L-Fuzzy Filters of a Poset

Many generalizations of ideals and filters of a lattice to an arbitrary poset have been studied by different scholars. The authors of this paper introduced several generalizations of L-fuzzy ideal of a lattice to an arbitrary poset in [1]. In this paper, we introduce several L-fuzzy filters of a poset which generalize the L-fuzzy filter of a lattice and give several characterizations of them.


I. INTRODUCTION
W E have found several generalizations of ideals and filters of a lattice to arbitrary poset (partially ordered set) in a literature. Birkhoff in [2, p. 59] introduced a closed or normal ideals who gives accredit to the work of Stone in [3]. Next, in 1954 the second type of ideal and filter of a poset called Frink ideal and Frink filter have been introduced by O. Frink [4]. Following this P. V. Venkatanarasimhan developed the theory of semi ideals and semi filter in [5] and ideals and filters for a poset in [6], in 1970. These ideals (respectively, filters) are called ideals (respectively, filter) in the sense of Venkatanarasimhan or V-ideals (V-filters) for short. Later Halaś [7], in 1994, introduced a new ideal and filter of a poset which seems to be a suitable generalization of the usual concept of ideal and filter in a lattice. We will simply call it ideal (respectively, filter) in the sense of Halaš. Moreover, the concept of fuzzy ideals and filters of a lattice has been studied by different authors in series of papers [8], [9], [10], [11] and [12]. The aim of this paper is to notify several generalizations of L-fuzzy filters of a lattice to an arbitrary poset whose truth values are in a complete lattice satisfying the infinite meet distributive law and give several characterizations of them. We also prove that the set of all L-fuzzy filters of a poset forms a complete lattice with respect to point-wise ordering "⊆". Throughout this work, L means a non-trivial complete lattice satisfying the infinite meet distributive law: x ∧ sup S = sup{x ∧ s : s ∈ S} for all x ∈ L and for any subset S of L.

II. PRELIMINARIES
We briefly recall certain necessary concepts, terminologies and notations from [2], [13] and [14]. A binary relation " ≤ " on a non-empty set Q is called a partial order if it is reflexive, anti-symmetric and transitive. A pair (Q, ≤) is mihretmahlet@yahoo.com called a partially ordered set or simply a poset if Q is a nonempty set and " ≤ " is a partial order on Q. When confusion is unlikely, we use simply the symbol Q to denote a Poset (Q, ≤). Let Q be a poset and S ⊆ Q. An element x in Q is called a lower bound (respectively, an upper bound) of S if x ≤ a (respectively, x ≥ a ) for all a ∈ S. We denote the set of all lower bounds and upper bounds of S by S l and S u , respectively. That is S l = {x ∈ Q : x ≤ a ∀ a ∈ S} and S u = {x ∈ Q : x ≥ a ∀ a ∈ S}. S ul shall mean {S u } l and S lu shall mean {S l } u . Let a, b ∈ Q. Then {a} u is simply denoted by a u and {a, b} u is denoted by (a, b) u . Similar notations are used for the set of lower bounds. We note that S ⊆ S ul and S ⊆ S lu and if S ⊆ T in Q then S l ⊇ T l and S u ⊇ T u . Moreover, S lul = S l , S ulu = S u , {a u } l = a l and {a l } u = a u . An element x 0 in Q is called the least upper bound of S or supremum of S, denoted by supS (respectively, the greatest lower bound of S or infimum of S, denoted by in f S) if x 0 ∈ S u and x 0 ≤ x ∀x ∈ S u (respectively, if x 0 ∈ S l and x ≤ x 0 ∀x ∈ S l ). An element x 0 in Q is called the largest (respectively, the smallest) element if x ≤ x 0 (respectively, x 0 ≤ x) for all x ∈ Q. The largest (respectively, the smallest) element if it exists in Q is denoted by 1 (respectively, by 0). A poset (Q ≤) is called bounded if it has 0 and 1. Note that if S = / 0 we have S lu = ( / 0 l ) u = Q u which is equal to the empty set or the singleton set {1} if Q has the largest element 1 Now we recall definitions of filters of a poset that are introduced by different scholars.
Definition 2.7: Let A be any subset of a poset Q. Then the smallest filter containing A is called a filter generated by A and is denoted by [A). The filter generated by a singleton set {a}, is called a principal filter and is denoted by [a) Note that for any subset S of Q if inf S exists then S lu = [inf S).
The followings are some characterizations of filters generated by a subset S of a poset Q. We write T ⊂⊂ S to mean T is a finite subset of S.
1) The closed or normal filter generated by S, denoted by Definition 2.10 ( [16]): An L-fuzzy subset η of a poset Q is a function from Q into L. Note that if L is a unit interval of real numbers [0,1], then the L-fuzzy subset η is the fuzzy subsets of Q which is introduced by L. Zadeh [17]. The set of all L-fuzzy subsets of Q is denoted by L Q .
Definition 2.11 ([11]): Let η ∈ L Q . Then for each α ∈ L the set η α = {x : η(x) ≥ α} is called the level subset or level cut of η at α. Lemma 2.12 ([9]): Let η ∈ L Q . Then η(x) = sup{α ∈ L : x ∈ η α } for all x ∈ Q. Definition 2.13 ( [16]): Let ν, σ ∈ L Q . Define a binary relation "⊆" on L Q by ν ⊆ σ if and only ν( It is simple to verify that the binary relation " ⊆ " on L Q is a partial order and it is called the point wise ordering. Definition 2.14 ( [18]): Let θ and η be in L Q . Then the union of fuzzy subsets θ and η of X, denoted by θ ∪ η, is a fuzzy subset of Q defined by (θ ∪ η)(x) = θ (x) ∨ η(x) for all x ∈ Q and the intersection of fuzzy subsets θ and η of Q, denoted by θ ∩ η, is a fuzzy subset of X defined by More generally, the union and intersection of any family {η i } i∈∆ of L-fuzzy subsets of Q, denoted by i∈∆ η i and i∈∆ η i respectively, are defined by: Definition 2.15 ( [10]): An L-fuzzy subset η of a lattice Q with 1 is said to be an L -fuzzy filter of Q; if η(1) = 1 and Definition 2.16: Let η be Lfuzzy subset of a poset Q. The smallest fuzzy filter of Q containing η is called a fuzzy filter generated by η and is denoted by [η).

III. L-FUZZY FILTERS OF A POSET
In this section, we notify the concept of L-fuzzy filters of a poset and give several characterizations of them. Throughout this paper, Q stands for a poset (Q, ≤) with 1 unless otherwise stated. We begin with the following Definition 3.1: An L-fuzzy subset η of Q is called an Lfuzzy closed filter if it fulfills the following conditions: This implies χ F (x) = 1 and hence x ∈ F. Therefore, F lu ⊆ F and hence F is a closed filter. This proves the result.
Again let S be any subset of Q.
Therefore η is an L-fuzzy closed filter of Q. This proves the result.
Lemma 3.4: Let η be fuzzy closed filter of a poset Q. Then η is iso-tone, in the sense that η(x) ≤ η(y) whenever x ≤ y.
Proof: Let x, y ∈ Q such that x ≤ y. Put η(x) = α. Since η is a fuzzy closed filter, η α is a closed filter of Q and . This proves the result.
and hence η is an L-fuzzy filter in the lattice Q. Conversely suppose µ be an L-fuzzy filter in the lattice Q. Then η(1) = 1 and η(a Therefore η is an L-fuzzy closed filter in the poset Q. This proves the result.
Lemma 3.6: The intersection of any family of L-fuzzy closed filters is an L-fuzzy closed filter.
Theorem 3.7: Let [S) C be a closed filter generated by a subset S of Q and χ S be its characteristic functions. Then the Proof: Since [S) C is a closed filter of Q containing S, by Lemma 3.2, we have χ [S) C is a fuzzy closed filter. Again since S ⊆ [S) C , clearly we have χ S ⊆ χ [S) C . Now, we show that it is the smallest Lfuzzy closed filter containing χ S . Let η be an L-fuzzy closed filter such that χ S ⊆ η. Then η(a) = 1 for all a ∈ S. Now we claim This proves the theorem.
In the following theorem we characterize a fuzzy closed filter generated by a fuzzy subset of Q in terms of its level closed filters.
Theorem 3.8: Let η ∈ L Q . Then the L-fuzzy subsetη of Q defined byη(x) = sup{α ∈ L : x ∈ [η α ) C } for all x ∈ Q is a fuzzy closed filter of Q generated by η, where [µ α ) C is a closed filter generated by η α .
In the following, we give an algebraic characterization of L-fuzzy Closed filter generated by fuzzy subset of Q. Theorem 3.9: Let η ∈ L Q . Then the fuzzy subset η defined by is a fuzzy closed filter of Q generated by η.
Proof: It is enough to show thatη =η whereη is an L-fuzzy subset given in the above theorem. Let we have x ∈ S lu for some subset S of η α . This implies η(a) ≥ α for all a ∈ S and hence inf{η(a) : a ∈ S} ≥ α. Thus β = inf{η(a) : a ∈ S} ∈ A x . Thus for each α ∈ B x we get β ∈ A x such that α ≤ β and hence sup A x ≥ sup B x . Therefore sup A x = sup B x and hence η =η.
The above result yields the following. Theorem 3.10: Let F C F (Q) be the set of all L-fuzzy closed filters of Q. Then (F C F (Q), ⊆) forms a complete lattice with respect to the point wise ordering " ⊆ ", in which the supremum sup i∈∆ µ i and the inifimum inf i∈∆ η i of any family {η i : i ∈ ∆} in F C F (Q) are given by: sup i∈∆ η i = i∈∆ {η i } and inf i∈∆ η i = i∈∆ η i . Corollary 3.11: For any L-fuzzy closed filters η and ν of Q, the supremum η ∨ ν and the infimum η ∧ ν of η and ν in F C F (Q) respectively are: η ∨ ν = η ∪ ν and η ∧ ν = η ∩ ν. Now we introduce the fuzzy version of a filter (dual ideal) of a poset introduced by O. Frink [4]. Definition 3.12: An L-fuzzy subset η of Q is an L-fuzzy Frink filter if it satisfies the following conditions: 1) η(1) = 1 and 2) for any finite subset F of Q, η(x) ≥ inf{η(a) : a ∈ F} ∀x ∈ F lu Lemma 3.13: Let η ∈ L Q . Then η is an L-fuzzy Frink filter of Q if and only if η α is a Frink filter of Q for all α ∈ L. Lemma 3.14: Let η be fuzzy Frink filter of a poset Q. Then η is iso-tone, in the sense that η(x) ≤ η(y) whenever x ≤ y.
Frink filter generated by η α , where [η α ) F is a Frink filter generated by η α . Thenη is an L-fuzzy Frink filter of Q generated by η. In the following, we give an algebraic characterization of Lfuzzy Frink filters generated by fuzzy subset of Q. Theorem 3.20: Let η be a fuzzy subset of Q. Then the fuzzy subset − → η defined by is a Frink fuzzy filter of Q generated by η.
Theorem 3.21: Let F F F (Q) be the of all L-fuzzy Frink filter of Q. Then (F F F (Q), ⊆) forms a complete lattice with respect to point wise ordering " ⊆ ", in which the supremum and the infimum of any family {η i : i ∈ ∆} in F F F (Q) respectively are: sup i∈∆ η i = − −−−−− → i∈∆ {η i } and inf i∈∆ η i = i∈∆ η i . Corollary 3.22: For any L-fuzzy Frink ideals η and ν of Q in the supremum η ∨ ν and the infimum η ∧ ν of η and ν in F F F (Q) respectively are: η ∨ ν = − −− → η ∪ ν and η ∧ ν = η ∩ ν. Now we introduce the fuzzy version of semi-filters and Vfilters of a poset introduced by P.V. Venkatanarasimhan [5] and [6].
Definition 3.23: η in L Q is said to be an L-fuzzy semi-filter or L-fuzzy order filter if η(x) ≤ η(y) whenever x ≤ y in Q. Definition 3.24: η in L Q is said to be an Lfuzzy V -filter if it satisfies the following conditions: 1) for any x, y ∈ Q η(x) ≤ η(y) whenever x ≤ y and 2) for any non-empty finite subset B of Q, if inf B exists then η(inf B) ≥ inf{η(b) : b ∈ B}. Theorem 3.25: Every L-fuzzy Frink filter is an L-fuzzy Vfilter.
Proof: Let η be an L-fuzzy Frink filter and let x, y ∈ Q such that x ≤ y. Put η(x) = α. Since η is an L-fuzzy Frink filter, η α is a Frink filter of Q. Now η(x) = α ⇒ x ∈ η α ⇒ {x} lu ⊆ η α . Now x ≤ y ⇒ y ∈ x u = x lu ⊆ η α ⇒ η(x) = α ≤ η(y). Again let B be any nonempty subset of Q such that inf B exists in Q. Then inf B ∈ B lu and hence η(inf B) ≥ inf{η(a) : a ∈ B}. Therefore η is an L-fuzzy V -filter. Now we introduce the fuzzy version filters of a poset introduced by Halaš [7] which seems to be a suitable generalization of the usual concept of L-fuzzy filter of a lattice. Definition 3.26: η ∈ L Q is called an Lfuzzy filter in the sense of Halaš if it fulfills the followings: 1) η(1) = 1 and 2) for any a, b ∈ Q, η(x) ≥ η(a) ∧ η(b) for all x ∈ (a, b) lu In the rest of this paper, an L-fuzzy filter of a poset will mean an L-fuzzy filter in the sense of Halaš.
Lemma 3.27: η ∈ L Q is an L-fuzzy filter of Q if and only if η α is a filter of Q in the sense of Halaš for all α ∈ L. Corollary 3.28: A subset S of Q is a filter of Q in the sense of Halaš if and only if its characteristic map χ S is an L-fuzzy filter of Q. Lemma 3.29: If η is an L-fuzzy filter of Q, then the following assertions hold: 1) for any x, y ∈ Q η(x) ≤ η(y) whenever x ≤ y.
Theorem 3.30: Let (Q, ≤) be a lattice. Then an L-fuzzy subset η of Q is an L-fuzzy filter in the poset Q if and only if an L-fuzzy filter is in the lattice Q. Theorem 3.31: Let [S) H be a filter generated by subset S of Q in the sense of Halaš and χ S be its characteristic functions.
Lemma 3.32: The intersection of any family of L-fuzzy filters is an Lfuzzy filter. Now we give characterization of an Lfuzzy filter generated by a fuzzy subset of a poset Q. Definition 3.33: Let η be a fuzzy subset of Q and N be a set of positive integers. Define fuzzy subsets of Q inductively as follows: Theorem 3.34: The set {B η n : n ∈ N } forms a chain and the fuzzy subsetη defined byη(x) = sup{B η n (x) : n ∈ N } is a fuzzy filter generated by η.
Proof: Let x ∈ Q and n ∈ N . Then Therefore B η n ⊆ B η n+1 for each n ∈ N and hence {B η n : n ∈ N } is a chain. Now we showη is the smallest fuzzy filter containing η.
Thereforeη is a fuzzy filter. Again let θ be any L-fuzzy filter of Q such that η ⊆ θ . Now let a, b ∈ Q and x ∈ (a, b) lu . Then . Thus by induction we have θ (x) ≥ B η n (x) ∀n ∈ N and ∀ x ∈ (a, b) lu . Thus for any x ∈ Q, we havê Therefore θ ⊇η . This proves the theorem.
The above result yields the following.  Then σ is an L-fuzzy semi-filter but not an L-fuzzy filter.
Theorem 3.42: Let x ∈ Q and α ∈ L. Define an Lfuzzy subset α x of Q by for all y ∈ Q. Then α x is an L-fuzzy filter of Q. Proof: By the definition of α x , we clearly have α x (1) = 1. Let a, b ∈ Q and y ∈ (a, b) . Therefore in either cases we have α x (y) ≥ α x (a) ∧ α x (b) for all y ∈ (a, b) lu and hence α x is an L-fuzzy filter.
Definition 3.43: The L-fuzzy filter α x defined above is called the α-level principal fuzzy filter corresponding to x. Definition 3.44: An L-fuzzy filter µ of a poset Q is called an l-L-fuzzy filter if for any a, b ∈ Q, there exists x ∈ (a, b) l such that µ(x) = µ(a) ∧ µ(b).
Lemma 3.45: An L-fuzzy filter µ of Q is an l-L-fuzzy filter of Q if and only if µ α is an l-filter of Q for all α ∈ L.
Corollary 3.46: Let (Q, ≤) be a poset with 0 and let x ∈ Q and α ∈ L. Then the α-level principal fuzzy filter corresponding to x is an l-L-fuzzy filter.   Proof: Suppose η is an l-L-fuzzy filter. Let F be a finite subset of Q. Then there exists y ∈ F l such that η(y) = in f {η(a) : a ∈ F}.
Proof: Let σ be an L-fuzzy subset of Q defined by σ (x) = sup{η(a) ∧ θ (b) : x ∈ (a, b) lu } ∀x ∈ Q. Now we claim σ is the smallest L-fuzzy filter of Q containing η ∪ θ . Let x ∈ Q. x ∈ (r, s) lu } ≤ σ (x) for all x ∈ (a, b) lu and hence σ is an L-fuzzy filter. Let φ be any L-fuzzy filter of Q such that η ∪ θ ⊆ φ . Now for any x ∈ Q, we have and hence σ ⊆ φ . Therefore σ = (η ∪ θ ] = η ∨ θ , that is σ is the supremum of η and θ in F F (Q).