α-Ideals of Fuzzy Lattices - Semantic Scholar
α-Ideals of Fuzzy Lattices Benjamin Bedregal and Regivan H. N. Santiago
Ivan Mezzomo Department of Mathematical Sciences, Technology and Human – DCETH Rural Federal University of SemiArid – UFERSA Angicos – RN, Brazil, 59.515-000 and Department of Informatics and Applied Mathematics – DIMAp Federal University of Rio Grande do Norte – UFRN Natal – Rio Grande do Norte, Brazil, 59.072-970 Email: [email protected]
Group for Logic, Language, Information, Theory and Applications - LOLITA Department of Informatics and Applied Mathematics – DIMAp Federal University of Rio Grande do Norte – UFRN Natal – Rio Grande do Norte, Brazil, 59.072-970 Email: {bedregal, regivan}@dimap.ufrn.br
Abstract—We consider the fuzzy lattice notion introduced by Chon (Korean J. Math 17 (2009), No. 4, 361-374), define an αideals and α-filters for fuzzy lattices and characterize α-ideals and α-filters of fuzzy lattices by using its support and its level set. Moreover, we prove some similar properties to the classical theory of α-ideals and α-filters, such as, the class of α-ideals and α-filters are closed under union and intersection.
I. I NTRODUCTION The concept of fuzzy set was introduced by Zadeh [21] which in his seminal paper also defined the notion of fuzzy relations. From then, several mathematical concepts such as number, group, topology, differential equation, etc., had been fuzzified. In particular for the case of order and lattice notions different definitions has been proposed, for example [3], [4], [5], [8], [10]. Yuan and Wu [20] introduced the concepts of fuzzy sublattices and fuzzy ideals of a lattice. Ajmal and Thomas [1] defined a fuzzy lattice as a fuzzy algebra and characterized fuzzy sublattices. In 2009, Chon [6] characterized a fuzzy partial order relation using its level set and defined a fuzzy lattice as a fuzzy relation, he also discovered some basic properties of fuzzy lattices and showed that a fuzzy totally ordered 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 ⊆ X endowed with the fuzzy order A|Y ×Y . In paper [15], we define both ideal and filter of a fuzzy lattice (X, A) and some kinds of ideals and filters, we also study the intersection of families for each kind of ideal and filter together with some of its consequences. Finally, in paper [16], we define a new notion of fuzzy ideal and fuzzy filter for fuzzy lattice and define some 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 the classical theory of fuzzy ideals (filters), such as, the class of proper fuzzy ideals (filters) is closed under fuzzy union and fuzzy intersection. As a step forward of such investigations, we
978-1-4799-0348-1/13/$31.00 ©2013 IEEE
define α-ideals and α-filters of fuzzy lattices using the fuzzy partial order relation and fuzzy lattices defined by Chon. In section II, we provide preliminary results on some basic concepts like ideal, filter and lattice. In section III, we consider Chon’s approach [6] on fuzzy partial order relation. We also characterize, a fuzzy lattice (X, A) as a classical set X under a fuzzy partial order relation A and fuzzy ideals and fuzzy 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 support and its level set, and study some properties analogous to the classical theory of α-ideals and α-filters, such as, the class of α-ideals and α-filters are closed under union and intersection. II. P RELIMINARIES In this section, we will briefly review some basic concepts of lattices, ideals and filters both from the algebraic theoretical view and order as necessary for the development. This presentation is quite introductory and can be found in many books on lattice theory. The reader familiar with such concepts can proceed to the next section. When exist the top and bottom elements of a set P , they are denoted by > and ⊥, respectively. Definition 2.1 ([7], Definition 2.4): Let (P, ≤) be a nonempty 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 sup S and inf S exist for all S ⊆ P , then (P, ≤) is called a complete lattice. We introduced lattices as ordered sets of a special type. However, we may adopt an alternative viewpoint. Given a lattice 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 let x, y ∈ L. Then the following are equivalent:
157
(i) x ≤ y; (ii) x ∨ y = y; (iii) x ∧ y = x.
Let X be a nonempty set and x, y, z ∈ X. A fuzzy binary relation 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, A is transitive if A(x, z) ≥ sup min[A(x, y), A(y, z)], and A
We have shown that lattices can be completely characterized in terms of the join and meet operations. We may henceforth say “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 special kind 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 view these 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 that a = a ∧ 1 for all x ∈ L. Dually, we say L has a bottom element if there exists 0 ∈ L such that x = x ∨ 0 for all x ∈ L. The lattice (L, ∨, ∧) has a 1 iff (L, ≤) has a top element > and, in that case, 1 = >. A dual statement holds for 0 and ⊥. A lattice (L, ∨, ∧) possessing 0 and 1 is called bounded. A finite lattice is automatically bounded, with 1 = sup L and 0 = inf L. In [12] was defined ideals and filters of a lattice L. Let L be a nonempty set and L = (L, ∧, ∨, 0, 1) stand for a bounded distributive lattice. Definition 2.3 ([7], Definition 2.20): A nonempty subset I of 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 F of 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. F UZZY L ATTICES In this section we define a fuzzy lattice as a fuzzy partial order relation and develop some properties for them. Let X be a universal set. A fuzzy set A on X is a function µA : X → [0, 1], where [0, 1] means real numbers between 0 and 1 (including 0 and 1). Given two fuzzy set A and B on X, we say that A ⊆ B if, for all x ∈ X, µA (x) ≤ µB (x). In particular, 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) = 1 for all x ∈ X. For more detailed study refer to [13], [21]. Let X and Y be nonempty sets and x ∈ X and y ∈ Y . A fuzzy relation A is a mapping from the Cartesian space X ×Y to the interval [0, 1], where the strength of the mapping is expressed by the membership function of the relation A, that is, A : X × Y → [0, 1]. If X = Y then we say that A is a binary fuzzy relation on X.
y∈X
is antisymmetric if A(x, y) > 0 and A(y, x) > 0 implies x = y. These definitions of reflexivity, symmetry, transitivity and antisymmetry can be found in several books and papers as [6], [11], [13], [22]. A function A : X × X −→ [0, 1] is called a fuzzy equivalence relation in X if A is reflexive, transitive and symmetric. A fuzzy relation A is a fuzzy partial order relation if A is reflexive, antisymmetric and transitive. A fuzzy partial order relation A is a fuzzy total order relation if A(x, y) > 0 or A(y, x) > 0 for all x, y ∈ X. If A is a fuzzy partial order relation on a set X, then (X, A) is called a fuzzy partially ordered set or fuzzy poset. If A is a fuzzy total order relation in a set X, then (X, A) is called fuzzy totally ordered set or a fuzzy chain. For more detailed study refer to [6]. In the literature there are several other ways to define a fuzzy which is reflexive, symmetric or transitive relation c.f. [8], [9]. Also, we can find several other forms to define fuzzy partial order relations, as we can be see in [3], [4], [19]. Remark 3.1: When A is reflexive, then the transitivity property can be rewritten by replacing the ”≥” by ”=”. In other words, A is transitive iff A(x, z) = sup min[A(x, y), A(y, z)], for all x, y, z ∈ X. y∈X
The statement that is claimed in the last remark can be easily proved. First, we know that A(x, z) ≥ supy∈X min[A(x, y), A(y, z)] and second, trivially, supy∈X min[A(x, y), A(y, z)] ≥ min[A(x, x), A(x, z)] = min[1, A(x, z)] = A(x, z). Therefore, we have that A(x, z) = supy∈X min[A(x, y), A(y, z)]. Proposition 3.1 ([16] Proposition 3.1): Let (X, A) be fuzzy 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 and develop some properties of fuzzy lattices. Definition 3.1 ([6], Definition 3.1): Let (X, A) be a fuzzy poset and let Y ⊆ X. An element u ∈ X is said to be an upper bound for a subset Y if A(y, u) > 0 for all y ∈ Y . An upper bound u0 for Y is the least upper bound (or supremum) of Y if A(u0 , u) > 0 for every upper bound u for Y . An element v ∈ X is said to be a lower 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 (or infimum) of Y if A(v, v0 ) > 0 for every lower bound v for Y . The least upper bound of Y will be denoted by sup Y or LU B Y and the greatest lower bound by inf Y or GLB Y .
158
We denote the least upper bound of the set {x, y} by x ∨ y and 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, is unique. 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, and A(w, z) = 0.1. Then it is easily to verify that A is a fuzzy total 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. The following diagram show us the fuzzy order relation.
? xO W/ ~~ // ~ ~ / ~~ // ~~ // 0.5 y W0m Z W S / 0.2 O K // 00 00 0.8 F // A/ 00 0 0.4 0 >z 00 }} } 00 } 0 }}} 0.1 w 0.3
× w z y x
w 1.0 0.0 0.0 0.0
z 0.1 1.0 0.0 0.0
y 0.4 0.2 1.0 0.0
x 0.8 0.5 0.3 1.0
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 fuzzy lattice. We can build a fuzzy lattice using the idea of support as follows: 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 on X, 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 lattice and x, y ∈ X. Then x ∨ y in terms of (X, A) coincides with x ∨ y in terms of (X, S(A)). On the other hand, x ∧ y in 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 a fuzzy 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.
Now, let Y = {z, w} be a subset of X. Then, x, y and z are upper bounds of Y and since A(z, w) = 0 and A(w, z) > 0, the LU B Y is z and the GLB Y is w.
For more detailed study we refer to [6], [14], [16]. IV. α-I DEALS AND α-F ILTERS
Proposition 3.2 ([6], Proposition 2.2): Let (X, A) be a fuzzy poset (or chain) and Y ⊆ X. If B = A|Y ×Y , that is, 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).
In this section, we propose the notions of α-ideals and α-filters of a fuzzy lattice and characterize them by using its support and its level set. we define α-ideals and α-filters of a fuzzy lattice as follows:
Definition 3.3 ([14] Definition 3.3): Let (X, A) be a fuzzy lattice. (Y, B) is a fuzzy sublattice of (X, A) if Y ⊆ X, B = A|Y ×Y and (Y, B) is a fuzzy lattice.
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 .
We define, for any α ∈ (0, 1], the α-level set Aα = {(x, y) ∈ X × X : A(x, y) ≥ α} of a fuzzy relation A and the support of a fuzzy relation A by S(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 order relation on X iff for each α ∈ (0, 1], the α-level set Aα is a partial order relation in X.
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 any x ∈ X, if A(x, y) ≥ β, then A(x, y) ≥ α, so by Definition
159
4.1 (i), x ∈ Y . On the other hand, if x, y ∈ Y , then by Definition 4.1 (ii), x ∨ y ∈ Y . Therefore, Y is a β-ideal of (X, A).
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 fuzzy lattice (X, A) iff for each α ∈ (0, 1], Yα is an ideal of (X, Aα ).
Dually, we prove that if α ≤ β, then any β-filter is a α-filter.
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.
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 fuzzy lattice (X, A) is an α-filter, for all α ∈ (0, 1]. In paper [15], we defined an ideal and a filter of a fuzzy lattice (X, A), respectively, as follows: Definition 4.3 ([15], Definition 41): Let (X, A) be a fuzzy lattice 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 fuzzy lattice 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 is an α-ideal. Dually, all filter in the sense of Definition 4.4 is an α-filter.
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 fuzzy lattice (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 as follow: Definition 4.5: A fuzzy poset (Y, A) is called fuzzy sup-lattice if each pair of element has supremum on Y . Dually, a fuzzy poset (Y, A) is called fuzzy inf-lattice if each pair of element has infimum on Y . Notice that a structure is a fuzzy lattice iff it is simultaneously fuzzy sup-lattice and fuzzy inf-lattice.
Proof: Straightforward from Proposition 4.1. Proposition 4.2: Let α ∈ (0, 1]. If Y is an ideal of the lattice (X, S(A)), then for all α ∈ (0, 1], Y is an α-ideal of fuzzy 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 4.1 and the condition (ii) is satisfied because it does not depend on the value of α. Proposition 4.3: Let α ∈ (0, 1]. If Y is a filter of the lattice (X, S(A)), then for all α ∈ (0, 1], Y is an α-filter of fuzzy 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 4.5: 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-ideal of (X, A). Proof: (i) Let α ∈ (0, 1], z ∈⇓ Yα and w ∈ X such that A(w, z) ≥ α. How z ∈⇓ Yα , then exists x ∈ Y such that A(z, x) ≥ α, and by transitivity, A(w, x) ≥ α, for some α ∈ (0, 1]. Therefore, w ∈⇓ Yα . (ii) Suppose x, y ∈⇓ Yα , then exist z1 , z2 ∈ Y such that A(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 a filter of (X, A).
Proposition 4.4: Let Y be an α-ideal of (X, A) and B = A|Y ×Y . The set Yα = {x ∈ Y : B(x, y) ≥ α for any y ∈ Y } is an ideal of (X, Aα ). Proof: Straightforward from Definition 4.1.
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
160
(ii) Y ⊆ W ⇒⇓ Yα ⊆⇓ Wα (iii) ⇓⇓ Yα =⇓ Yα
We will define a kind of α-ideals of fuzzy lattice called principal α-ideal generated by x ∈ X.
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, exists z ∈ 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 exists x ∈⇓ Yα such that A(y, x) ≥ α, for any α ∈ (0, 1]. Since x ∈⇓ Yα , then exists z ∈ Y such that A(x, z) ≥ α. So, by transitivity, 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
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 an α-ideal ⇓ Yα and principal α-ideals ⇓ yα . We denote by Pα (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].
Proof: Analogous to Proposition 4.7. Corollary 4.2: Let α ∈ (0, 1]. ⇓ Yα (⇑ Yα ) is the lowest α-ideal (α-filter) containing Y . The family of all α-ideals of a fuzzy lattice (X, A), for some α ∈ (0, 1], will be denoted by Iα (X). Dually, will denote 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) andTW be a nonempty set of Iα (X), then (i) SZ ∈ Iα (X); (ii) W ∈ Iα (X). Proof: (i) Let Z ⊆ Iα (X), for some α ∈ (0, 1]. Suppose T x ∈ Z and A(y, x) ≥ α, then x ∈ Zj for all Zj ∈T Z. How A(y, x) ≥T 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 T Z = X. S (ii) Let W ⊆ Iα (X), for some α ∈ (0, 1]. Suppose x ∈ W and A(y, x) ≥ α, then exists Wj ∈ W such S that x ∈ Wj , and how W ∈ I (X), then y ∈ W . So y ∈ W and therefore, j α j S W ∈ Iα (X). Proposition 4.10: Let α ∈ (0, 1], Z be a subset of Fα (X) andTW be a nonempty set of Fα (X), then (i) SZ ∈ Fα (X); (ii) W ∈ Fα (X). Proof: Analogous to Proposition 4.9. Corollary 4.3: X ∈ Iα (X) only if α has the lowest value of the range (0, 1]. Similarly, X ∈ Fα (X) only if α has the lowest value of the range (0, 1].
Proposition 4.11: [ For all Y α ∈ (0, 1], ⇓ Yα = ⇓ yα .
∈
Pα (X) and for all
y∈Y
Proof: Let Y ∈ Pα (X), for all α ∈ (0, 1]. Then, x ∈⇓ Yα iff exists y ∈ Y such [ that A(x, y) ≥ α iff exists y ∈ Y such that x ∈⇓ yα iff x ∈ ⇓ yα . y∈Y
Proposition 4.12: [ For all Y α ∈ (0, 1], ⇑ Yα = ⇑ yα .
∈
Pα (X) and for all
y∈Y
Proof: Analogous to Proposition 4.12.
V. C ONCLUSION In this paper, we have studied the notion of fuzzy lattice using a fuzzy order relation and introduced a new notion of α-ideals and α-filters of fuzzy lattice. We established that the α-ideal theorem of a fuzzy lattice through its level set and its support. We also defined, some properties of α-ideals and α-filters of fuzzy lattice analogous the classical theory. We can found several other forms to define fuzzy partial order relations, 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 in several other forms in the literature. One of the most promising ideas could be the investigation of operations among fuzzy lattices and its consequences. As future work we consider the idea of Palmeira and Bedregal [17] and Palmeira, Bedregal, Mesiar and Fernandez [18] to extend fuzzy ideals and fuzzy filters from a fuzzy lattice to a sup-lattice. Thus, for further research we hope to think of building bounded interval fuzzy lattice, using the idea of
161
Bedregal and Santos [2], from bounded fuzzy lattices. R EFERENCES [1] N. AJMAL, K.V. THOMAS, Fuzzy lattices, Information Sciences 79 (1994) 271-291. [2] B.C. BEDREGAL, H.S. SANTOS, T-norms on bounded lattices: tnorm morphisms and operators, IEEE International Conference on Fuzzy Systems - 2006, 22-28. DOI: 10.1109/FUZZY.2006.1681689. [3] I. BEG, On fuzzy order relations, Journal of Nonlinear Science and Applications, 5 (2012), 357-378. ´ [4] R. BELOHLAVEK, Concept lattices and order in fuzzy logic, Annals of Pure and Applied Logic 128 (2004), 277-298. [5] U. BODENHOFER, J. KUNG, Fuzzy orderings in flexible query answering systems, Soft Computing 8 (2004) 512-522. [6] I. CHON, Fuzzy partial order relations and fuzzy lattices, Korean Journal Mathematics 17 (2009), No. 4, pp 361-374. [7] B.A. DAVEY, H.A. PRIESTLEY, Introduction to Lattices and Order, Cambridge University Press, 2 edition, 2002. [8] J. FODOR, M. ROUBENS, Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer Academic Publisher, Dordrecht, 1994. [9] J. FODOR, R.R. YAGER, Fuzzy Set-Theoretic Operators and Quantifiers, In: Fundamentals of Fuzzy Sets, D. Dubois and H. Prade (eds.), Kluwer Academic Publisher, Dordrecht, 2000. [10] G. GERLA, Representation theorems for fuzzy orders and quasi-metrics, Soft Computing 8 (2004) 571-580. [11] G. J. KLIR, T. A. Folger, Fuzzy Sets, Uncertainty, and Information, Prentice Hall, Englewood Cliffs, 1988. [12] B.B.N. KOGUEP, C. NKUIMI, C. LELE, On Fuzzy Prime Ideals of Lattice, SJPAM, 3 (2008), 1-11. [13] K.H. LEE, First Course on Fuzzy Theory and Applications, Springer (2005). [14] I. MEZZOMO, B.C. BEDREGAL, R.H.N. SANTIAGO, On fuzzy ideals of fuzzy lattice, IEEE International Conference on Fuzzy Systems - 2012, 1-5. DOI: 10.1109/FUZZ-IEEE.2012.6251307. [15] I. MEZZOMO, B.C. BEDREGAL, R.H.N. SANTIAGO, Kinds of ideals of fuzzy lattice, Second Brasilian Congress on Fuzzy Systems - 2012, 657-671. [16] I. MEZZOMO, B.C. BEDREGAL, R.H.N. SANTIAGO, Types of fuzzy ideals of fuzzy lattices, Preprint Submitted to Journal of Intelligent and Fuzzy Systems. [17] E.S. PALMEIRA, B.C. BEDREGAL, B.C. Extension of fuzzy logic operators defined on bounded lattices via retractions, Computers and Mathematics with Applications 63 (2012), 1026-1038. [18] E.S. PALMEIRA, B.C. BEDREGAL, R. MESIAR, J. FERNANDEZ, A new way to extend t-norms, t-conorms and negations, Fuzzy Set and Systems (to appear). [19] W. YAO, L. LU, Fuzzy Galois connections on fuzzy poset, Mathematical Logic Quarterly 55 (2009), No 1, 105-112. [20] B. YUAN, W. WU, Fuzzy ideals on a distributive lattice, Fuzzy Sets and Systems 35 (1990) 231-240. [21] L.A. ZADEH, Fuzzy sets, Information and Control 8 (1965) 338-353. [22] H. J. ZIMMERMANN, Fuzzy Set Theory and Its Applications, 2nd ed., Kluwer Academic Publishers, Boston, 1991.
162
Ivan Mezzomo Department of Mathematical Sciences, Technology and Human – DCETH Rural Federal University of SemiArid – UFERSA Angicos – RN, Brazil, 59.515-000 and Department of Informatics and Applied Mathematics – DIMAp Federal University of Rio Grande do Norte – UFRN Natal – Rio Grande do Norte, Brazil, 59.072-970 Email: [email protected]
Group for Logic, Language, Information, Theory and Applications - LOLITA Department of Informatics and Applied Mathematics – DIMAp Federal University of Rio Grande do Norte – UFRN Natal – Rio Grande do Norte, Brazil, 59.072-970 Email: {bedregal, regivan}@dimap.ufrn.br
Abstract—We consider the fuzzy lattice notion introduced by Chon (Korean J. Math 17 (2009), No. 4, 361-374), define an αideals and α-filters for fuzzy lattices and characterize α-ideals and α-filters of fuzzy lattices by using its support and its level set. Moreover, we prove some similar properties to the classical theory of α-ideals and α-filters, such as, the class of α-ideals and α-filters are closed under union and intersection.
I. I NTRODUCTION The concept of fuzzy set was introduced by Zadeh [21] which in his seminal paper also defined the notion of fuzzy relations. From then, several mathematical concepts such as number, group, topology, differential equation, etc., had been fuzzified. In particular for the case of order and lattice notions different definitions has been proposed, for example [3], [4], [5], [8], [10]. Yuan and Wu [20] introduced the concepts of fuzzy sublattices and fuzzy ideals of a lattice. Ajmal and Thomas [1] defined a fuzzy lattice as a fuzzy algebra and characterized fuzzy sublattices. In 2009, Chon [6] characterized a fuzzy partial order relation using its level set and defined a fuzzy lattice as a fuzzy relation, he also discovered some basic properties of fuzzy lattices and showed that a fuzzy totally ordered 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 ⊆ X endowed with the fuzzy order A|Y ×Y . In paper [15], we define both ideal and filter of a fuzzy lattice (X, A) and some kinds of ideals and filters, we also study the intersection of families for each kind of ideal and filter together with some of its consequences. Finally, in paper [16], we define a new notion of fuzzy ideal and fuzzy filter for fuzzy lattice and define some 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 the classical theory of fuzzy ideals (filters), such as, the class of proper fuzzy ideals (filters) is closed under fuzzy union and fuzzy intersection. As a step forward of such investigations, we
978-1-4799-0348-1/13/$31.00 ©2013 IEEE
define α-ideals and α-filters of fuzzy lattices using the fuzzy partial order relation and fuzzy lattices defined by Chon. In section II, we provide preliminary results on some basic concepts like ideal, filter and lattice. In section III, we consider Chon’s approach [6] on fuzzy partial order relation. We also characterize, a fuzzy lattice (X, A) as a classical set X under a fuzzy partial order relation A and fuzzy ideals and fuzzy 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 support and its level set, and study some properties analogous to the classical theory of α-ideals and α-filters, such as, the class of α-ideals and α-filters are closed under union and intersection. II. P RELIMINARIES In this section, we will briefly review some basic concepts of lattices, ideals and filters both from the algebraic theoretical view and order as necessary for the development. This presentation is quite introductory and can be found in many books on lattice theory. The reader familiar with such concepts can proceed to the next section. When exist the top and bottom elements of a set P , they are denoted by > and ⊥, respectively. Definition 2.1 ([7], Definition 2.4): Let (P, ≤) be a nonempty 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 sup S and inf S exist for all S ⊆ P , then (P, ≤) is called a complete lattice. We introduced lattices as ordered sets of a special type. However, we may adopt an alternative viewpoint. Given a lattice 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 let x, y ∈ L. Then the following are equivalent:
157
(i) x ≤ y; (ii) x ∨ y = y; (iii) x ∧ y = x.
Let X be a nonempty set and x, y, z ∈ X. A fuzzy binary relation 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, A is transitive if A(x, z) ≥ sup min[A(x, y), A(y, z)], and A
We have shown that lattices can be completely characterized in terms of the join and meet operations. We may henceforth say “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 special kind 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 view these 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 that a = a ∧ 1 for all x ∈ L. Dually, we say L has a bottom element if there exists 0 ∈ L such that x = x ∨ 0 for all x ∈ L. The lattice (L, ∨, ∧) has a 1 iff (L, ≤) has a top element > and, in that case, 1 = >. A dual statement holds for 0 and ⊥. A lattice (L, ∨, ∧) possessing 0 and 1 is called bounded. A finite lattice is automatically bounded, with 1 = sup L and 0 = inf L. In [12] was defined ideals and filters of a lattice L. Let L be a nonempty set and L = (L, ∧, ∨, 0, 1) stand for a bounded distributive lattice. Definition 2.3 ([7], Definition 2.20): A nonempty subset I of 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 F of 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. F UZZY L ATTICES In this section we define a fuzzy lattice as a fuzzy partial order relation and develop some properties for them. Let X be a universal set. A fuzzy set A on X is a function µA : X → [0, 1], where [0, 1] means real numbers between 0 and 1 (including 0 and 1). Given two fuzzy set A and B on X, we say that A ⊆ B if, for all x ∈ X, µA (x) ≤ µB (x). In particular, 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) = 1 for all x ∈ X. For more detailed study refer to [13], [21]. Let X and Y be nonempty sets and x ∈ X and y ∈ Y . A fuzzy relation A is a mapping from the Cartesian space X ×Y to the interval [0, 1], where the strength of the mapping is expressed by the membership function of the relation A, that is, A : X × Y → [0, 1]. If X = Y then we say that A is a binary fuzzy relation on X.
y∈X
is antisymmetric if A(x, y) > 0 and A(y, x) > 0 implies x = y. These definitions of reflexivity, symmetry, transitivity and antisymmetry can be found in several books and papers as [6], [11], [13], [22]. A function A : X × X −→ [0, 1] is called a fuzzy equivalence relation in X if A is reflexive, transitive and symmetric. A fuzzy relation A is a fuzzy partial order relation if A is reflexive, antisymmetric and transitive. A fuzzy partial order relation A is a fuzzy total order relation if A(x, y) > 0 or A(y, x) > 0 for all x, y ∈ X. If A is a fuzzy partial order relation on a set X, then (X, A) is called a fuzzy partially ordered set or fuzzy poset. If A is a fuzzy total order relation in a set X, then (X, A) is called fuzzy totally ordered set or a fuzzy chain. For more detailed study refer to [6]. In the literature there are several other ways to define a fuzzy which is reflexive, symmetric or transitive relation c.f. [8], [9]. Also, we can find several other forms to define fuzzy partial order relations, as we can be see in [3], [4], [19]. Remark 3.1: When A is reflexive, then the transitivity property can be rewritten by replacing the ”≥” by ”=”. In other words, A is transitive iff A(x, z) = sup min[A(x, y), A(y, z)], for all x, y, z ∈ X. y∈X
The statement that is claimed in the last remark can be easily proved. First, we know that A(x, z) ≥ supy∈X min[A(x, y), A(y, z)] and second, trivially, supy∈X min[A(x, y), A(y, z)] ≥ min[A(x, x), A(x, z)] = min[1, A(x, z)] = A(x, z). Therefore, we have that A(x, z) = supy∈X min[A(x, y), A(y, z)]. Proposition 3.1 ([16] Proposition 3.1): Let (X, A) be fuzzy 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 and develop some properties of fuzzy lattices. Definition 3.1 ([6], Definition 3.1): Let (X, A) be a fuzzy poset and let Y ⊆ X. An element u ∈ X is said to be an upper bound for a subset Y if A(y, u) > 0 for all y ∈ Y . An upper bound u0 for Y is the least upper bound (or supremum) of Y if A(u0 , u) > 0 for every upper bound u for Y . An element v ∈ X is said to be a lower 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 (or infimum) of Y if A(v, v0 ) > 0 for every lower bound v for Y . The least upper bound of Y will be denoted by sup Y or LU B Y and the greatest lower bound by inf Y or GLB Y .
158
We denote the least upper bound of the set {x, y} by x ∨ y and 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, is unique. 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, and A(w, z) = 0.1. Then it is easily to verify that A is a fuzzy total 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. The following diagram show us the fuzzy order relation.
? xO W/ ~~ // ~ ~ / ~~ // ~~ // 0.5 y W0m Z W S / 0.2 O K // 00 00 0.8 F // A/ 00 0 0.4 0 >z 00 }} } 00 } 0 }}} 0.1 w 0.3
× w z y x
w 1.0 0.0 0.0 0.0
z 0.1 1.0 0.0 0.0
y 0.4 0.2 1.0 0.0
x 0.8 0.5 0.3 1.0
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 fuzzy lattice. We can build a fuzzy lattice using the idea of support as follows: 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 on X, 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 lattice and x, y ∈ X. Then x ∨ y in terms of (X, A) coincides with x ∨ y in terms of (X, S(A)). On the other hand, x ∧ y in 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 a fuzzy 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.
Now, let Y = {z, w} be a subset of X. Then, x, y and z are upper bounds of Y and since A(z, w) = 0 and A(w, z) > 0, the LU B Y is z and the GLB Y is w.
For more detailed study we refer to [6], [14], [16]. IV. α-I DEALS AND α-F ILTERS
Proposition 3.2 ([6], Proposition 2.2): Let (X, A) be a fuzzy poset (or chain) and Y ⊆ X. If B = A|Y ×Y , that is, 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).
In this section, we propose the notions of α-ideals and α-filters of a fuzzy lattice and characterize them by using its support and its level set. we define α-ideals and α-filters of a fuzzy lattice as follows:
Definition 3.3 ([14] Definition 3.3): Let (X, A) be a fuzzy lattice. (Y, B) is a fuzzy sublattice of (X, A) if Y ⊆ X, B = A|Y ×Y and (Y, B) is a fuzzy lattice.
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 .
We define, for any α ∈ (0, 1], the α-level set Aα = {(x, y) ∈ X × X : A(x, y) ≥ α} of a fuzzy relation A and the support of a fuzzy relation A by S(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 order relation on X iff for each α ∈ (0, 1], the α-level set Aα is a partial order relation in X.
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 any x ∈ X, if A(x, y) ≥ β, then A(x, y) ≥ α, so by Definition
159
4.1 (i), x ∈ Y . On the other hand, if x, y ∈ Y , then by Definition 4.1 (ii), x ∨ y ∈ Y . Therefore, Y is a β-ideal of (X, A).
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 fuzzy lattice (X, A) iff for each α ∈ (0, 1], Yα is an ideal of (X, Aα ).
Dually, we prove that if α ≤ β, then any β-filter is a α-filter.
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.
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 fuzzy lattice (X, A) is an α-filter, for all α ∈ (0, 1]. In paper [15], we defined an ideal and a filter of a fuzzy lattice (X, A), respectively, as follows: Definition 4.3 ([15], Definition 41): Let (X, A) be a fuzzy lattice 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 fuzzy lattice 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 is an α-ideal. Dually, all filter in the sense of Definition 4.4 is an α-filter.
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 fuzzy lattice (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 as follow: Definition 4.5: A fuzzy poset (Y, A) is called fuzzy sup-lattice if each pair of element has supremum on Y . Dually, a fuzzy poset (Y, A) is called fuzzy inf-lattice if each pair of element has infimum on Y . Notice that a structure is a fuzzy lattice iff it is simultaneously fuzzy sup-lattice and fuzzy inf-lattice.
Proof: Straightforward from Proposition 4.1. Proposition 4.2: Let α ∈ (0, 1]. If Y is an ideal of the lattice (X, S(A)), then for all α ∈ (0, 1], Y is an α-ideal of fuzzy 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 4.1 and the condition (ii) is satisfied because it does not depend on the value of α. Proposition 4.3: Let α ∈ (0, 1]. If Y is a filter of the lattice (X, S(A)), then for all α ∈ (0, 1], Y is an α-filter of fuzzy 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 4.5: 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-ideal of (X, A). Proof: (i) Let α ∈ (0, 1], z ∈⇓ Yα and w ∈ X such that A(w, z) ≥ α. How z ∈⇓ Yα , then exists x ∈ Y such that A(z, x) ≥ α, and by transitivity, A(w, x) ≥ α, for some α ∈ (0, 1]. Therefore, w ∈⇓ Yα . (ii) Suppose x, y ∈⇓ Yα , then exist z1 , z2 ∈ Y such that A(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 a filter of (X, A).
Proposition 4.4: Let Y be an α-ideal of (X, A) and B = A|Y ×Y . The set Yα = {x ∈ Y : B(x, y) ≥ α for any y ∈ Y } is an ideal of (X, Aα ). Proof: Straightforward from Definition 4.1.
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
160
(ii) Y ⊆ W ⇒⇓ Yα ⊆⇓ Wα (iii) ⇓⇓ Yα =⇓ Yα
We will define a kind of α-ideals of fuzzy lattice called principal α-ideal generated by x ∈ X.
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, exists z ∈ 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 exists x ∈⇓ Yα such that A(y, x) ≥ α, for any α ∈ (0, 1]. Since x ∈⇓ Yα , then exists z ∈ Y such that A(x, z) ≥ α. So, by transitivity, 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
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 an α-ideal ⇓ Yα and principal α-ideals ⇓ yα . We denote by Pα (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].
Proof: Analogous to Proposition 4.7. Corollary 4.2: Let α ∈ (0, 1]. ⇓ Yα (⇑ Yα ) is the lowest α-ideal (α-filter) containing Y . The family of all α-ideals of a fuzzy lattice (X, A), for some α ∈ (0, 1], will be denoted by Iα (X). Dually, will denote 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) andTW be a nonempty set of Iα (X), then (i) SZ ∈ Iα (X); (ii) W ∈ Iα (X). Proof: (i) Let Z ⊆ Iα (X), for some α ∈ (0, 1]. Suppose T x ∈ Z and A(y, x) ≥ α, then x ∈ Zj for all Zj ∈T Z. How A(y, x) ≥T 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 T Z = X. S (ii) Let W ⊆ Iα (X), for some α ∈ (0, 1]. Suppose x ∈ W and A(y, x) ≥ α, then exists Wj ∈ W such S that x ∈ Wj , and how W ∈ I (X), then y ∈ W . So y ∈ W and therefore, j α j S W ∈ Iα (X). Proposition 4.10: Let α ∈ (0, 1], Z be a subset of Fα (X) andTW be a nonempty set of Fα (X), then (i) SZ ∈ Fα (X); (ii) W ∈ Fα (X). Proof: Analogous to Proposition 4.9. Corollary 4.3: X ∈ Iα (X) only if α has the lowest value of the range (0, 1]. Similarly, X ∈ Fα (X) only if α has the lowest value of the range (0, 1].
Proposition 4.11: [ For all Y α ∈ (0, 1], ⇓ Yα = ⇓ yα .
∈
Pα (X) and for all
y∈Y
Proof: Let Y ∈ Pα (X), for all α ∈ (0, 1]. Then, x ∈⇓ Yα iff exists y ∈ Y such [ that A(x, y) ≥ α iff exists y ∈ Y such that x ∈⇓ yα iff x ∈ ⇓ yα . y∈Y
Proposition 4.12: [ For all Y α ∈ (0, 1], ⇑ Yα = ⇑ yα .
∈
Pα (X) and for all
y∈Y
Proof: Analogous to Proposition 4.12.
V. C ONCLUSION In this paper, we have studied the notion of fuzzy lattice using a fuzzy order relation and introduced a new notion of α-ideals and α-filters of fuzzy lattice. We established that the α-ideal theorem of a fuzzy lattice through its level set and its support. We also defined, some properties of α-ideals and α-filters of fuzzy lattice analogous the classical theory. We can found several other forms to define fuzzy partial order relations, 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 in several other forms in the literature. One of the most promising ideas could be the investigation of operations among fuzzy lattices and its consequences. As future work we consider the idea of Palmeira and Bedregal [17] and Palmeira, Bedregal, Mesiar and Fernandez [18] to extend fuzzy ideals and fuzzy filters from a fuzzy lattice to a sup-lattice. Thus, for further research we hope to think of building bounded interval fuzzy lattice, using the idea of
161
Bedregal and Santos [2], from bounded fuzzy lattices. R EFERENCES [1] N. AJMAL, K.V. THOMAS, Fuzzy lattices, Information Sciences 79 (1994) 271-291. [2] B.C. BEDREGAL, H.S. SANTOS, T-norms on bounded lattices: tnorm morphisms and operators, IEEE International Conference on Fuzzy Systems - 2006, 22-28. DOI: 10.1109/FUZZY.2006.1681689. [3] I. BEG, On fuzzy order relations, Journal of Nonlinear Science and Applications, 5 (2012), 357-378. ´ [4] R. BELOHLAVEK, Concept lattices and order in fuzzy logic, Annals of Pure and Applied Logic 128 (2004), 277-298. [5] U. BODENHOFER, J. KUNG, Fuzzy orderings in flexible query answering systems, Soft Computing 8 (2004) 512-522. [6] I. CHON, Fuzzy partial order relations and fuzzy lattices, Korean Journal Mathematics 17 (2009), No. 4, pp 361-374. [7] B.A. DAVEY, H.A. PRIESTLEY, Introduction to Lattices and Order, Cambridge University Press, 2 edition, 2002. [8] J. FODOR, M. ROUBENS, Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer Academic Publisher, Dordrecht, 1994. [9] J. FODOR, R.R. YAGER, Fuzzy Set-Theoretic Operators and Quantifiers, In: Fundamentals of Fuzzy Sets, D. Dubois and H. Prade (eds.), Kluwer Academic Publisher, Dordrecht, 2000. [10] G. GERLA, Representation theorems for fuzzy orders and quasi-metrics, Soft Computing 8 (2004) 571-580. [11] G. J. KLIR, T. A. Folger, Fuzzy Sets, Uncertainty, and Information, Prentice Hall, Englewood Cliffs, 1988. [12] B.B.N. KOGUEP, C. NKUIMI, C. LELE, On Fuzzy Prime Ideals of Lattice, SJPAM, 3 (2008), 1-11. [13] K.H. LEE, First Course on Fuzzy Theory and Applications, Springer (2005). [14] I. MEZZOMO, B.C. BEDREGAL, R.H.N. SANTIAGO, On fuzzy ideals of fuzzy lattice, IEEE International Conference on Fuzzy Systems - 2012, 1-5. DOI: 10.1109/FUZZ-IEEE.2012.6251307. [15] I. MEZZOMO, B.C. BEDREGAL, R.H.N. SANTIAGO, Kinds of ideals of fuzzy lattice, Second Brasilian Congress on Fuzzy Systems - 2012, 657-671. [16] I. MEZZOMO, B.C. BEDREGAL, R.H.N. SANTIAGO, Types of fuzzy ideals of fuzzy lattices, Preprint Submitted to Journal of Intelligent and Fuzzy Systems. [17] E.S. PALMEIRA, B.C. BEDREGAL, B.C. Extension of fuzzy logic operators defined on bounded lattices via retractions, Computers and Mathematics with Applications 63 (2012), 1026-1038. [18] E.S. PALMEIRA, B.C. BEDREGAL, R. MESIAR, J. FERNANDEZ, A new way to extend t-norms, t-conorms and negations, Fuzzy Set and Systems (to appear). [19] W. YAO, L. LU, Fuzzy Galois connections on fuzzy poset, Mathematical Logic Quarterly 55 (2009), No 1, 105-112. [20] B. YUAN, W. WU, Fuzzy ideals on a distributive lattice, Fuzzy Sets and Systems 35 (1990) 231-240. [21] L.A. ZADEH, Fuzzy sets, Information and Control 8 (1965) 338-353. [22] H. J. ZIMMERMANN, Fuzzy Set Theory and Its Applications, 2nd ed., Kluwer Academic Publishers, Boston, 1991.
162
