α-Ideals of Fuzzy LatticesIvan Mezzomo

Department of Mathematical Sciences,Technology and Human – DCETH

Rural Federal University of SemiArid – UFERSAAngicos – RN, Brazil, 59.515-000 and

Department of Informaticsand Applied Mathematics – DIMAp

Federal University of Rio Grande do Norte – UFRNNatal – Rio Grande do Norte, Brazil, 59.072-970

Email: imezzomo@ufersa.edu.br

Benjamin Bedregal andRegivan H. N. Santiago

Group for Logic, Language, Information,Theory and Applications - LOLITA

Department of Informatics andApplied Mathematics – DIMAp

Federal University of Rio Grande do Norte – UFRNNatal – Rio Grande do Norte, Brazil, 59.072-970

Email: {bedregal, regivan}@dimap.ufrn.br

Abstract—We consider the fuzzy lattice notion introduced byChon (Korean J. Math 17 (2009), No. 4, 361-374), define an α-ideals and α-filters for fuzzy lattices and characterize α-idealsand α-filters of fuzzy lattices by using its support and its levelset. Moreover, we prove some similar properties to the classicaltheory of α-ideals and α-filters, such as, the class of α-ideals andα-filters are closed under union and intersection.

I. INTRODUCTION

The concept of fuzzy set was introduced by Zadeh [21]which in his seminal paper also defined the notion of fuzzyrelations. From then, several mathematical concepts such asnumber, group, topology, differential equation, etc., had beenfuzzified. In particular for the case of order and lattice notionsdifferent definitions has been proposed, for example [3], [4],[5], [8], [10].

Yuan and Wu [20] introduced the concepts of fuzzy sub-lattices and fuzzy ideals of a lattice. Ajmal and Thomas [1]defined a fuzzy lattice as a fuzzy algebra and characterizedfuzzy sublattices. In 2009, Chon [6] characterized a fuzzypartial order relation using its level set and defined a fuzzylattice as a fuzzy relation, he also discovered some basicproperties of fuzzy lattices and showed that a fuzzy totallyordered set is a distributive fuzzy lattice. Recently, in paper[14], we define fuzzy ideals and fuzzy filters of a fuzzy lattice(X,A), in the sense of Chon [6], as a crisp set Y ⊆ Xendowed with the fuzzy order A|Y×Y . In paper [15], we defineboth ideal and filter of a fuzzy lattice (X,A) and some kindsof ideals and filters, we also study the intersection of familiesfor each kind of ideal and filter together with some of itsconsequences. Finally, in paper [16], we define a new notionof fuzzy ideal and fuzzy filter for fuzzy lattice and definesome types of fuzzy ideals and fuzzy filters of fuzzy lattice,such as, fuzzy principal ideals (filters), proper fuzzy ideals(filters), fuzzy prime ideals (filters) and fuzzy maximal ideals(filters). In addition, we prove some properties analogous theclassical theory of fuzzy ideals (filters), such as, the class ofproper fuzzy ideals (filters) is closed under fuzzy union andfuzzy intersection. As a step forward of such investigations, we

define α-ideals and α-filters of fuzzy lattices using the fuzzypartial order relation and fuzzy lattices defined by Chon.

In section II, we provide preliminary results on somebasic concepts like ideal, filter and lattice. In section III, weconsider Chon’s approach [6] on fuzzy partial order relation.We also characterize, a fuzzy lattice (X,A) as a classical setX under a fuzzy partial order relation A and fuzzy ideals andfuzzy filters of fuzzy lattice via its p-level set and its support.In section IV, we define α-ideal and α-filter of fuzzy lattice,characterize an α-ideal of fuzzy lattice, using its supportand its level set, and study some properties analogous to theclassical theory of α-ideals and α-filters, such as, the class ofα-ideals and α-filters are closed under union and intersection.

II. PRELIMINARIES

In this section, we will briefly review some basic conceptsof lattices, ideals and filters both from the algebraic theoreticalview and order as necessary for the development. This presen-tation is quite introductory and can be found in many bookson lattice theory. The reader familiar with such concepts canproceed to the next section.

When exist the top and bottom elements of a set P , theyare denoted by > and ⊥, respectively.

Definition 2.1 ([7], Definition 2.4): Let (P,≤) be a non-empty partially ordered set.(i) If sup{x, y} and inf{x, y} exist for all x, y ∈ P , then(P,≤) is called a lattice.(ii) If supS and inf S exist for all S ⊆ P , then (P,≤) iscalled a complete lattice.

We introduced lattices as ordered sets of a special type.However, we may adopt an alternative viewpoint. Given alattice L = (L,≤), we may define binary operations called:join and meet on L by x∨y = sup{x, y} and x∧y = inf{x, y}.

Lemma 2.1 ([7], Lemma 2.8): Let L be a lattice and letx, y ∈ L. Then the following are equivalent:

157978-1-4799-0348-1/13/$31.00 ©2013 IEEE

(i) x ≤ y;(ii) x ∨ y = y;(iii) x ∧ y = x.

We have shown that lattices can be completely characterizedin terms of the join and meet operations. We may henceforthsay “let L be a lattice”, replacing L by (L,≤) or by (L,∨,∧)if we want to emphasize that we are thinking of it as a specialkind of ordered set or as an algebraic structure.

It may happen that (L,≤) has top and bottom elements.When thinking of L as (L,∨,∧), it is appropriate to viewthese elements from a more algebraic standpoint.

Definition 2.2 ([7], Definition 2.12): Let L be a lattice.We say L has a top element if there exists 1 ∈ L such thata = a ∧ 1 for all x ∈ L. Dually, we say L has a bottomelement if there exists 0 ∈ L such that x = x ∨ 0 for allx ∈ L. The lattice (L,∨,∧) has a 1 iff (L,≤) has a topelement > and, in that case, 1 = >. A dual statement holdsfor 0 and ⊥. A lattice (L,∨,∧) possessing 0 and 1 is calledbounded.

A finite lattice is automatically bounded, with 1 = supLand 0 = inf L.

In [12] was defined ideals and filters of a lattice L. LetL be a nonempty set and L = (L,∧,∨, 0, 1) stand for abounded distributive lattice.

Definition 2.3 ([7], Definition 2.20): A nonempty subset Iof L is called an ideal of L if for all x, y ∈ L(i) if y ∈ I with x ≤ y, then x ∈ I .(ii) x, y ∈ I implies x ∨ y ∈ I .

Definition 2.4 ([7], Definition 2.21): A nonempty subset Fof L is called a filter of L if for all x, y ∈ L(i) if y ∈ F with y ≤ x, then x ∈ F .(ii) x, y ∈ F implies x ∧ y ∈ F .

III. FUZZY LATTICES

In this section we define a fuzzy lattice as a fuzzy partialorder relation and develop some properties for them.

Let X be a universal set. A fuzzy set A on X is a functionµA : X → [0, 1], where [0, 1] means real numbers between 0and 1 (including 0 and 1). Given two fuzzy set A and B onX , we say that A ⊆ B if, for all x ∈ X , µA(x) ≤ µB(x). Inparticular, we define the fuzzy empty set ∅ on X by µ∅(x) = 0

and we define the fuzzy universe set X on X by µX(x) = 1for all x ∈ X . For more detailed study refer to [13], [21].

Let X and Y be nonempty sets and x ∈ X and y ∈ Y . Afuzzy relation A is a mapping from the Cartesian space X×Yto the interval [0, 1], where the strength of the mapping isexpressed by the membership function of the relation A, thatis, A : X × Y → [0, 1]. If X = Y then we say that A is abinary fuzzy relation on X .

Let X be a nonempty set and x, y, z ∈ X . A fuzzy binaryrelation A in X is reflexive if A(x, x) = 1 for all x ∈ X ,A is symmetric if A(x, y) = A(y, x) for any x, y ∈ X , Ais transitive if A(x, z) ≥ sup

y∈Xmin[A(x, y), A(y, z)], and A

is antisymmetric if A(x, y) > 0 and A(y, x) > 0 impliesx = y. These definitions of reflexivity, symmetry, transitivityand antisymmetry can be found in several books and papersas [6], [11], [13], [22].

A function A : X × X −→ [0, 1] is called a fuzzyequivalence relation in X if A is reflexive, transitive andsymmetric.

A fuzzy relation A is a fuzzy partial order relation if A isreflexive, antisymmetric and transitive. A fuzzy partial orderrelation A is a fuzzy total order relation if A(x, y) > 0 orA(y, x) > 0 for all x, y ∈ X . If A is a fuzzy partial orderrelation on a set X , then (X,A) is called a fuzzy partiallyordered set or fuzzy poset. If A is a fuzzy total order relationin a set X , then (X,A) is called fuzzy totally ordered set ora fuzzy chain. For more detailed study refer to [6].

In the literature there are several other ways to define afuzzy which is reflexive, symmetric or transitive relation c.f.[8], [9]. Also, we can find several other forms to define fuzzypartial order relations, as we can be see in [3], [4], [19].

Remark 3.1: When A is reflexive, then thetransitivity property can be rewritten by replacingthe ”≥” by ”=”. In other words, A is transitive iffA(x, z) = sup

y∈Xmin[A(x, y), A(y, z)], for all x, y, z ∈ X .

The statement that is claimed in the lastremark can be easily proved. First, we know thatA(x, z) ≥ supy∈Xmin[A(x, y), A(y, z)] and second, trivially,supy∈Xmin[A(x, y), A(y, z)] ≥ min[A(x, x), A(x, z)] =min[1, A(x, z)] = A(x, z). Therefore, we have thatA(x, z) = supy∈Xmin[A(x, y), A(y, z)].

Proposition 3.1 ([16] Proposition 3.1): Let (X,A) befuzzy poset and x, y, z ∈ X . If A(x, y) > 0 and A(y, z) > 0,then A(x, z) > 0.

Now we define a fuzzy lattice as a fuzzy partial order anddevelop some properties of fuzzy lattices.

Definition 3.1 ([6], Definition 3.1): Let (X,A) be afuzzy poset and let Y ⊆ X . An element u ∈ X is saidto be an upper bound for a subset Y if A(y, u) > 0 forall y ∈ Y . An upper bound u0 for Y is the least upperbound (or supremum) of Y if A(u0, u) > 0 for everyupper bound u for Y . An element v ∈ X is said to be alower bound for a subset Y if A(v, y) > 0 for all y ∈ Y .A lower bound v0 for Y is the greatest lower bound (orinfimum) of Y if A(v, v0) > 0 for every lower bound v for Y .

The least upper bound of Y will be denoted by supY orLUB Y and the greatest lower bound by inf Y or GLB Y .

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We denote the least upper bound of the set {x, y} by x ∨ yand denote the greatest lower bound of the set {x, y} by x∧y.

Remark 3.2 ([14] Remark 3.2): Since A is antisymmetric,then the least upper (greatest lower) bound, if it exists, isunique.

Definition 3.2 ([6], Definition 3.2): A fuzzy poset (X,A)is called a fuzzy lattice if x∨y and x∧y exist for all x, y ∈ X .

Example 3.1: Let X = {x, y, z, w} and let A : X×X −→[0, 1] be a fuzzy relation such that A(x, x) = A(y, y) =A(z, z) = A(w,w) = 1, A(x, y) = A(x, z) = A(x,w) =A(y, z) = A(y, w) = A(z, w) = 0, A(y, x) = 0.3, A(z, x) =0.5, A(w, x) = 0.8, A(z, y) = 0.2, A(w, y) = 0.4, andA(w, z) = 0.1. Then it is easily to verify that A is a fuzzytotal order relation. Also, x ∨ y = x, x ∨ z = x, x ∨ w =x, y ∨ z = y, y ∨ w = y, z ∨ w = z, x ∧ y = y, x ∧ z =z, x ∧ w = w, y ∧ z = z, y ∧ w = w, and z ∧ w = w. Thefollowing diagram show us the fuzzy order relation.

x

y

0.3

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z

0.5

WW//////////////

0.2mm

AF

KOSWZ

w

0.4

WW00000000000000 0.1

>>}}}}}}}

0.8

OO����������

× w z y x

w 1.0 0.1 0.4 0.8z 0.0 1.0 0.2 0.5y 0.0 0.0 1.0 0.3x 0.0 0.0 0.0 1.0

Now, let Y = {z, w} be a subset of X . Then, x, y and z areupper bounds of Y and since A(z, w) = 0 and A(w, z) > 0,the LUB Y is z and the GLB Y is w.

Proposition 3.2 ([6], Proposition 2.2): Let (X,A) be afuzzy poset (or chain) and Y ⊆ X . If B = A|Y×Y , thatis, B is a fuzzy relation on Y such that for all x, y ∈ Y ,B(x, y) = A(x, y), then (Y,B) is a fuzzy poset (or chain).

Definition 3.3 ([14] Definition 3.3): Let (X,A) be a fuzzylattice. (Y,B) is a fuzzy sublattice of (X,A) if Y ⊆ X ,B = A|Y×Y and (Y,B) is a fuzzy lattice.

We define, for any α ∈ (0, 1], the α-level setAα = {(x, y) ∈ X × X : A(x, y) ≥ α} of a fuzzyrelation A and the support of a fuzzy relation A byS(A) = {(x, y) ∈ X ×X : A(x, y) > 0}.

Proposition 3.3 ([14] Proposition 3.2): Let A : X×X −→[0, 1] be a fuzzy relation. Then, A is a fuzzy partial orderrelation on X iff for each α ∈ (0, 1], the α-level set Aα is apartial order relation in X .

Proposition 3.4 ([6], Proposition 3.5): Let A : X × X →[0, 1] be a fuzzy relation and let Aα an α-level set. If (X,Aα)is a lattice for every α with α ∈ (0, 1], then (X,A) is a fuzzylattice.

We can build a fuzzy lattice using the idea of support asfollows:

Lemma 3.1 ([14] Proposition 3.4): Let A : X×X → [0, 1]be a fuzzy relation. If A is a fuzzy partial order relation onX , then S(A) is a partial order relation on X .

Proposition 3.5 ([16] Proposition 3.5 and 3.6): Let(X,A) be a fuzzy lattice, (X,S(A)) be a limited latticeand x, y ∈ X . Then x ∨ y in terms of (X,A) coincideswith x ∨ y in terms of (X,S(A)). On the other hand, x ∧ yin terms of (X,A) coincides with x∧y in terms of (X,S(A)).

Proposition 3.6 ([6], Proposition 3.3): Let (X,A) be afuzzy lattice and let x, y, z ∈ X . Then

1) A(x, x ∨ y) > 0, A(y, x ∨ y) > 0, A(x ∧ y, x) >0, A(x ∧ y, y) > 0.

2) A(x, z) > 0 and A(y, z) > 0 implies A(x ∨ y, z) > 0.3) A(z, x) > 0 and A(z, y) > 0 implies A(z, x ∧ y) > 0.4) A(x, y) > 0 iff x ∨ y = y.5) A(x, y) > 0 iff x ∧ y = x.6) If A(y, z) > 0, then A(x ∧ y, x ∧ z) > 0 and

A(x ∨ y, x ∨ z) > 0.

Corollary 3.1 ([16] Corollary 3.1): Let A : X × X →[0, 1] be a fuzzy relation. If (X,A) is a fuzzy lattice, then(X,S(A)) is a lattice.

For more detailed study we refer to [6], [14], [16].

IV. α-IDEALS AND α-FILTERS

In this section, we propose the notions of α-ideals andα-filters of a fuzzy lattice and characterize them by using itssupport and its level set. we define α-ideals and α-filters of afuzzy lattice as follows:

Definition 4.1: Let (X,A) be a fuzzy lattice, α ∈ (0, 1]and Y ⊆ X . Y is an α-ideal of (X,A),(i) If x ∈ X , y ∈ Y and A(x, y) ≥ α, then x ∈ Y ;(ii) If x, y ∈ Y , then x ∨ y ∈ Y .

Definition 4.2: Let (X,A) be a fuzzy lattice, α ∈ (0, 1]and Y ⊆ X . Y is a α-filter of (X,A),(i) If x ∈ X , y ∈ Y and A(y, x) ≥ α, then x ∈ Y ;(ii) If x, y ∈ Y , then x ∧ y ∈ Y .

Proposition 4.1: If α ≤ β, then any α-ideal is a β-ideal.

Proof: Let Y be a β-ideal and α ≤ β. Then for anyx ∈ X , if A(x, y) ≥ β, then A(x, y) ≥ α, so by Definition

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4.1 (i), x ∈ Y . On the other hand, if x, y ∈ Y , then byDefinition 4.1 (ii), x ∨ y ∈ Y . Therefore, Y is a β-ideal of(X,A).

Dually, we prove that if α ≤ β, then any β-filter is a α-filter.

Remark 4.1: Notice that the set X of fuzzy lattice (X,A)is an α-ideal, for all α ∈ (0, 1]. Dually, the set X of fuzzylattice (X,A) is an α-filter, for all α ∈ (0, 1].

In paper [15], we defined an ideal and a filter of a fuzzylattice (X,A), respectively, as follows:

Definition 4.3 ([15], Definition 41): Let (X,A) be a fuzzylattice and Y ⊆ X . Y is an ideal of (X,A)(i) If x ∈ X , y ∈ Y and A(x, y) > 0, then x ∈ Y ;(ii) If x, y ∈ Y , then x ∨ y ∈ Y .

Definition 4.4 ([15], Definition 42): Let (X,A) be a fuzzylattice and Y ⊆ X . Y is a filter of (X,A)(i) If x ∈ X , y ∈ Y and A(y, x) > 0, then x ∈ Y ;(ii) If x, y ∈ Y , then x ∧ y ∈ Y .

Corollary 4.1: All ideal in the sense of Definition 4.3 isan α-ideal. Dually, all filter in the sense of Definition 4.4 isan α-filter.

Proof: Straightforward from Proposition 4.1.

Proposition 4.2: Let α ∈ (0, 1]. If Y is an ideal of thelattice (X,S(A)), then for all α ∈ (0, 1], Y is an α-ideal offuzzy lattice (X,A).

Proof: Let Y be an ideal of (X,S(A)) and y ∈ Y .Consider α fixed. If (x, y) ∈ S(A), then because Y is an ideal,x ∈ Y . So, trivially satisfy the condition (i) of Definition 4.1and the condition (ii) is satisfied because it does not dependon the value of α.

Proposition 4.3: Let α ∈ (0, 1]. If Y is a filter of thelattice (X,S(A)), then for all α ∈ (0, 1], Y is an α-filter offuzzy lattice (X,A).

Proof: Analogous to Proposition 4.2.

Let Aα be the α-level set Aα = {(x, y) ∈ X × X :A(x, y) ≥ α}, for any α ∈ (0, 1].

Proposition 4.4: Let Y be an α-ideal of (X,A) andB = A|Y×Y . The set Yα = {x ∈ Y : B(x, y) ≥ α for anyy ∈ Y } is an ideal of (X,Aα).

Proof: Straightforward from Definition 4.1.

Theorem 4.1: Let (X,A) be a fuzzy lattice, α ∈ (0, 1]such that (X,Aα) is a lattice. Y ⊆ X is an α-ideal of fuzzylattice (X,A) iff for each α ∈ (0, 1], Yα is an ideal of (X,Aα).

Proof: (⇒) Let Y be an α-ideal of (X,A) and let y ∈ Y .(i) If x ∈ Yα, then exists y ∈ Y such that A(x, y) ≥ α. So,by Definition 4.1 item (i), x ∈ Y .(ii) If x ∈ Y and y ∈ Y , then by Definition 4.1 item (ii),x ∨ y ∈ Y .(⇐) (i) Let x ∈ X and y ∈ Y and suppose that A(x, y) ≥ α,then x ∈ Yα.(ii) Trivially.

Theorem 4.2: Let (X,A) be a fuzzy lattice, α ∈ (0, 1]such that (X,Aα) is a lattice. Y ⊆ X is an α-filter of fuzzylattice (X,A) iff for each α ∈ (0, 1], Yα is a filter of (X,Aα).

Proof: Analogous to Proposition 4.2.

We define the fuzzy sup-lattice and fuzzy inf-lattice asfollow:

Definition 4.5: A fuzzy poset (Y,A) is called fuzzysup-lattice if each pair of element has supremum on Y .Dually, a fuzzy poset (Y,A) is called fuzzy inf-lattice if eachpair of element has infimum on Y .

Notice that a structure is a fuzzy lattice iff it issimultaneously fuzzy sup-lattice and fuzzy inf-lattice.

Proposition 4.5: Let (X,A) be a fuzzy lattice, α ∈ (0, 1],(Y,A) be a fuzzy sup-lattice and Y ⊆ X . The set⇓ Yα = {x ∈ X : A(x, y) ≥ α for some y ∈ Y } is an α-idealof (X,A).

Proof: (i) Let α ∈ (0, 1], z ∈⇓ Yα and w ∈ X suchthat A(w, z) ≥ α. How z ∈⇓ Yα, then exists x ∈ Y suchthat A(z, x) ≥ α, and by transitivity, A(w, x) ≥ α, for someα ∈ (0, 1]. Therefore, w ∈⇓ Yα.(ii) Suppose x, y ∈⇓ Yα, then exist z1, z2 ∈ Y such thatA(x, z1) ≥ α and A(y, z2) ≥ α, for some α ∈ (0, 1]. So,A(x, z1∨z2) ≥ α and A(y, z1∨z2) ≥ α. By hypothesis (Y,A)is a fuzzy sup-lattice, then z1∨z2 ∈ Y and A(x∨y, z1∨z2) ≥α, for some α ∈ (0, 1]. Therefore, x ∨ y ∈⇓ Yα.

Proposition 4.6: Let (X,A) be a fuzzy lattice, α ∈ (0, 1],(Y,A) be a fuzzy inf-lattice and Y ⊆ X . The set⇑ Yα = {x ∈ X : A(y, x) ≥ α for any y ∈ Y } is afilter of (X,A).

Proof: Analogous to Proposition 4.5.

Proposition 4.7: Let (X,A) be a fuzzy lattice and Y ⊆ X ,then ⇓ Yα satisfies the following properties:(i) Y ⊆⇓ Yα

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(ii) Y ⊆W ⇒⇓ Yα ⊆⇓Wα

(iii) ⇓⇓ Yα =⇓ Yα

Proof: (i) If y ∈ Y and how A(y, y) = 1, i.e., A(y, y) ≥α, for any α ∈ (0, 1]. Therefore, y ∈⇓ Yα.(ii) Suppose Y ⊆ W and y ∈⇓ Yα, then by definition, existsz ∈ Y such that A(z, y) ≥ α, for any α ∈ (0, 1]. How Y ⊆W ,then z ∈W and A(z, y) ≥ α. So y ∈⇓Wα.(iii) (⇒) ⇓⇓ Yα ⊆⇓ Yα. Suppose y ∈⇓⇓ Yα, then existsx ∈⇓ Yα such that A(y, x) ≥ α, for any α ∈ (0, 1]. Sincex ∈⇓ Yα, then exists z ∈ Y such that A(x, z) ≥ α. So, bytransitivity, A(y, z) ≥ α. Therefore, y ∈⇓ Yα.(⇐) Straightforward from (i).

Proposition 4.8: Let (X,A) be a fuzzy lattice and Y ⊆ X ,then ⇑ Yα satisfies the following properties:(i) Y ⊆⇑ Yα(ii) Y ⊆W ⇒⇑ Yα ⊆⇑Wα

(iii) ⇑⇑ Yα =⇑ Yα

Proof: Analogous to Proposition 4.7.

Corollary 4.2: Let α ∈ (0, 1]. ⇓ Yα (⇑ Yα) is the lowestα-ideal (α-filter) containing Y .

The family of all α-ideals of a fuzzy lattice (X,A), forsome α ∈ (0, 1], will be denoted by Iα(X). Dually, willdenote by Fα(X) the family of all α-filters of a fuzzy lattice(X,A), for some α ∈ (0, 1].

Proposition 4.9: Let α ∈ (0, 1], Z be a subset of Iα(X)and W be a nonempty set of Iα(X), then(i)

⋂Z ∈ Iα(X);

(ii)⋃W ∈ Iα(X).

Proof: (i) Let Z ⊆ Iα(X), for some α ∈ (0, 1]. Supposex ∈

⋂Z and A(y, x) ≥ α, then x ∈ Zj for all Zj ∈ Z. How

A(y, x) ≥ α, then y ∈ Zj for each Zj ∈ Z. So y ∈⋂Z and

therefore,⋂Z ∈ Iα(X). Notice that if Z is an empty set then⋂

Z = X .(ii) Let W ⊆ Iα(X), for some α ∈ (0, 1]. Suppose x ∈

⋃W

and A(y, x) ≥ α, then exists Wj ∈W such that x ∈Wj , andhow Wj ∈ Iα(X), then y ∈ Wj . So y ∈

⋃W and therefore,⋃

W ∈ Iα(X).

Proposition 4.10: Let α ∈ (0, 1], Z be a subset of Fα(X)and W be a nonempty set of Fα(X), then(i)

⋂Z ∈ Fα(X);

(ii)⋃W ∈ Fα(X).

Proof: Analogous to Proposition 4.9.

Corollary 4.3: X ∈ Iα(X) only if α has the lowest valueof the range (0, 1]. Similarly, X ∈ Fα(X) only if α has thelowest value of the range (0, 1].

We will define a kind of α-ideals of fuzzy lattice calledprincipal α-ideal generated by x ∈ X .

Definition 4.6: Let (X,A) be a fuzzy lattice and x ∈ X .Then, the set defined by ⇓ xα = {y ∈ X : A(y, x) ≥ α,for some α ∈ (0, 1]} is called principal α-ideal of (X,A)generated by x.

Definition 4.7: Let (X,A) be a fuzzy lattice and x ∈ X .Then, the set defined by ⇑ xα = {y ∈ X : A(x, y) ≥ α,for some α ∈ (0, 1]} is called principal α-filter of (X,A)generated by x.

Remark 4.2: Notice ⇓ xα =⇓ {xα} and ⇑ xα = ⇑ {xα}.

The following propositions prove the relation between anα-ideal ⇓ Yα and principal α-ideals ⇓ yα. We denote byPα(X) the set of parts of all α-ideals, α ∈ (0, 1], that is,Iα(X) ⊆ Pα(X) and dually, Fα(X) ⊆ Pα(X), α ∈ (0, 1].

Proposition 4.11: For all Y ∈ Pα(X) and for allα ∈ (0, 1], ⇓ Yα =

⋃y∈Y⇓ yα.

Proof: Let Y ∈ Pα(X), for all α ∈ (0, 1]. Then, x ∈⇓ Yαiff exists y ∈ Y such that A(x, y) ≥ α iff exists y ∈ Y suchthat x ∈⇓ yα iff x ∈

⋃y∈Y⇓ yα.

Proposition 4.12: For all Y ∈ Pα(X) and for allα ∈ (0, 1], ⇑ Yα =

⋃y∈Y⇑ yα.

Proof: Analogous to Proposition 4.12.

V. CONCLUSION

In this paper, we have studied the notion of fuzzy latticeusing a fuzzy order relation and introduced a new notion ofα-ideals and α-filters of fuzzy lattice. We established that theα-ideal theorem of a fuzzy lattice through its level set andits support. We also defined, some properties of α-ideals andα-filters of fuzzy lattice analogous the classical theory. Wecan found several other forms to define fuzzy partial orderrelations, as we can see in [3], [4], [6], [19]. The same way,one should observe that the concept of fuzzy partial order,fuzzy partially ordered set and fuzzy lattice can be found inseveral other forms in the literature.

One of the most promising ideas could be the investigationof operations among fuzzy lattices and its consequences. Asfuture work we consider the idea of Palmeira and Bedregal[17] and Palmeira, Bedregal, Mesiar and Fernandez [18] toextend fuzzy ideals and fuzzy filters from a fuzzy lattice toa sup-lattice. Thus, for further research we hope to thinkof building bounded interval fuzzy lattice, using the idea of

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Bedregal and Santos [2], from bounded fuzzy lattices.

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