^{1}

^{*}

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The convergence theory of ideals in generalized topological molecular lattices is studied. Some properties of this kind of convergence are investigated. Finally, the relations between convergence theories of both molecular nets and ideals in GTMLs are discussed together with the
*GT*
_{2} separation axiom.

After Wang [

In [

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.

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

a)

b) If

The greatest minimal family of a is denoted by

Throughout this paper, the entry

a) For

b) For all

c)

Definition 2.1 [

(T_{1})

(T_{2})

The pair

Definition 2.2 [

Recall that according to the definition of ideals, the family

For a GTML

Definition 2.3 [

It is clear that a is an adherence point of A if and only if

Definition 2.4 [

a)

b) f is join preserving, i.e;

c)

Definition 2.5 [

1) continuous GOH, if for every

2) continuous at a molecule

For a directed set D and

The molecular net S is said to be:

1) eventually in A if there exists

2) frequently in A if for all

Definition 2.6 [

1) a is called a limit point of S, if

i.e,

2) a is called a cluster point of S, if

i.e,

Definition 2.7 [_{2}, if

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

Definition 3.1 Let

1) a is said to be a limit point of I if

The join of all limit points of i will be denoted by

2) a is said to be a cluster point of I if

The join of all cluster points of i will be denoted by

As a consequence, we obtain the following proposition:

Proposition 1 Let

1)

2)

3)

4)

Proof.

1) Let

Therefore, we have

2) Let

Since

Therefore, we have

3) Let

Therefore, we have

4) Let

Theorem 2 Let

1)

2)

Proof.

1) Let

Conversely, let

2) Let

Now, let

Corollary 1 Let

1)

2)

Theorem 3 Let

Proof.

Put

Lemma 1 Let

is an ideal in

Proof. It is easily to check the conditions of ideals. □

Theorem 4 Let

Proof. Let f be a continuous GOH at

we have

Therefore,

Theorem 5 Let

Proof.

Since f is a GOH, then f preserves arbitrary joins and hence

Assume that

So,

Then,

Therefore, f is continuous GOH. □

In [

For a generalized topological molecular lattice

is a directed set with respect to the relation “

Set

is a molecular net in

Now, let

is an ideal in l called the ideal generated by S.

Theorem 6 Let

1)

2)

3)

Proof. 1) Case I: Let

Therefore,

Conversely, let

Case II: Let

Thus,

Conversely, we need to show that

Now,

Therefore,

2) Let

Conversely, let

3) Let

Thus,

Now, let

Therefore,

According to Theorem 6, one can get directly the following result:

Corollary 2 Let

1)

2)

3)

Theorem 7 Let

Proof. Let

Now,

Therefore,

In 1986, Yang [

Definition 4.1 [

Definition 4.2 [

Proposition 8 Let

Proof. It is clear that

Let

Then J is an ideal in L and clearly that

Since I is a maximal ideal in L, we get

So,

Theorem 9 Let

(i) For every ideal I,

(ii) For every maximal ideal I,

(iii) For every universal net S,

Proof.

So,

Lastly, we conclude this section by studying the relation between the ideal convergence and the GT_{2} separa- tion axiom in GTMLs.

Theorem 10 Let _{2}, if and only if for every ideal I in L,

Proof. _{2}, I be an ideal in L. Assume that

Therefore,

_{2}, then

Then I is an ideal in L with _{2}. □

Corollary 3 Let

a) _{2}.

b) For every molecular net S and every

c) For every ideal I in L and every

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 GT_{2} separation axiom.

KamalEl-Saady,FatimaAl-Nabbat, (2015) Ideal Convergence in Generalized Topological Molecular Lattices. Advances in Pure Mathematics,05,653-659. doi: 10.4236/apm.2015.511059