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ABSTRACT
In this paper, we introduce the concept of filters in a pseudo-complemented distributive lattices. We study the special class of fuzzy filters called
filters, which is isomorphic to the set of all fuzzy ideals of the lattice of boosters. We observe that every
filters is the intersection of all prime
filters containing it. We also topologize the set of all prime
filters of a pseudo-completed distributive lattice. Properties of the space also studied. We show that there is a one to one correspondence between the class of
filters and the lattice of all open sets in
. It is proved that the space
is a T0 space.
1. Introduction
Classical set theory is based on the fundamental concept of a set, in which individuals are either a member or not a member. A sharp, crisp, and unambiguous distinction exists between a member and a non-member for any well-defined set of entities in this theory, and there is a very precise and clear boundary to indicate if an entity belongs to a set or not. Thus, in classical set theory an element is not allowed to be in a set or not in a set at the same time. This means that many real-world problems cannot be handled by classical set theory. Many of the collections and categories we commonly employ, however, do not exhibit this characteristic. Instead their boundaries seem vague, and the transition from member to nonmember appears gradual rather than abrupt. Thus fuzzy set introduces vagueness by eliminating the sharp boundary dividing members of the class from nonmembers. Real situations are very often not crisp and deterministic and they can not be described precisely. Such situations in our real life which are characterized by vagueness or imprecision can not be answered just in yes or no. In 1965 (Zadeh, Citation1965) mathematically formulated the fuzzy subset concept. He defined fuzzy subset of a non-empty 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. In 1971, A. Rosenfeld used the notion of a fuzzy subset of a set to introduce the notion of a fuzzy subgroup of a group (Rosenfeld, Citation1971) and his paper inspired the development of fuzzy abstract algebra. Since then, several authors have developed interesting results on fuzzy theory; see (Abou-Zaid, Citation1993; Addis & Engidaw, Citation2020; Ajmal & Thomas, Citation1994; Alemayehu & Wondifraw, Citation2021; Asaad, Citation1991; Liu, Citation1982; Muhiuddin et al., Citation2021; Norahun & Dvorák, Citation2020; Norahun & Zeleke, Citation2021; Yuan & Wu, Citation1990).
The theory of pseudo-complementation was introduced and extensively studied in semi-lattices and particularly in distributive lattices by (Frink, Citation1962) and (Birkhoff, Citation1948). Later, pseudo-complement in Stone algebra has been studied by several authors like (Balbes & Horn, Citation1970), G. Grtzer (Frink, Citation1963) etc. In (Badawy, Citation2016), A. E. Badawy studied the concept of filters of p-algebras with respect to a closure operator. Motivated by Badawy’s work, in this paper, we study the fuzzy aspect of
filters in a p-algebra.
In this paper, we introduce the concept of filters in pseudo-complemented distributive lattice. Basic properties of
filters also investigated. We observe that there is a homomorphism mapping between the class of all fuzzy filters and the class of all fuzzy ideals of the boosters. We also study the special class of fuzzy filters called
filters. We prove that these fuzzy filters forms a complete distributive lattice and isomorphic to the set of fuzzy ideals of the lattice of boosters
. Furthermore, we show that there is a one to one correspondence between the class of prime
filters of a p-algebra L and the set of all prime ideals of
. We observe that every
filter is the intersection of all prime
filters containing it. We give the definition of prime fuzzy filter using the concept of fuzzy points. Moreover, we study the space of all prime
filters in a pseudo-complemented distributive lattice. The set of prime
filters of L is denoted by
. For a
filter ν of L, open subset of
is of the form
and
is a closed set. We also show that the set of all open sets of the form
forms a basis for the open sets of
. Finally, we observe that there is a one to one correspondence between the class of all
filters of L and the lattice of all open sets of prime
filters
.
2. Preliminaries
We refer (Badawy, Citation2016; Birkhoff, Citation1948) for the elementary concepts of lattices and γ-filters of a pseudo-complemented distributive lattice. We also take a reference (Swamy & Raju, Citation1998; Zadeh, Citation1965), for the elementary concepts of fuzzy set theory and fuzzy ideals of a lattice.
An algebra is of type
is a pseudo-complemented distributive lattice , if the following conditions hold:
(1) |
| ||||
(2) | for all |
Definition 2.1.
(Badawy, Citation2016) Let L be a pseudo-complemented distributive lattice. Then for any , define the booster of a as follows:
Then it can be easily observed that is a filter of L containing a.
The set of all boosters of a pseudo-complemented lattice is denoted by . (Badawy, Citation2016).
Lemma 2.2.
For any , the following conditions hold.
(1) | |||||
(2) | |||||
(3) | |||||
(4) |
|
Definition 2.3.
(Swamy & Raju, Citation1998) A fuzzy subset µ of a lattice L is called a fuzzy ideal of L if, for all the following condition satisfies:
(1) | |||||
(2) | |||||
(3) |
Definition 2.4.
(Swamy & Raju, Citation1998) A fuzzy subset µ of a lattice L is called a fuzzy filter of L if, for all the following condition satisfies:
(1) | |||||
(2) | |||||
(3) |
The set of all fuzzy filters of L is denoted by FF(L).
Let µ be a fuzzy subset of a lattice L. The smallest fuzzy filter of L containing µ is called a fuzzy filter of L induced by µ and denoted by and
The binary operations ”+” and “·” on the set of all fuzzy subsets of a distributive lattice L as:
If µ and θ are fuzzy ideals of L, then and
.
If µ and θ are fuzzy filters of L, then and
Lemma 2.5.
(Alaba & Norahun, Citation2018) For any two fuzzy subsets µ and θ of a distributive lattice L, we have
The above result works dually, that is
For any two fuzzy subsets µ and θ of a distributive lattice L, we have
3. ![](//:0)
filters
In this section, we introduce the concept of filters in a pseudo-complemented distributive lattice. We study some basic properties of the class of
filters. We prove that the class of
filters forms a complete distributive lattice and isomorphic to the class of fuzzy filters of
. We also show that there is a one to one correspondence between the set of all prime
filters of L and prime fuzzy ideals of
. Finally, we observe that every
filters is the intersection of all prime
filters containing it.
Throughout the rest of this paper L stands for a pseudo-complemented distributive lattice unless otherwise mentioned.
Theorem 3.1.
For any fuzzy filter η of L, the fuzzy subset of
defined by:
is a fuzzy ideal of .
Proof.
Let η be a fuzzy filter of L. Then it can be easily verified that . Now, for any
,
Thus, .
On the other hand,
This shows that . Hence
is a fuzzy ideal of
.
Lemma 3.2.
Let λ be a fuzzy ideal of . Then the fuzzy subset
of L defined as
is a fuzzy filter of L.
Proof.
Since is the smallest element of
, we get
. For any
,
Thus is a fuzzy filter of L.
Lemma 3.3.
If λ and ν are fuzzy filters of L, then implies
.
Lemma 3.4.
If are fuzzy ideals of
, then
implies
.
Theorem 3.5.
The set of all fuzzy ideals of
forms a complete distributive lattice, where the infimum and supremum of any family
of fuzzy ideals is given by:
Theorem 3.6.
The mapping γ is a homomorphism of FF(L) into .
Proof.
Let be two fuzzy filters of L. Then by Lemma 3.3, we have
and
. For any
,
Thus .
And
Thus . So γ is a homomorphism.
Corollary 3.7.
For any two fuzzy filters λ and ν of L,
Lemma 3.8.
Let λ be a fuzzy ideal of . Then
.
Proof.
Since λ is a fuzzy ideal of , by Lemma 3.2,
is a fuzzy filter of L and
is a fuzzy ideal of
. Now,
. Thus
.
Now we define filter.
Definition 3.9.
A fuzzy filter θ of L is called a filter of L if
.
Example 3.10.
Consider the p-algebra whose Hasse diagram is given below.
Define fuzzy subsets λ and ν of L as follows:
Then it can be easily verified that λ is a filter of L. But ν is a fuzzy filter of L but not a
filter of L.
Theorem 3.11.
For a nonempty fuzzy subset θ of L , θ is a filter if and only if each level subset of θ is a γ-filter of L.
Proof.
Let θ be a filter of L. Then
. To prove each level subset of θ is a γ-filter of L, it is enough to show
for all
. Clearly,
. Let
. Then
and there is
such that
. Which implies
. This shows that
. Thus,
and hence each level subset of θ is a γ-filter of L.
Conversely, assume that each level subset of θ is a γ-filter. Then θ is a fuzzy filter and . Let
. Then for each ϵ > 0, there is
such that
and
.
Corollary 3.12.
For a nonempty subset F of L, F is a γ-filter if and only if χF is a filter of L.
Theorem 3.13.
Let λ be a fuzzy filter of L. Then λ is a filter if and only if for each
imply
.
Lemma 3.14.
Let λ be a filter of L. Then
for all
.
Proof.
Let λ be a filter of L. For any
, we know that
. Thus, by Theorem 3.13, we get that
.
Lemma 3.15.
For any fuzzy filter λ of L, the map is a closure operator on FF(L). That is.
(1) | |||||
(2) | |||||
(3) |
|
filters are simply the closed elements of FF(L) with respect to the closure operator.
Let us denote the set of all filters of L by
.
Lemma 3.16.
If , then the supremum of λ and ν is given by:
Proof.
Clearly is a
filter of L. Now we proceed to show that
is the least upper bound of
. Since
,
is an upper bound of
. Let η be any upper bound for
in
. Then
. Which implies that
. Therefore,
is the supremum of both
in
.
Theorem 3.17.
The class of all γ-fuzzy filters of L forms a complete distributive lattice with respect to set inclusion.
Proof.
Clearly is a partially ordered set. For
, define
Then clearly . Hence
is a lattice.
We now prove the distributivity. Let . Then
Thus is a distributive lattice. Next we prove the completeness. Since
and L are γ-filters,
and χL are least and greatest elements of
respectively. Let
. Then
is a fuzzy filter of L and
.
Thus . So
is a complete distributive lattice.
Theorem 3.18.
The set is isomorphic to the lattice of fuzzy ideals of
.
Proof.
Define
Let and
. Then
. Thus
. So
. Hence f is one to one.
Let . Then by Lemma 3.2,
is a fuzzy filter of L. Now we proceed to show that
is a
filter of L. Let
. Then
. Thus by Lemma 3.8, we get that
. So
. Thus for each
. Therefore, f is onto.
Now for any ,
. Similarly
. Therefore, f is an isomorphism of
onto the lattice of fuzzy filters of
Theorem 3.19.
The following are equivalent for each non-constant filter λ of L.
(1) | For all | ||||
(2) | For any fuzzy points xα and yβ of L, | ||||
(3) | For all |
Proof.
: Let
such that
. Then
. Since L is a distributive lattice, by Lemma 3.3 of (Norahun & Zeleke, Citation2021), we have
. Since
and
are fuzzy filters of L, by the assumption,
or
. This shows that
or
.
: Let
such that
. Now we need to show
or
. Suppose not. Then
and
. Which implies there exist
such that
and
. Put
and
. Then
and
. Since
, we have
. By the assumption, we get that
or
. Which is a contradiction. Thus
or
.
: Suppose
such that
. Then by Corollary 3.7 we have
. Since
and
are
filters, by the assumption, we get that
or
. Which implies
or
.
Definition 3.20.
By a prime filter, we mean a non-constant
filter of L satisfying (1) and hence all of the condition of Theorem 3.19.
We have proved in Theorem 3.18 that, there is an order isomorphism between the class of filters and the set of fuzzy ideals of
. Now, we show that there is an isomorphism between the prime
filters and the prime fuzzy ideals of the lattice of boosters of L.
Theorem 3.21.
There is an isomorphism between the prime filters and the prime fuzzy ideals of the lattice of booster.
Proof.
By Theorem 3.18 the map f is an isomorphism from into
. Let σ be a prime
filter of L. Then
. Now we prove
is a prime fuzzy ideal of
. Let
such that
. Since f is onto, there exist
such that
and
. Thus
. Since
is an isotone, we have
. Thus
. Since σ is a prime fuzzy filter, either
or
. This shows that either
or
. Thus
or
. Hence
is a prime fuzzy ideal of
.
Conversely, suppose that θ is a prime fuzzy ideal in . Since f is onto, there exists a
filter σ in
such that
. Let
such that
. Since γ is an isotone, we get
. Thus
. Since
is a prime fuzzy ideal of
, either
or
. This implies
or
. Since σ is a
filter, we get
or
. Thus σ is prime fuzzy filter in
. So the prime
filters corresponds to prime fuzzy ideals of
.
Theorem 3.22.
Let θ be a filter of L and η be a fuzzy ideal of L such that
,
. Then there exists a prime
filter λ of L such that
and
.
Proof.
Put . Since
,
is nonempty and it forms a poset together with the inclusion ordering of fuzzy sets. Let
be any chain in
. Then clearly
is a
filter. Since
for each
, we get that
. Thus
. By applying Zorn’s lemma we get a maximal element, let say
; that is σ is a
filter of L such that
and
.
Now we proceed to show σ is a prime fuzzy filter. Assume that σ is not prime fuzzy filter. Let such that
and
. If we put
and
, then both σ1 and σ2 are
filters of L properly containing σ. Since σ is maximal in
, we get
. Thus
and
. This implies there exist
such that
and
. Which implies
. This shows that
. This is a contradiction. Thus σ is prime
filter of L.
Corollary 3.23.
Let θ be a filter of L,
and
. If
, then there exists a prime
filter η of L such that
and
.
Proof.
Put . Since
,
is nonempty and it forms a poset together with the inclusion ordering of fuzzy sets. Let
be any chain in
. Clearly
is a
filter. Since
for each
, α is an upper bound of
. Thus
. So
is a
filter containing θ and
. Hence
. By applying Zorn’s lemma we get a maximal element, let say
; that is σ is a
filter of L such that
and
.
Now we proceed to show σ is a prime fuzzy filter. Assume that σ is not prime fuzzy filter. Let and
and
. If we put
and
, then both σ1 and σ2 are
filters of L properly containing σ. Since σ is maximal in
, we get
. Thus
and
. Now
. This is a contradiction. Hence σ is prime
filter.
Corollary 3.24.
For any filter of L is the intersection of all prime
filters containing it.
Proof.
Let θ be a proper filter of L. Consider the following.
Clearly . Assume that
. Then there is
such that
. Let
. Consider the set
By the above corollary we can find a prime filter µ of L such that
and
. This implies
. This shows that
. Which is a contradiction. Thus
. So
.
4. The space of prime γ-fuzzy filters
In this section, we study the space of prime filters of a pseudo-complemented distributive lattice and some properties of the space also studied.
Let be the set of all prime
filters of a pseudo-complemented distributive lattice. Let
where λ is a fuzzy subset of L and
. We let
, i.e.
.
Lemma 4.1.
Let η and ν be fuzzy filters of L. Then
(1) | |||||
(2) | |||||
(3) |
Proof.
1. Let and
. Then
and
. Thus
.
2. By (1), we have . To show the other inclusion, let
. Then
. Thus either
or
. So
. Hence
.
3. It can be easily verified that . To show the other inclusion, let
. Then
. Since δ is a prime fuzzy filter, we have
. Thus
. So
.
Lemma 4.2.
Let λ be a fuzzy subset of L. Then .
Proof.
Since , we have
. Let
. Then
. We need to show
. Assume that
. Then
. Which is a contradiction. Thus
. So
.
Theorem 4.3.
Let and
. Then
1. | |||||
2. | |||||
3. |
|
Proof.
(1) If , then either
or
. Which implies either
or
. This shows that
. Thus
. Hence
. To show the other inclusion, let
. Then
. This implies either
or
. Thus
.
(2) If , then
and
. This implies
and
. This shows that
. Since ν is prime fuzzy filter, card Im ν = 2 and
is prime. Thus
. Which implies
. Thus
and hence
.
Conversely, let . Then
. Which implies
. Thus
and
. This shows that
and
. Thus
. So
. Therefore,
.
(3) To show , let
. Since ν is prime fuzzy filter, ν is two valued. Thus
. This implies that there is
such that
. Let us take some
such that
. Then
. Which implies that
. Thus
and hence
. Therefore,
.
Lemma 4.4.
Let ;
and
. Then
Proof.
If , then
and
. This implies that
and
. Since
is prime filter and
, we have
and
. This shows that
. Thus
and hence
. To show the other inclusion, let
. Then
. This implies
and
. Thus
and
. So
. Hence
.
Lemma 4.5.
Let be any family of fuzzy filters of L. Then
Proof.
Since for each
, we have
for each
. Thus
.
Conversely, let . Then
for each
. This implies
. Thus for any
,
is an upper bound of
. This implies that
. This shows that
and
. So
. Thus
. Hence
.
Theorem 4.6.
The collection .
Proof.
Consider the fuzzy subsets of L defined as:
and
for all
. Clearly
and ν2 are fuzzy filters of L. Again,
, for all
. Then
and thus
. Since each
is non-constant,
for all
. So
. Hence
.
Next, let . Since ν1 and ν2 are fuzzy filters of L, then by Lemma 4.1, we get that
. Then
is closed under finite intersection.
Finally, let be any family of fuzzy filters of L. Then by Lemma 4.5 we have
Which implies . Thus by Lemma 4.2, we get that
So is closed under arbitrary union. Consequently,
is a topology on
. The space
will be called the space of prime
filters in L.
In the above theorem we proved that, the family of is a topology on
. In the following result, we show that the set of all open sets of the form
is a basis for the topology on
.
Theorem 4.7.
The collection forms base for some topology
.
Proof.
Let be any open set in
and
. Then
and there is
such that
. Put
, then
and
. To show
, let
. Then
and
. Which implies
. Thus
. Hence for any open set
in
we can find
in
such that
. Therefore,
is a base for
.
Theorem 4.8.
The space is a T0-space.
Proof.
Let such that
. Then either
or
. Without loss of generality, we can assume that
. Then
and
. Thus
is a T0-space.
Theorem 4.9.
For any fuzzy filter λ of L, .
Proof.
For any fuzzy filter λ of L, we have and
. Let
. Then
. Suppose
, then
. This implies
. Which is impossible. Thus
and hence
.
In the following result, we show that there is a one to one correspondence between the class of filters and the lattice of all open sets in
.
Theorem 4.10.
The lattice is isomorphic with the lattice of all open sets in
.
Proof.
The lattice of all open sets in Xγ is . Define the mapping
Since and
is a
filter, every open subset of
is of the form
for some
. Hence the map is onto.
Let . If
, then there is
such that either
or
. Without loss of generality, we can assume that
. Put
. Then by Corollary 3.23, we can find a prime
filter η such that
and
. Thus
and
. So
and
. This is a contradiction. Hence
.
Now we show that f is homomorphism. Let . Then
. Similarly,
. This shows that f is a homomorphism. Hence f is an isomorphism.
For any fuzzy subset λ of L, is an open set of
and
is a closed set of
. In the following result we prove the closure of a fuzzy set.
Theorem 4.11.
For any family , closure of
is given by
.
Proof.
We know that closure of is the smallest closed set containing
. To prove our claim, it is enough to show that
is the smallest closed set containing
. Since the set of all
filter is a complete distributive lattice,
is a
filter and
is a closed set in
. If
, then
. Thus
. This implies that
. Let
be any closed set in
containing
. Then
, for each
. Thus
and
. So
is the smallest closed set containing
. Hence
.
5. Conclusion
In this work, we studied the concept of filters of a pseudo-complemented distributive lattice. We have shown that the set of all
filters of a distributive lattice forms a complete distributive lattice and isomorphic to the set of all fuzzy ideals of
. We observed that every
filter is the intersection of all
filters containing it. We also studied the space of all prime
filters in a distributive lattice. Our future work will focus on σ-fuzzy ideals in a 0–1 distributive lattice.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Data availability statement
No data were used to support this study.
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