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The aim of this paper is to introduce the concept of generalized topological molecular lattices briefly GTMLs as a generalization of Wang’s topological molecular lattices TMLs, Császár’s setpoint generalized topological spaces and lattice valued generalized topological spaces. Some notions such as continuous GOHs, convergence theory and separation axioms are introduced. Moreover, the relations among them are investigated.

In 1992 Wang [

In this paper, we aim to introduce a generalization of TMLs under the name of generalized topological molecular lattice (briefly, GTML). In the same manner, we study several notions in these GTMLs, investigate some properties and set the relations among these notions including GOHs, convergence theories and separation axioms.

Throughout this work,

By an L-generalized topology [

(T_{1}) For all

(T_{2})

The pair

For

Let

and its inverse

A mapping

Let us recall that a non-zero element a in a lattice L is said to be a molecule, if for every

Definition 1.1. [

(i)

(ii) If

Denote the greatest minimal family of a by

It is easily to see that both

Definition 1.2. [

(i)

(ii) f is join preserving, i.e.;

(iii)

Theorem 1. [

(1) f and

(2)

(3)

(4)

(5)

(6)

Theorem 2. [

(i) If

(ii) If B is a minimal family of a in

Proposition 3. [

(1) f is an isomorphism.

(2) f is a bijective GOH.

(3)

This section is devoted to introduce the concept of generalized topological molecular lattices and other concepts which play an essential role in these GTMLs.

To denote a molecular lattice, the entry

Definition 2.1. Let

(T_{1})

(T_{2})

A generalized co-topology

(T_{3})

(T_{4})

The pair

Example 1. Let

Example 2. Let

Definition 2.2. [

In a GTML, if

Definition 2.3. [

(i) For

(ii) For all

(iii)

Generally, one can get that

Then

Definition 2.4. Let

By the definition of

Proposition 4 Let

(1)

(2)

(3)

Proof.

(1) For

i.e.,

(2) For all

(3) For all

Since any generalized co-topology is not necessarily closed under finite join, then the finite join is not necessarily be a closed L-fuzzy set, so some relations that are valid in topological molecular lattices do not remain true in generalized topological ones, for example the equation

is not necessarily true in generalized topological lattice as shown in the following example:

Example 3. Let

Definition 2.5. Let

Since

Proposition 5. Let

(1)

(2)

(3)

Proof.

(1) For all

(2) For

i.e.,

(3) For all

The relation

Example 4. For

Definition 2.6. Let

(1) continuous GOH, if for every

(2) continuous at a molecule

It is clear that the generalized topological molecular lattices GTMLs and the continuous GOHs form a category denoted by GTML.

Theorem 6. Let

(i) f is a continuous GOH.

(ii)

(iii)

Proof. The proof the same as given for Theorem 5.2 [

For an L-generalized continuous mapping

Definition 2.7. Let

Definition 2.8. Let

(1) closed, if for every

(2) open, if for every

Remark 1. In the case

As given in [

Proposition 7 The compositions of closed (resp.,open) GOHs are closed (resp.,open) GOHs.

Definition 2.9. [

Definition 2.10. [

(i)

(ii)

Definition 2.11. Let

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

In symbol,

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

In symbol,

Corollary 1 [

From the Definition 1.1, similarly to the case of TMLs, the following proposition is hold:

Proposition 8. Let

(1)

(2)

Proposition 9. Let

(1)

(2)

Proof.

(1) We only prove the sufficiency. Suppose that

(2) The proof is similar to that of (1) and is omitted.

Remark 2. Let

(1) If

(2) If

Theorem 10 [

(1) If

(2) If

Corollary 2 [

In this section, we introduce some kinds of separation axioms in GTMLs and investigate their properties. Moreover, we discuss the relations among them, isomorphic GOHs.

Definition 3.1. Let

(1)

(2)

(3)

(4)

According to the above definitions, we can directly obtain the following results:

Corollary 3 For a GTML, we have the following implications:

In general, we have that

Example 5. Take

Definition 3.2. [

Lemma 1. [

Theorem 11. Let

Proof.

Assume that there exists

Let

Therefore,

Theorem 12. Let

Proof.

Let

Let

Theorem 13. Let

Proof.

Let

Let

Theorem 14. Let

Proof.

Let

Assume that

Then S is a molecular net with both a and b are limit points of S. Hence,

Theorem 15. Let

Proof. We only show the case of

Thus,

Therefore,

Similarly, one can check the other cases.

Analogously to [

Definition 3.3. Let

Clearly, if

Definition 3.4. Let

Theorem 16. Let

Proof. Let

But

Therefore,

The concept of generalized topological molecular lattices GTMLs has been defined. Some notions have been extended to such spaces namely continuous GOHs, convergence theory in terms of molecular nets and and separation axioms.