μ -Fuzzy Filters in Distributive Lattices

. In this paper, we introduce the concept of μ -fuzzy ﬁlters in distributive lattices. We study the special class of fuzzy ﬁlters called μ -fuzzy ﬁlters, which is isomorphic to the set of all fuzzy ideals of the lattice of coannihilators. We observe that every μ -fuzzy ﬁlter is the intersection of all prime μ -fuzzy ﬁlters containing it. We also topologize the set of all prime μ -fuzzy ﬁlters of a distributive lattice. Properties of the space are also studied. We show that there is a one-to-one correspondence between the class of μ -fuzzy ﬁlters and the lattice of all open sets in X μ . It is proved that the space X μ is a T 0 space.


Introduction
In 1970, Mandelker [1] introduced the concept of relative annihilators as a natural generalization of relative pseudocomplement and he characterized distributive lattices with the help of these annihilators. e concept of coannihilators and μ-filters in a distributive lattice with greatest element "1" was introduced by Rao and Badawy [2] and they characterized μ-filters in terms of coannihilators. For a filter F in L, μ(F) � (x) ++ : x ∈ F is an ideal in the set A + (L) of all coannihilators, and conversely μ ← (I) � x ∈ L: (x) ++ ∈ I is a filter in L when I is any ideal in A + (L). A filter F of L is called a μ-filter if μ ← μ(F) � F. In 1965, Zadeh [3] mathematically formulated the fuzzy subset concept. He defined fuzzy subset of a nonempty set as a collection of objects with grade of membership in a continuum, with each object being assigned a value between 0 and 1 by a membership function. Fuzzy set theory was guided by the assumption that classical sets were not natural, appropriate, or useful notions in describing the real-life problems, because every object encountered in this real physical world carries some degree of fuzziness. A lot of work on fuzzy sets has come into being with many applications to various fields such as computer science, artificial intelligence, expert systems, control systems, decisionmaking, medical diagnosis, management science, operations research, pattern recognition, neural network, and others (see [4][5][6][7]).
Alaba and Norahun [29] studied the concept of α-fuzzy ideals of a distributive lattice in terms of annulates. ey also studied the space of prime α-fuzzy ideals of a distributive lattice. In this paper, we introduce the dual of the concept of α-fuzzy ideals which is called μ-fuzzy filters in a distributive lattice with greatest element "1." We study the special class of fuzzy filters called μ-fuzzy filters. We prove that the set of all μ-fuzzy filters of a distributive lattice forms a complete distributive lattice isomorphic to the set of all fuzzy ideals of A + (L). We also show that there is a one-to-one correspondence between the class of prime μ-fuzzy filters of L and the set of all prime ideals of A + (L). We prove that every μ-fuzzy filter is the intersection of all prime μ-fuzzy filters containing it. Moreover, we study the space of all prime μ-fuzzy filters in a distributive lattice. e set of prime μ-fuzzy filters of L is denoted by X μ . For a μ-fuzzy filter θ of L, open subset of X μ is of the form X(θ) � η ∈ X μ : θ ⊈ η and V(θ) � η ∈ X μ : θ ⊆ η is a closed set. We also show that the set of all open sets of the form X(x β ) � η ∈ X μ : x β ⊈ η, x ∈ L, β ∈ (0, 1] forms a basis for the open sets of X μ . e set of all μ-fuzzy filters of L is isomorphic with the set of all open sets in X μ .

Preliminaries
We refer to Birkhoff [30] for the elementary properties of lattices.
Definition 1 (see [2]). For any set S of a lattice L, define S + as follows: Lemma 1 (see [2]). For any x, y ∈ L, the following conditions hold.
e set of all coannihilator denotes A + (L). Each coannihilator is a coannihilator filter, and hence, for two coannihilators (x) + and (y) + , their supremum and infimum in respectively. In a distributive lattice L with 1, the set of all coannihilators A + (L) of L is a lattice (A + (L), ∩ , ∨ ) and a sublattice of the Boolean algebra of coannihilator filters of L.
For a filter F in L, is an ideal in A + (L) and the set Definition 2 (see [3]). Let X be any nonempty set. A mapping μ: X ⟶ [0, 1] is called a fuzzy subset of X. e unit interval [0, 1] together with the operations min and max form a complete lattice satisfying the infinite meet distributive law. We often write ∧ for minimum or infimum and ∨ for maximum or supremum.
For any collection, μ i : i ∈ I of fuzzy subsets of X, where I is a nonempty index set, and the least upper bound ∪ i∈I μ i and the greatest lower bound ∩ i∈I μ i of the μ i 's are given for each x ∈ X, respectively.
For each t ∈ [0, 1], the set is called the level subset of μ at t [3].
Definition 4 (see [27]). A fuzzy subset μ of a lattice L is called a fuzzy ideal of L if; for all x, y ∈ L, the following conditions are satisfied: Definition 5 (see [27]). A fuzzy subset μ of a lattice L is called a fuzzy filter of L if, for all x, y ∈ L, the following condition is satisfied: In [27], Swamy and Raju observed the following: (1) A fuzzy subset μ of a lattice L is a fuzzy ideal of L if and only if (2) A fuzzy subset μ of a lattice L is a fuzzy filter of L if and only if μ(1) � 1 and μ(x ∧ y) � μ(x) ∧ μ(y), for all x, y ∈ L.
Let μ be a fuzzy subset of a lattice L. e smallest fuzzy filter of L containing μ is called a fuzzy filter of L induced by μ and denoted by [μ) and Lemma 2 (see [23]). For any two fuzzy subsets μ and θ of a distributive lattice L, we have e above result works dually. For any two fuzzy subsets μ and θ of a distributive lattice L, we have e binary operations "+" and "·" on the set of all fuzzy subsets of a distributive lattice L are as follows: If μ and θ are fuzzy ideals of L, then μ · θ � μ ∧ θ � μ ∩ θ and μ + θ � μ ∨ θ If μ and θ are fuzzy filters of L, then μ + θ � μ ∧ θ and μ · θ � μ ∨ θ e set of all fuzzy filters of L is denoted by FF(L).

μ-Fuzzy Filters
In this section, we introduce the concept of μ-fuzzy filters in a distributive lattice with greatest element "1." We study some basic properties of the class of μ-fuzzy filters. We prove that the class of μ-fuzzy filters forms a complete distributive lattice isomorphic to the class of fuzzy filters of A + (L). We also show that there is a one-to-one correspondence between the set of all prime μ-fuzzy filters of L and prime fuzzy ideals of A + (L). Finally, we observe that every μ-fuzzy filter is the intersection of all prime μ-fuzzy filters containing it.
roughout the rest of this paper, L stands for the distributive lattice with greatest element "1" unless otherwise mentioned. Theorem 1. Let θ be a fuzzy filter of L. en, the fuzzy subset On the other hand, Hence, μ(θ) is a fuzzy ideal of A + (L).

Theorem 2.
e set FI(A + (L)) of all fuzzy ideals of A + (L) forms a complete distributive lattice, where the infimum and supremum of any family θ j : j ∈ J of fuzzy ideals are given by

Corollary 1. For any two fuzzy filters θ and η of
Proof. For any

Lemma 6. For any fuzzy ideal θ of
us, μμ Lemma 7. For any fuzzy filter θ of L, the map θ ⟶ μ ← μ(θ) is a closure operator on FF(L). at is, , for any two fuzzy filters θ, η of L Now, we define μ-fuzzy filter.
us, μ-fuzzy filters are simply the closed elements with respect to the closure operator of Lemma 7, and μ ← μ(θ) is the smallest μ-fuzzy filter containing θ, for any fuzzy filter θ of L.

Theorem 4. For a nonempty fuzzy subset θ of L, θ is a μfuzzy filter if and only if each level subset of θ is a μ-filter of L.
Proof. Let θ be a μ-fuzzy filter of L. en, en, (x) ++ ∈ μ(θ t ), and there is y ∈ θ t such that (x) ++ � (y) ++ . us, μ Conversely, assume that each level subset of θ is a μ-filter. en, θ is a fuzzy filter and θ ⊆ μ

Corollary 2. For a nonempty subset F of L, F is a μ-filter if and only if χ F is a μ-fuzzy filter of L.
Theorem 5. Let θ be a fuzzy filter of L. en, θ is a μ-fuzzy filter if and only if, for each x, y ∈ L, (x) + � (y) + implies θ(x) � θ(y).

Theorem 6. A fuzzy filter θ of L is a μ-fuzzy filter if and only if
Proof. Suppose a fuzzy filter θ of L is a μ-fuzzy filter. en, by eorem 4, every level subset is a μ-filter of L. Let x ∈ L such that θ(x) � t. Since θ t is a μ-filter of L, then (x) ++ ⊆ θ t , which implies a ∈ θ t for all a ∈ (x) ++ . us, θ(a) ≥ θ(x) for each a ∈ (x) ++ . So, Suppose conversely that the condition holds. To prove θ is a μ-fuzzy filter, it suffices to show that μ Let us denote the set of all μ-fuzzy filters of L by FF μ (L). Proof. Clearly, (FF μ (L), ⊆ ) is a partially ordered set. For η, θ ∈ FF μ (L), define en, clearly η ∧ θ, η ∨ θ ∈ FF μ (L). We need to show η ∨ θ is the least upper bound of η, θ . Since θ, η ⊆ η ∨ θ ⊆ η ∨ θ, η ∨ θ is an upper bound of μ, θ . Let λ be any upper bound for η, θ in FF μ (L). en, η ∨ θ ⊆ λ, which implies that μ We now prove the distributivity. Let η, θ, λ ∈ FF μ (L). en, us, FF μ (L) is a distributive lattice. Now, we proceed to show the completeness. Since 1 { } and L are μ-filters, χ 1 { } and χ L are least and greatest elements of FF μ (L), respectively. Let θ i : i ∈ I ⊆ FF μ (L). en, ∩ i∈I θ i is a fuzzy filter of L and ∩ i∈I θ

Theorem 8. e set FF μ (L) is isomorphic to the lattice of fuzzy ideals of A + (L).
Proof. Define Hence, f is one to one.
Let λ ∈ FI(A + (L)). en, by Lemma 3, μ ← (λ) is a fuzzy filter of L. Now, we proceed to show that μ . us, by Lemma 6, we get that μμ erefore, f is an isomorphism of FF μ (L) onto the lattice of fuzzy filters of A + (L).

Theorem 9.
e following are equivalent for each nonconstant μ-fuzzy filter λ of L.
We have proved in eorem 8 that there is an order isomorphism between the class of μ-fuzzy filters and the set of fuzzy ideals of A + (L). Now, we show that there is an isomorphism between the prime μ-fuzzy filters and the prime fuzzy ideals of the lattice of coannihilators.

Theorem 10.
ere is an isomorphism between the prime μ-fuzzy filters and the prime fuzzy ideals of the lattice of coannihilator.
Proof. By eorem 8, the map f is an isomorphism from FF μ (L) into FI(A + (L)). Let σ be a prime μ-fuzzy filter of L.
Proof. Put P � σ ∈ FF μ (L): σ ⊆ η and η ∩ σ ≤ α . Since θ ∈ P, P is nonempty, and it forms a poset together with the inclusion ordering of fuzzy sets. Let A � θ i i∈I be any chain in P. en, clearly ∪ i∈I θ i is a μ-fuzzy filter. Since By applying Zorn's lemma, we get a maximal element, say σ ∈ P; that is, σ is a μ-fuzzy filter of L such that θ ⊆ σ and σ ∩ η ≤ α. Now, we proceed to show σ is a prime fuzzy filter. Assume that σ is not prime fuzzy filter. Let c 1 ∩ c 2 ⊆ σ such that c 1 ⊈σ and c 2 ⊈σ, c 1 , c 2 ∈ FF(L). If we put , then both σ 1 and σ 2 are μ-fuzzy filters of L properly containing σ. Since σ is maximal in P, we get σ 1 , σ 2 ∉ P.
us, σ 1 ∩ η≰α and is is a contradiction. us, σ is prime μ-fuzzy filter of L. Proof. Put P � σ ∈ FF μ (L): σ ⊆ η and η ∩ σ ≤ α . Since θ ∈ P, P is nonempty, and it forms a poset together with the inclusion ordering of fuzzy sets. Let A � θ i i∈I be any chain in P. Clearly, ∪ i∈I θ i is a μ-fuzzy filter. Since θ i (a) ≤ α for each i ∈ I, α is an upper bound of θ i (a): i ∈ I . us, ∪ i∈I θ i (a) ≤ α. So, ∪ i∈I θ i is a μ-fuzzy filter containing θ and ∪ i∈I θ i (a) ≤ α. Hence, ∪ i∈I θ i ∈ P. By applying Zorn's lemma, we get a maximal element, say σ ∈ P; that is, σ is a μfuzzy filter of L such that θ ⊆ σ and σ(a) ≤ α. Now, we proceed to show σ is a prime fuzzy filter. Assume that σ is not prime fuzzy filter. Let c 1 ∩ c 2 ⊆ σ and c 1 ⊈σ and c 2 ⊈σ, c 1 , c 2 ∈ FF(L). If we put and σ 2 � μ ← μ(c 2 ∨ σ), then both σ 1 and σ 2 are μ-fuzzy filters of L properly containing σ. Since σ is maximal in P, we get σ 1 , σ 2 ∉ P.

The Space of Prime μ-Fuzzy Filters
In this section, we study the space of prime μ-fuzzy filters of a distributive lattice and some properties of the space also.
is shows that x, y ∉ λ * . Since λ is prime fuzzy filter, card Im λ � 2 and λ * is prime.
Lemma 11. Let θ i : i ∈ I be any family of fuzzy filters of L. en, Conversely, let λ ∈ ∪ i∈I V(θ i ). en, λ ∈ V(θ i ) for each i ∈ I.
is implies θ i ⊆ μ. us, for any x ∈ L, μ(x) is an upper bound of θ i (x): i ∈ I . is implies that is shows that ∪ i∈I θ i ⊆ λ and

13.
e collection T � X(θ): θ { is a fuzzy filter of L} is a topology on X μ .
Next, let X(θ 1 ), X(θ 2 ) ∈ T. en, by Lemma 8, we get that is shows that T is closed under finite intersection. Now, we proceed to show that T is closed under arbitrary union. Let θ i : i ∈ I be any family of fuzzy filters of L. en, by Lemma 11 we have which implies ∪ i∈I X(θ i ) � X([ ∪ i∈I θ i )). us, by Lemma 9, we get that So, T is closed under arbitrary union. erefore, T is a topology on X μ . e space (X μ , T) will be called the space of prime μ-fuzzy filters in L.
In the above theorem, we proved that the family of X(θ) is a topology on X μ . In the following result, we show that the set of all open sets of the form X(x β ) is a basis for the topology on X μ .
For any fuzzy subset η of L, X(θ) � η ∈ X μ : θ⊈η is an open set of X μ and V(θ) � η ∈ X μ : θ ⊆ η � X μ − V(θ) is a closed set of X μ . In the following result, we prove the closure of a fuzzy set.
Theorem 18. For any family F ⊆ X μ , closure of F is given by F � V( ∩ λ∈F λ).

Conclusion
In this work, we studied the concept of μ-fuzzy filters of a distributive lattice. We proved that the set of all μ-fuzzy filters of a distributive lattice forms a complete distributive lattice isomorphic to the set of all fuzzy ideals of A + (L). We observed that every μ-fuzzy filter is the intersection of all μ-fuzzy filters containing it. We also studied the space of all prime μ-fuzzy filters in a distributive lattice. Our future work will focus on α-fuzzy ideals of a C-algebra.

Data Availability
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Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.