a nonconstant fuzzy filter of M.
Assume thatμ is fuzzy prime; then x∨y∈μμ(x∨y), so μμ(x∨y) is M or a prime filter; thus x∈μμ(x∨y) or y∈μμ(x∨y); this implies μ(x)≥μ(x∨y) or μ(y)≥μ(x∨y); so μ(x)≥μ(y) or μ(x)≤μ(y).
Conversely, assume that the conclusion holds. Letα∈L such that μα is a proper filter, and x,y∈M such that x∨y∈μα. That is, μ(x∨y)≥α; since μ(x∨y)∈{μ(x),μ(y)}, we obtain x∈μα or y∈μα. So μα is a prime filter of M.


## 4. Fuzzyn-Fold Obstinate Filter

Definition 17.
LetM be a residuated lattice, and n≥1: (i)
A proper filterF is n-fold obstinate if, for all x∈L, x∉F if and only if xn¯∈F and xn~∈F.(ii)
M is said to be an n-fold obstinate residuated lattice if, for all x≠1, xn=0.Remark 18.
(i) In Example4, the filter F1 is n-fold obstinate for n≥2, but the filter F is not, since d∉F and dn¯=c∉F for n≥2.
(ii) IfG is an n-fold obstinate filter and x∉G, then xn¯∈G; so xn¯⊗xm=0 for all m≥n, showing that G is a maximal filter.It is easy to show the following result.Proposition 19.
Letn≥1: (i)
A commutative filterG is n-fold obstinate if and only if M/G is n-fold obstinate.(ii)
M is n-fold obstinate if and only if {1} is an n-fold obstinate filter.Definition 20.
Letn≥1.
A proper fuzzy filterμ of M is said to be fuzzy n-fold obstinate if, for all α∈L, μα is an n-fold obstinate filter of M whenever it is proper.Example 21.
LetM be the residuated lattice of Example 4 and L a lattice. Then, for all α<β in L, the fuzzy filter (F1)αβ is a fuzzy n-fold obstinate filter for n≥2, but the fuzzy filter θ≔(F)αβ is not, since d∉θθ(1) and dn¯∉θθ(1) for n≥1.Theorem 22.
A fuzzy filterμ:M→L is fuzzy n-fold obstinate if and only if, for any x∈M,  μ(x)≠μ(1) if and only if μ(xn¯)∧μ(xn~)=μ(1).Proof.
We first note that, for a proper filterF and x∈M, xn¯∈F (resp., xn~∈F) implies x∉F.
(
⇒
)
μ
(
x
)
≠
μ
(
1
) implies that μ(x)⪇μ(1); so x∉μμ(1), and the latter is a proper filter as μ is proper; so xn¯,xn~∈μμ(1), and μ(xn¯)=μ(1)=μ(xn~).
(
⇐
) Let α∈L such that μα is a proper filter of M. Let x∈M such that x∉μα; since 1∈μα, we have μ(x)≠μ(1); so μ(xn¯)∧μ(xn~)=μ(1), and xn¯,xn~∈μμ(1)⊆μα. Thus μα is an n-fold obstinate filter of M.Remark 23.
Sincexn¯≤xn+1¯ and xn~≤xn+1~, any fuzzy n-fold obstinate filter is fuzzy (n+1)-fold obstinate.Corollary 24.
A proper filterF of M is n-fold obstinate if and only if, for all α<β in L, (F)αβ is a fuzzy n-fold obstinate filter of M. In particular, M is n-fold obstinate if and only if ({1})αβ is a fuzzy n-fold obstinate filter.Definition 25 (see [25]).
(i) A filterF of M is n-fold Boolean if x∨xn~∈F and x∨xn¯∈F for all x∈M.
(ii) A residuated latticeM is n-fold Boolean if xn¯∨x=1=xn~∨x for all x∈M.Example 26.
LetM={0,a,b,c,1} be a lattice such that 0<a<b<c<1. Define the operations ⊗, →, and ⇝ by the three tables in (4). Then M is a residuated lattice which is not a pseudo-BL-algebra, since (c→b)⊗c=c⊗c=a≠b=b∧c. Consider(4)
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1F1={1} is the only proper filter of M and it is an n-fold Boolean filter for n≥3.
M is an n-fold Boolean residuated lattice for all n≥3.
One can observe thatM is not a Boolean algebra since a∨a¯=c≠1.Definition 27.
A fuzzy filterθ is fuzzy n-fold Boolean if, for all x∈M, θ((x∨xn~)∧(x∨xn¯))=θ(1).
In particular, a fuzzy 1-fold Boolean filter is a fuzzy Boolean filter [12].Proposition 28.
The following conditions are equivalent for any fuzzy filterθ and any n≥1:(i)
θ is fuzzy n-fold Boolean.(ii)
For allx∈M, θ(xn¯→x)∨θ(xn~⇝x)≤θ(x).(iii)
For allx∈M, θ(xn¯⇝x)∨θ(xn~→x)≤θ(x).(iv)
For allx,y∈M, θ((xn→y)⇝x)∨θ((xn⇝y)→x)≤θ(x).(v)
For allx,y∈M, θ((xn→y)→x)∨θ((xn⇝y)⇝x)≤θ(x).(vi)
For allα∈L, θα≠∅ implies θα is an n-fold Boolean filter.Proof.
(
i
)
⇒
(
i
i
) Suppose that θ is fuzzy n-fold Boolean. Let x∈M. Since (x∨xn~)⇝x=(x⇝x)∧(xn~⇝x)=xn~⇝x, we have θ(xn~⇝x)∧θ(x∨xn~)=θ((x∨xn~)⇝x)∧θ(x∨xn~)≤θ(x), so θ(xn~⇝x)≤θ(x), as θ is fuzzy n-fold Boolean. Similarly, θ(xn¯→x)≤θ(x) and then θ(xn¯→x)∨θ(xn~⇝x)≤θ(x).
(
i
i
)
⇒
(
i
) Let x∈L and t=x∨xn¯; and we must show that θ(t)=θ(1). Since x≤t, we have xn≤tn and then tn¯≤xn¯≤xn¯∨x=t. Hence tn¯→t=1 and θ(tn¯→t)=θ(1); from this, we have θ(1)=θ(tn¯→t)∨θ(tn~⇝t)≤θ(t), by (ii) and θ(x∨xn¯)=θ(t)=θ(1). Similarly, we can prove that θ(x∨xn~)=θ(1).
(
i
)
⇔
(
i
i
i
) By the same arguments as for (i)⇔(ii).
(
i
i
i
)
⇒
(
i
v
) Let x,y∈M. By Proposition 1(4), we have xn⇝y≥xn~ and then (xn⇝y)→x≤xn~→x; this implies θ((xn⇝y)→x)≤θ(xn~→x) (a).
Similarly, we show thatθ((xn→y)⇝x)≤θ(xn¯⇝x) (b).
Now the result follows from (a), (b), and (iii)
(
i
v
)
⇒
(
i
i
i
) Let y=0 in conditions (iv).
(
i
i
)
⇔
(
v
) It is similar to the proof of (iii)⇔(iv).
(
i
)
⇔
(
v
i
) It follows from Definition 27.Theorem 29.
The following assertions are equivalent for a fuzzy filterμ:M→L:(i)
μ is fuzzy n-fold obstinate.(ii)
μ is fuzzy n-fold Boolean and fuzzy prime.Proof.
(
i
)
⇒
(
i
i
) From Remark 18 and Definitions 17 and 25, if α∈L with μα a proper filter, then μα is n-fold obstinate, hence n-fold Boolean and maximal, and thus n-fold Boolean and prime. So μ is fuzzy n-fold Boolean and fuzzy prime.
(
i
i
)
⇒
(
i
) Let α∈L such that μα is a proper filter. Then μα is prime and n-fold Boolean. For each x∈M, we have x∨xn¯, x∨xn~∈μα; so x∉μα if and only if xn¯,xn~∈μα.Remark 30.
Since(x∨xn~)∧(x∨xn¯)≤(x∨xn+1~)∧(x∨xn+1¯), it follows that any fuzzy n-fold Boolean filter is also fuzzy (n+1)-fold Boolean.


## 5. Fuzzyn-Fold Implicative Filter

Definition 31 (see [25]).
LetM be a residuated lattice.
(i) A filterF of M is said to be n-fold implicative if (xn→x2n)∧(xn⇝x2n)∈F for all x,y∈M.
In particular a 1-fold implicative filter is an implicative filter.
(ii)M is said to be n-fold implicative if {1} is an n-fold implicative filter.It is easy to see thatM is n-fold implicative iff, for all x∈M, xn=xn+1.Theorem 32.
The following conditions are equivalent for a filterF of M and n≥1:(i)
F is n-fold implicative.(ii)
∀x∈M, (xn→xn+1)∧(xn⇝xn+1)∈F.(iii)
∀x,y∈M, xn+1→y∈F implies xn→y∈F, and xn+1⇝y∈F implies xn⇝y∈F.Proof.
We give the proof for the binary operation→, since the case of ⇝ is similar.
(
i
)
⇒
(
i
i
) Note that x2n≤xn+1; so xn→x2n≤xn→xn+1. Thus, xn→xn+1∈F.
(
i
i
)
→
(
i
i
i
)  xn→xn+1≤(xn+1→y)⇝(xn→y), by Proposition 1(12). So from (ii) and the definition of a filter, we obtain that xn+1→y∈F implies xn→y∈F.
(
i
i
i
)
⇒
(
i
) Since xn+1→xn+1=1∈F, we have xn→xn+1∈F.
From Proposition1((4) and (14)), we have u→v≤u⊗x→v⊗x, and taking u=xn and v=xn+1 we obtain xn→xn+1≤xn+1→xn+2, showing that xn+1→xn+2∈F. Then (xn+1→xn+2)⊗(xn→xn+1)∈F. Now by Proposition 1(3), we have xn→xn+2. By repeating this process, we obtain that xn→x2n∈F.Corollary 33.
(i) Anyn-fold implicative filter is (n+1)-fold implicative.
(ii) Anyn-fold Boolean filter in n-fold implicative.Proof.
(i) This clearly appear in the proof of(iii)⇒(i) above.
(ii) SupposeF is an n-fold Boolean filter; then x∨xn¯∈F for all x∈M. Now, (x∨xn¯)→(xn→xn+1)=[x→(xn→xn+1)]∧[xn¯→(xn→xn+1)]=(xn+1→xn+1)∧(0→xn+1) = xn+1→xn+1=1. Thus xn→xn+1∈F.
The case ofxn⇝xn+1∈F is similar.The converse of the above lemma is not true, as illustrated by the following example.Example 34.
LetM be the residuated lattice of Example 4:(i)
The filterF={1} is n-fold implicative for all n≥2. But F is not implicative, since a→a2=a→0=c≠1.(ii)
The filter{1} is n-fold implicative for n≥2, but not n-fold Boolean, since d∨dn¯=d≠1.Definition 35.
LetM be a residuated lattice.
A fuzzy subsetθ is called a fuzzy n-fold implicative filter if it satisfies the following conditions:(i)
For allx∈M, θ(x)≤θ(1).(ii)
For allx,y,z∈M, θ((xn⊗y)⇝z)∧θ(xn⇝y)≤θ(xn⇝z).(iii)
For allx,y,z∈M, θ((y⊗xn)→z)∧θ(xn→y)≤θ(xn→z).By takingx=1 in the definition, we see by Proposition 10 that any fuzzy n-fold implicative filter is a fuzzy filter.Proposition 36.
Letθ be a fuzzy filter. Then, the following conditions are equivalent:(i)
θ is a fuzzy n-fold implicative filter.(ii)
For allx∈M, θ((xn→x2n)∧(xn⇝x2n))=θ(1).(iii)
For allx∈M, θ((xn→xn+1)∧(xn⇝xn+1))=θ(1).(iv)
For allx,y∈M, θ(xn+1→y)≤θ(xn→y) and θ(xn+1⇝y)≤θ(xn⇝y).(v)
For allα∈L, θα≠∅ implies θα is an n-fold implicative filter.Proof.
(
i
)
⇒
(
i
i
) Let y=xn and z=x2n in condition (i).
(
i
i
)
⇒
(
i
i
i
) We have xn→x2n≤(x2n→xn+1)⇝(xn→xn+1) = xn→xn+1. So θ(xn→xn+1)=θ(1). By a similar argument, we have θ(xn⇝xn+1)=θ(1).
(
i
i
i
)
⇒
(
i
v
) Clearly, xn→xn+1≤(xn+1→y)⇝(xn→y); so θ(1)=θ(xn→xn+1)≤θ((xn+1→y)⇝(xn→y)), showing that θ(xn+1→y)≤θ(xn→y) by Proposition 11. A similar argument shows that θ(xn+1⇝y)≤θ(xn⇝y).
(
i
v
)
⇒
(
i
i
) Note that x2n→x2n=xn+1→(xn-1→x2n); thus θ(1)=θ(x2n→x2n)=θ(xn+1→(xn-1→x2n))≤θ(xn→(xn-1→x2n))=θ(x2n-1→x2n).
Ifn=1, then θ(x→x2)=θ(1); and similarly, θ(x⇝x2)=θ(1).
Ifn≥2, then θ(1)=θ(x2n-1→x2n)=θ(xn+1→(xn-2→x2n))≤θ(xn→(xn-2→x2n))=θ(x2n-2→x2n). By iteration, we obtain θ1=θx2n-2→x2n≤θx2n-3→x2n≤⋯≤θ(x2n-(n-1)→x2n)=θ(xn+1→x2n)≤θ(xn→x2n). So θ(xn→x2n)=θ(1).
A similar argument givesθ(xn⇝x2n)=θ(1).
(
i
i
)
⇒
(
v
) Let θα be a filter; then for each x∈M, θ(xn→x2n)∧(xn⇝x2n)=θ(1)≥α; and (xn→x2n)∧(xn⇝x2n)∈θα.
(
v
)
⇒
(
i
) Let x,y,z∈M, and α = θ(xn⇝y)∧θxn⊗y⇝z=θxn⇝y⊗xn⊗y⇝z. Then, (xn⇝y)⊗xn⊗y⇝z=(xn⇝y)⊗y⇝xn⇝z≤xn⇝xn⇝z=x2n⇝z and x2n⇝z∈θα. Now, xn⇝x2n∈θα, and xn⇝x2n≤(x2n⇝z)→(xn⇝z); thus xn⇝z∈θα.
A similar argument shows thatθ((y⊗xn)→z)∧θ(xn→y)≤θ(xn→z).From Proposition28, Corollary 33, and Proposition 36, we have the following.Corollary 37.
(i) Any fuzzyn-fold implicative filter is fuzzy (n+1)-fold implicative.
(ii) Any fuzzyn-fold Boolean filter is fuzzy n-fold implicative.The converse of this corollary is not true, as it can be seen from Example38.Example 38.
LetM be the residuated lattice of Example 4, L a lattice, and α≤β in L:(i)
The fuzzy filterθ≔({1})αβ is a fuzzy n-fold implicative filter for all n≥2, but not a fuzzy implicative filter, since θ(a→a2)=θ(a→0)=θ(c)=α≠β=θ(1).(ii)
θ is not n-fold Boolean, since θ(d∨dn¯)=θ(d)=α≠β=θ(1).


## 6. Fuzzyn-Fold Extended Involutive Filter

Definition 39 (see [25]).
LetM be a residuated lattice, and n≥1.
(i) A filterF is an n-fold normal filter if it satisfies the following conditions for all x,y∈M:(xn⇝y)→y∈F implies (y⇝x)→x∈F,(xn→y)⇝y∈F implies (y→x)⇝x∈F.
(ii)M is an n-fold normal residuated lattice if it satisfies the following conditions:yn→x=x implies y=x→y, and yn⇝x=x implies y=x⇝y.Takingx=0 in the above definition, we have the following.Definition 40.
(i) A filterF is called an n-fold extended involutive filter of the residuated lattice M if xn¯~∈F implies x∈F and xn~¯∈F implies x∈F, for each x∈M.
(ii)M is called an n-fold extended involutive residuated lattice if, for each x∈M, xn¯~=1 implies x=1 and xn~¯=1 implies x=1.So, anyn-fold normal filter is an n-fold extended involutive filter.Proposition 41 (see [25]).
LetF be a commutative filter of M. Then F is an n-fold normal filter if and only if M/F is an n-fold normal residuated lattice.From the above result, it is easy to see that a residuated latticeM is n-fold normal (resp., n-fold extended involutive) if and only if {1} is an n-fold normal (resp., n-fold extended involutive) filter of M.Example 42.
LetL={0,a,b,c,1} be a lattice such that 0<a<b<c<1. Define the operations ⊗, →, and ⇝ by the three tables in (5). Then L is a residuated lattice which is not a pseudo-BL-algebra, since b⊗(b⇝a)=b⊗b=0≠a=b∧a. Consider(5)
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1F1={1} and F2={1,c} are the only proper filters of L. F2={1,c} is n-fold normal; but F1={1} is not n-fold normal since (cn→0)⇝0=1∈{1} and (0→c)⇝c=c∉{1}.


### 6.1. Fuzzyn-Fold Normal Filter

Definition 43.
Letn≥1. A fuzzy filter θ is called a fuzzy n-fold normal filter if, for any α∈L, θα≠∅ implies θα is an n-fold normal filter.Proposition 44.
Letθ be a fuzzy filter. Then, θ is a fuzzy n-fold normal filter if and only if, for all x,y∈M, θ((yn⇝x)→x)≤θ((x⇝y)→y) and θ((yn→x)⇝x)≤θ((x→y)⇝y).Proof.
Letθ be a fuzzy filter.
(
⇒
) Assume that θ is a fuzzy n-fold normal filter; let x,y∈M. Then F=θθ((yn⇝x)→x) and G=θθ((yn→x)⇝x) are n-fold normal filters. Now (yn⇝x)→x∈θθ((yn⇝x)→x)=F implies (x⇝y)→y∈F, and (yn→x)⇝x∈G implies (x→y)⇝y∈G. So, θ((yn⇝x)→x)≤θ((x⇝y)→y) and θ((yn→x)⇝x)≤θ((x→y)⇝y).
(
⇐
) Let α∈L such that θα≠∅. If (yn⇝x)→x∈θα, then θ((x⇝y)→y)≥θ((yn⇝x)→x) implies that (x⇝y)→y∈θα.
By a similar argument, we show that(yn→x)⇝x∈θα implies (x→y)⇝y∈θα.From the above proposition and Proposition41, we have the following result.Corollary 45.
(i) A commutative fuzzy filterμ is fuzzy n-fold normal if and only if M/μμ(1) is an n-fold normal residuated lattice.
(ii) A filterG of M is n-fold normal if and only if, for all α<β in L, (G)αβ is a fuzzy n-fold normal filter of M.
(iii)M is an n-fold normal residuated lattice if and only if ({1})01 is a fuzzy n-fold normal filter of M.Takingx=0 in Proposition 44, we have the following.Definition 46.
Letθ be a fuzzy filter.
θ is called a fuzzy n-fold extended involutive filter if θ(xn¯~)∨θ(xn~¯)≤θ(x), for each x∈M.So, any fuzzyn-fold normal filter is a fuzzy n-fold extended involutive filter.Example 47.
LetM be the residuated lattice of Example 42, L a lattice, and α≤β in L. The fuzzy filter ({1,c})αβ is a fuzzy n-fold normal filter and is hence fuzzy n-fold extended involutive; but the fuzzy filter δ≔({1})αβ is not a fuzzy n-fold extended involutive filter, since δ(cn¯~)∨δ(cn~¯)=β≰α=δ(c) for n≥1.


### 6.2.  Fuzzyn-Fold Fantastic Filter

Definition 48 (see [25]).
LetF be a filter of M, and n≥1.
(i)F is n-fold fantastic if, for all x,y∈M, ([(xn→y)⇝y]→(x∨y))∧([(xn⇝y)→y]⇝(x∨y))∈F.
In particular, any 1-fold fantastic filter is a fantastic filter.
(ii)M is called an n-fold fantastic residuated lattice if {1} is an n-fold fantastic filter of M.It is known that [25] any n-fold fantastic filter is n-fold normal; and a filter F is n-fold Boolean iff it is n-fold fantastic and n-fold implicative.Takingy=0 in the above definition, we have the following.Definition 49.
Letn≥1:(i)
F is an n-fold involutive filter of M if (xn¯~→x)∧(xn~¯⇝x)∈F, for all x∈M. In particular, any 1-fold involutive filter is an involutive filter.(ii)
M is called an n-fold involutive residuated lattice if {1} is an n-fold involutive filter of M.Example 50.
The filterF4={1,a,b,c,d,e} of the residuated lattice of Example 5 is an n-fold fantastic filter and is hence n-fold involutive.Definition 51.
Letn≥1. θ is a fuzzy n-fold fantastic filter if it satisfies the following conditions for all x,y∈M:(i)
θ(x)≤θ(1).(ii)
θ(y→x)≤θ([(xn→y)⇝y]→x), and θ(y⇝x)≤θ([(xn⇝y)→y]⇝x).
In particular, any fuzzy 1-fold fantastic filter is a fuzzy fantastic filter.
IfM is commutative, a fuzzy fantastic filter is called a fuzzy MV-filter [22].Example 52.
LetM be the residuated lattice of Example 5, L a lattice, and α<β in L. The fuzzy filter ({1,a,b,c,d,e})αβ is fuzzy n-fold fantastic, hence is fuzzy n-fold involutive. But the fuzzy filter η≔({1})αβ is not fuzzy n-fold fantastic because η(0→a)=β≰α=η([(an→0)⇝0]→a).Theorem 53.
Letn≥1; the following conditions are equivalent for a fuzzy filter θ of M:(i)
θ is fuzzy n-fold fantastic.(ii)
For allx,y∈M, θ([(xn→y)⇝y]→(x∨y))=θ(1) and θ([(xn⇝y)→y]⇝(x∨y))=θ(1).(iii)
Every nonemptyα-cut of θ is an n-fold fantastic filter.Proof.
(
i
)
⇒
(
i
i
) Assume that θ is fuzzy n-fold fantastic. Since x,y≤x∨y, we have θ(1)=θ(y→(x∨y))≤θ([((x∨y)n→y)⇝y]→(x∨y))≤θ([(xn→y)⇝y]→(x∨y))≤θ(1). By a similar argument, we have θ([(xn⇝y)→y]⇝(x∨y))=θ(1).
(
i
i
)
⇒
(
i
i
i
) If 1∈θα, then θ([(xn⇝y)→y]⇝(x∨y))=θ(1)≥α, showing that [(xn⇝y)→y]⇝(x∨y)∈θα. Similarly, we have [(xn→y)⇝y]→(x∨y)∈θα.
(
i
i
i
)
⇒
(
i
) Let x,y∈M; then y→x∈θθ(y→x)=F; so F is an n-fold fantastic filter, and [(xn→y)⇝y]→(x∨y)∈F. By Proposition 1(12), [(xn→y)⇝y]→(x∨y) ≤ ((x∨y)→x)⇝([(xn→y)⇝y]→x)=(y→x)⇝([(xn→y)⇝y]→x); so the latter belongs to F, and [(xn→y)⇝y]→x∈F; this means that θ([(xn→y)⇝y]→x)≥θ(y→x).
Similarly, we show thatθ([(xn⇝y)→y]⇝x)≥θ(y⇝x).From Proposition1(4) and the fact that xn+1≤xn, one easily see that any fuzzy n-fold fantastic filter is fuzzy (n+1)-fold fantastic.Definition 54.
Letθ be a fuzzy filter of M.
θ is called a fuzzy n-fold involutive filter if θ(xn¯~→x)=θ(1)=θ(xn~¯⇝x), for all x∈M.Corollary 55.
Letθ be a fuzzy filter of M:(i)
Ifθ is fuzzy n-fold fantastic, then it is fuzzy n-fold normal and fuzzy n-fold involutive.(ii)
θ is fuzzy n-fold Boolean if and only if it is fuzzy n-fold fantastic and fuzzy n-fold implicative.Proof.
This follows from the observation after Definition48.


## 6.1. Fuzzyn-Fold Normal Filter

Definition 43.
Letn≥1. A fuzzy filter θ is called a fuzzy n-fold normal filter if, for any α∈L, θα≠∅ implies θα is an n-fold normal filter.Proposition 44.
Letθ be a fuzzy filter. Then, θ is a fuzzy n-fold normal filter if and only if, for all x,y∈M, θ((yn⇝x)→x)≤θ((x⇝y)→y) and θ((yn→x)⇝x)≤θ((x→y)⇝y).Proof.
Letθ be a fuzzy filter.
(
⇒
) Assume that θ is a fuzzy n-fold normal filter; let x,y∈M. Then F=θθ((yn⇝x)→x) and G=θθ((yn→x)⇝x) are n-fold normal filters. Now (yn⇝x)→x∈θθ((yn⇝x)→x)=F implies (x⇝y)→y∈F, and (yn→x)⇝x∈G implies (x→y)⇝y∈G. So, θ((yn⇝x)→x)≤θ((x⇝y)→y) and θ((yn→x)⇝x)≤θ((x→y)⇝y).
(
⇐
) Let α∈L such that θα≠∅. If (yn⇝x)→x∈θα, then θ((x⇝y)→y)≥θ((yn⇝x)→x) implies that (x⇝y)→y∈θα.
By a similar argument, we show that(yn→x)⇝x∈θα implies (x→y)⇝y∈θα.From the above proposition and Proposition41, we have the following result.Corollary 45.
(i) A commutative fuzzy filterμ is fuzzy n-fold normal if and only if M/μμ(1) is an n-fold normal residuated lattice.
(ii) A filterG of M is n-fold normal if and only if, for all α<β in L, (G)αβ is a fuzzy n-fold normal filter of M.
(iii)M is an n-fold normal residuated lattice if and only if ({1})01 is a fuzzy n-fold normal filter of M.Takingx=0 in Proposition 44, we have the following.Definition 46.
Letθ be a fuzzy filter.
θ is called a fuzzy n-fold extended involutive filter if θ(xn¯~)∨θ(xn~¯)≤θ(x), for each x∈M.So, any fuzzyn-fold normal filter is a fuzzy n-fold extended involutive filter.Example 47.
LetM be the residuated lattice of Example 42, L a lattice, and α≤β in L. The fuzzy filter ({1,c})αβ is a fuzzy n-fold normal filter and is hence fuzzy n-fold extended involutive; but the fuzzy filter δ≔({1})αβ is not a fuzzy n-fold extended involutive filter, since δ(cn¯~)∨δ(cn~¯)=β≰α=δ(c) for n≥1.


## 6.2.  Fuzzyn-Fold Fantastic Filter

Definition 48 (see [25]).
LetF be a filter of M, and n≥1.
(i)F is n-fold fantastic if, for all x,y∈M, ([(xn→y)⇝y]→(x∨y))∧([(xn⇝y)→y]⇝(x∨y))∈F.
In particular, any 1-fold fantastic filter is a fantastic filter.
(ii)M is called an n-fold fantastic residuated lattice if {1} is an n-fold fantastic filter of M.It is known that [25] any n-fold fantastic filter is n-fold normal; and a filter F is n-fold Boolean iff it is n-fold fantastic and n-fold implicative.Takingy=0 in the above definition, we have the following.Definition 49.
Letn≥1:(i)
F is an n-fold involutive filter of M if (xn¯~→x)∧(xn~¯⇝x)∈F, for all x∈M. In particular, any 1-fold involutive filter is an involutive filter.(ii)
M is called an n-fold involutive residuated lattice if {1} is an n-fold involutive filter of M.Example 50.
The filterF4={1,a,b,c,d,e} of the residuated lattice of Example 5 is an n-fold fantastic filter and is hence n-fold involutive.Definition 51.
Letn≥1. θ is a fuzzy n-fold fantastic filter if it satisfies the following conditions for all x,y∈M:(i)
θ(x)≤θ(1).(ii)
θ(y→x)≤θ([(xn→y)⇝y]→x), and θ(y⇝x)≤θ([(xn⇝y)→y]⇝x).
In particular, any fuzzy 1-fold fantastic filter is a fuzzy fantastic filter.
IfM is commutative, a fuzzy fantastic filter is called a fuzzy MV-filter [22].Example 52.
LetM be the residuated lattice of Example 5, L a lattice, and α<β in L. The fuzzy filter ({1,a,b,c,d,e})αβ is fuzzy n-fold fantastic, hence is fuzzy n-fold involutive. But the fuzzy filter η≔({1})αβ is not fuzzy n-fold fantastic because η(0→a)=β≰α=η([(an→0)⇝0]→a).Theorem 53.
Letn≥1; the following conditions are equivalent for a fuzzy filter θ of M:(i)
θ is fuzzy n-fold fantastic.(ii)
For allx,y∈M, θ([(xn→y)⇝y]→(x∨y))=θ(1) and θ([(xn⇝y)→y]⇝(x∨y))=θ(1).(iii)
Every nonemptyα-cut of θ is an n-fold fantastic filter.Proof.
(
i
)
⇒
(
i
i
) Assume that θ is fuzzy n-fold fantastic. Since x,y≤x∨y, we have θ(1)=θ(y→(x∨y))≤θ([((x∨y)n→y)⇝y]→(x∨y))≤θ([(xn→y)⇝y]→(x∨y))≤θ(1). By a similar argument, we have θ([(xn⇝y)→y]⇝(x∨y))=θ(1).
(
i
i
)
⇒
(
i
i
i
) If 1∈θα, then θ([(xn⇝y)→y]⇝(x∨y))=θ(1)≥α, showing that [(xn⇝y)→y]⇝(x∨y)∈θα. Similarly, we have [(xn→y)⇝y]→(x∨y)∈θα.
(
i
i
i
)
⇒
(
i
) Let x,y∈M; then y→x∈θθ(y→x)=F; so F is an n-fold fantastic filter, and [(xn→y)⇝y]→(x∨y)∈F. By Proposition 1(12), [(xn→y)⇝y]→(x∨y) ≤ ((x∨y)→x)⇝([(xn→y)⇝y]→x)=(y→x)⇝([(xn→y)⇝y]→x); so the latter belongs to F, and [(xn→y)⇝y]→x∈F; this means that θ([(xn→y)⇝y]→x)≥θ(y→x).
Similarly, we show thatθ([(xn⇝y)→y]⇝x)≥θ(y⇝x).From Proposition1(4) and the fact that xn+1≤xn, one easily see that any fuzzy n-fold fantastic filter is fuzzy (n+1)-fold fantastic.Definition 54.
Letθ be a fuzzy filter of M.
θ is called a fuzzy n-fold involutive filter if θ(xn¯~→x)=θ(1)=θ(xn~¯⇝x), for all x∈M.Corollary 55.
Letθ be a fuzzy filter of M:(i)
Ifθ is fuzzy n-fold fantastic, then it is fuzzy n-fold normal and fuzzy n-fold involutive.(ii)
θ is fuzzy n-fold Boolean if and only if it is fuzzy n-fold fantastic and fuzzy n-fold implicative.Proof.
This follows from the observation after Definition48.



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*Source: 386210-2015-10-01.xml*