Ideal Convergence in Generalized Topological Molecular Lattices ()
1. Introduction
After Wang [1] introduced the theory of topological molecular lattices or TMLs for short, several authors established various kinds of convergence theory in TMLs by using a corresponding concept of remote neighborhoods (see e.g. [2] , [3] - [5] ). The theory of remote neighborhood has been established first by Wang [1] as a dual notion of Pu and Liu’s theory of the quasi-coincident neighborhoods in fuzzy topology [6] [7] .
In [8] , we introduced a generalization of Wang’s topological molecular lattice called generalized topological molecular lattice or briefly GTML and studied the convergence theory of molecular nets by using the concept of generalized remote neighborhoods in these spaces.
In this paper, we aim to study the convergence of ideals in GTMLs and investigate the relations among this notion and that of molecular nets. Moreover, we state the relations with other defined topological notions in GTMLs such as generalized order homomorphism or GOH for short.
The paper is organized as follows. In Section 2, we will review some useful concepts in the paper. In Section 3, we will study the convergence in GTMLs in terms of ideals and investigate some properties of such conver- gence. Furthermore, we show the relations between convergence of ideals and the continuity of GOHs. In Section 4, we will discuss the relations between convergence of molecular nets and convergence of ideals in TMLs. Finally, Section 5 presents our conclusions.
2. Preliminaries
This section is devoted to recall some useful concepts which is required in the sequel. Let L be a complete lattice with the smallest element
and the largest element
, an element
is said to be a molecule (some time called co-prime or join-irreducible) if for
then
or
. The set of all mole- cules in L is denoted by
. The subset
is called a minimal family of a [1] , if the following two conditions are hold:
a)
.
b) If
and
, then
such that
.
The greatest minimal family of a is denoted by
while
.
Throughout this paper, the entry
denotes a molecular lattice, that is a lattice L and the set of its molecules M. For a non empty subset I of a complete lattice L, I is said to be an ideal [9] , if it satisfies the following conditions:
a) For
and
.
b) For all
.
c)
.
Definition 2.1 [8] Let
be a molecular lattice. A subfamily
is said to be a generalized closed topology, or briefly, generalized co-topology, if
(T1)
is closed under arbitrary intersections;
(T2)
.
The pair
is called a generalized topological molecular lattice, or briefly, GTML.
Definition 2.2 [8] Let
be a GTML,
, and
. Then F is said to be a genera- lized remote neighborhood of a. The set of all generalized remote neighborhoods of a will be denoted by
.
Recall that according to the definition of ideals, the family
is not necessary be an ideal in GTMLs while the family
satisfies the ideal conditions.
For a GTML
and
, the intersection of all η-elements containing A is called the gene- ralized closure of A and denoted by
. that is,
![]()
Definition 2.3 [8] Let
be a GTML,
, then a is said to be an adherence point of A, if for all
, we have
.
It is clear that a is an adherence point of A if and only if
.
Definition 2.4 [1] Let
and
be complete lattices. A mapping
is said to be a generalized order homomorphism or GOH for short if
a)
if and only if
.
b) f is join preserving, i.e;
.
c)
is join preserving, where
,
.
Definition 2.5 [8] Let
and
be GTMLs and
be a GOH, then f is called:
1) continuous GOH, if for every
, we have
.
2) continuous at a molecule
, if for every
, we have
.
For a directed set D and
, the mapping
is called a molecular net and denoted by
. The molecular net s is said to be in A, if
.
The molecular net S is said to be:
1) eventually in A if there exists
such that
, we have
.
2) frequently in A if for all
there exists
such that
.
Definition 2.6 [8] Let
be a GTML,
be a molecular net and
, then:
1) a is called a limit point of S, if
eventually true, and denoted by
. The join of all limit points of S will be denoted by
.
i.e,
.
2) a is called a cluster point of S, if
frequently true, and denoted by
. The join of all cluster points of S will be denoted by
.
i.e,
.
Definition 2.7 [8] Let
be a GTML, then
is said to be a Gt2, if
, there exists
and
such that
.
3. Convergence of Ideals in GTMLs
The aim of this section is to study the convergence in GTMLs in terms of ideals and investigate some properties of such convergence. Furthermore, we show the relations between convergence of ideals and the continuity of GOHs.
For the sake of convenience and no confusion, throughout this section and forwards, we restrict the attention of generalized remote neighborhoods of an element a in GTMLs into the set
instead of
.
Definition 3.1 Let
be a GTML,
be an ideal of L and
, then
1) a is said to be a limit point of I if
, denoted by
. In this case, we say that i converges to a.
The join of all limit points of i will be denoted by
.
2) a is said to be a cluster point of I if
and
, we have
, denoted by
. In this case, we say that i accumulates to a.
The join of all cluster points of i will be denoted by
.
As a consequence, we obtain the following proposition:
Proposition 1 Let
be a GTML, i and J be ideals of L with
and
. Then we have:
1)
.
2)
.
3)
.
4)
.
Proof.
1) Let
, then
. Thus,
and hence
.
Therefore, we have
.
2) Let
, then
and
, we have
.
Since
, then
and hence
and
,
.
Therefore, we have
.
3) Let
, then
. Since
, then we get
. So,
.
Therefore, we have
.
4) Let
, then
and
,
. But
, then
and
, we have
. Therefore, we have
. □
Theorem 2 Let
be a GTML, I be an ideal of L and
, then
1)
if and only if
.
2)
if and only if
.
Proof.
1) Let
, by the definition of
, it is clear that
.
Conversely, let
and
, then
and hence
. So, there exists
such that
, then
. Thus, we have
but
, hence
. Therefore,
.
2) Let
, then similarly to 1),
is clear.
Now, let
and
, than
and hence
. So, there exists
such that
, then
. Thus, for all
, also, we have
. So,
. Therefore,
. □
Corollary 1 Let
be a GTML, I be an ideal of L and
, then
1)
if and only if
.
2)
if and only if
.
Theorem 3 Let
be a GTML,
, and
, then
if and only if there exists an ideal I in L such that
and
.
Proof.
Since
, then a is an adherence point of a, i.e;
.
Put
, then
is an ideal and clearly that
also, we have
which implies
.
Let
, then
, we have
, i.e;
. Since
, then
. So, by Definition 2.3, a is an adherence point of A and hence
. □
Lemma 1 Let
and
be GTMLs,
be a GOH, and I be an ideal in
. Then the set
![]()
is an ideal in
.
Proof. It is easily to check the conditions of ideals. □
Theorem 4 Let
and
be GTMLs,
be a continuous GOH at
and I be an ideal in
. If
, then
.
Proof. Let f be a continuous GOH at
and I be an ideal in
with
, then
,
we have
. Hence, we get that
and for every
. so,
which implies that
.
Therefore,
. □
Theorem 5 Let
and
be GTMLs,
be a GOH, then f is continuous GOH if and only if for every ideal I of
,
.
Proof.
Let I be an ideal of
such that
, hence
. We need to show that
. Since f is a continuous GOH, then f is continuous at
and
. Hence, by Theorem 4, we get
, i.e;
.
Since f is a GOH, then f preserves arbitrary joins and hence
.
We want to prove that f is continuous at every
, i.e;
, we have
.
Assume that
. Hence, there exists an ideal i such that
and
. Then
which implies that
. Thus,
.
So,
, we have
. By the definition of
, there exists
such that
equivalently that
. Hence,
but
, so
. Contradiction.
Then,
and hence
.
Therefore, f is continuous GOH. □
4. Relations between Convergence of Molecular Nets and Convergence of Ideals in GTMLs
In [3] and [5] , the authors introduced a comparison between convergence of molecular nets and convergence of ideals in TMLs. In similar way, we discuss the relations between them in GTMs.
For a generalized topological molecular lattice
, let I be an ideal in L, then the set
![]()
is a directed set with respect to the relation “
”defined as
![]()
Set
, then the set
![]()
is a molecular net in
called the molecular net generated by the ideal I.
Now, let
be a molecular net in L, then the set
![]()
is an ideal in l called the ideal generated by S.
Theorem 6 Let
be a GTML,
be an ideal in l and S be a molecular net in L, then we have
1)
, (resp.
).
2)
.
3)
.
Proof. 1) Case I: Let
, then
eventually, i.e; there exists
such that
, we have
. Hence,
we get
, so
but
which implies that
.
Therefore,
and
.
Conversely, let
, then
. Since
, then
and
such that
, we have
, but
, hence
. Thus,
.
Case II: Let
,
and
, then there exists
with
. Thus,
, since
, there exists
such that
and
. Since
, then
but
, so
.
Thus,
and
,
. Therefore,
.
Conversely, we need to show that
eventually. Let
, then
and
.
Now,
, we have
, therefore,
such that
and
. So,
and
,
.
Therefore,
frequently and
.
2) Let
, then
. By the definition of
, we have
eventually which means that
.
Conversely, let
, then
eventually. So,
, i.e,
which means
.
3) Let
, then there exists
such that
and
with
, we have
. But B ≥ A, hence
, i.e;
eventually.
Thus,
and
.
Now, let
, then
eventually, i.e; there exists
such that
, we have
. Since
and
, then
. Hence,
and
, then
and
.
Therefore,
. □
According to Theorem 6, one can get directly the following result:
Corollary 2 Let
be a GTML, I be an ideal in L and S be a molecular net in L, then the following statements hold:
1)
.
2)
.
3)
.
Theorem 7 Let
be a GTML, I be an ideal in L and S be a molecular net in L, then we have
.
Proof. Let
, then
. So, we need to show that
.
Now,
frequently. Also,
eventually and hence,
fre- quently. So,
and
, we get
.
Therefore,
and hence,
. □
In 1986, Yang [9] introduced the concepts of maximal ideals and universal nets.
Definition 4.1 [9] An ideal I in a complete lattice L is called a maximal ideal , if for each ideal J in L such that
, we have
.
Definition 4.2 [9] A molecular net S in a complete lattice L is called a universal net , if there exists a maximal ideal in L such that S is a subnet of
.
Proposition 8 Let
be a GTML and I be a maximal ideal in L, then
.
Proof. It is clear that
. Now, we prove that
.
Let
, then
. Put ![]()
Then J is an ideal in L and clearly that
and
.
Since I is a maximal ideal in L, we get
, hence
.
So,
and
. Therefore,
. □
Theorem 9 Let
be a GTML, then the following conditions are equivalent:
(i) For every ideal I,
such that
.
(ii) For every maximal ideal I,
such that
.
(iii) For every universal net S,
such that
.
Proof.
Let I be a maximal ideal, by (i),
such that
. Since, I is a maximal, then by Proposition 8, we have
.
Let I be an ideal, then there exists a maximal ideal J with
and
such that
. Hence,
.
So,
and
. Thus,
.
Let S be a universal net and
, then by the definition, there exists a maximal ideal I such that S is a subnet of
. By (ii), we have
and hence
. Therefore,
.
Let I be a maximal ideal, then
is a universal net, by (iii),
such that
. Then, we get
. □
Lastly, we conclude this section by studying the relation between the ideal convergence and the GT2 separa- tion axiom in GTMLs.
Theorem 10 Let
be a GTML, then it is GT2, if and only if for every ideal I in L,
contains no disjoint molecules.
Proof.
Let
be GT2, I be an ideal in L. Assume that
with
. Then there exists
and
such that
. Since
and
, we have that
and
. Hence,
which implies that
. Contradiction with the definition of I.
Therefore,
contains no disjoint molecules.
Assume that
is not GT2, then
with
and
, we have
. Put
![]()
Then I is an ideal in L with
and
. Hence, limI contains two disjoint molecules
which contradicts the assumption. Therefore,
is GT2. □
Corollary 3 Let
be a GTML, then the following statements are equivalents:
a)
is a GT2.
b) For every molecular net S and every
, we have
.
c) For every ideal I in L and every
, we have
.
5. Conclusion
In this paper, we introduced a convergence theory of ideals in generalized topological molecular lattices by using the concept of generalized remote neighborhoods and studied some of its characterization and properties. Then, we investigated the relations between the ideal convergence and the continuity of GOH in GTMLs. Finally, we discussed the relations among the convergence theories of both ideals and molecular nets and also the GT2 separation axiom.