1. Introduction
In 1992 Wang [1] , introduced his important theory called topological molecular lattice (briefly, TML) as a generalization of ordinary topological and fuzzy topological spaces in tools of molecules, remote neighborhoods and generalized order homomorphisms GOHs. Then many authors characterized some topological notions in such TMLs, such as convergence theories of molecular nets or ideals [1] - [3] , separation axioms [1] [4] and other notions.
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,
is a complete lattice with an order-reversing involution
, and with the smallest element
and the largest element
.
By an L-generalized topology [5] , on a non-empty ordinary set X, we mean a subfamily
of
with the following axioms:
(T1) For all
;
(T2)
.
The pair
is called an L-generalized topological space. Every element A of
is called t-open L-set and the pseudo complement
is called t-closed L-set. The concept of L-generalized co-topological space can be defined dually.
For
, the L-generalized interior of A is the largest L-open subset contained in A and denoted by
, so A is open if and only if
. The L-generalized closure of A is the smallest L-closed subset contains A and denoted by
, so A is closed if and only if
.
Let
be an ordinary mapping. The corresponding L-fuzzy mapping
is defined as follows:
![]()
and its inverse
defined as:
![]()
A mapping
between two L-generalized topological spaces is said to be an L-genera- lized continuous mapping if and only if
. The mapping
is called an L-generalized open (resp. L-generalized closed) mapping if
(resp.
is m-closed for all t-closed set F). The category of L-generalized topological spaces and their L-generalized continuous mappings is denoted by L-GTop.
Let us recall that a non-zero element a in a lattice L is said to be a molecule, if for every
such that
, implies
or
. Denote the set of all molecules of L by
or M for short, clearly, every element in L can be constructed by elements of M, since each element in L is a union of molecules.
Definition 1.1. [1] Let L ba a complete lattice,
. The subset
is called a minimal family of a if the following two conditions are hold:
(i)
.
(ii) If
and
, then
such that
.
Denote the greatest minimal family of a by
. Hence, let
.
It is easily to see that both
and
are minimal families of a.
Definition 1.2. [1] Let
and
be complete lattice. A mapping
is called a generalized order homomorphism or GOH for short if
(i)
if and only if
.
(ii) f is join preserving, i.e.;
.
(iii)
is join preserving, where for all
,
.
Theorem 1. [1] Let
be GOH, then the following properties are hold:
(1) f and
are order preserving, i.e.:
.
.
(2)
, for all
.
(3)
, for all
.
(4)
.
(5)
, for all
.
(6)
is meet preserving, i.e.:
.
Theorem 2. [1] Let
be a GOH, then:
(i) If
, then
.
(ii) If B is a minimal family of a in
, then
is a minimal family of
in
.
Proposition 3. [1] Let
, be a mapping between complete lattices. The following are equivalent:
(1) f is an isomorphism.
(2) f is a bijective GOH.
(3)
is a bijective GOH.
2. Main Notions in GTMLs
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
is used: it indicates both the lattice itself and the set of its molecules.
Definition 2.1. 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)
.
A generalized co-topology
is said to be a closed topology (or co-topology) [1] , if it satisfies the following additional conditions:
(T3)
is closed under finite union;
(T4)
.
The pair
is called a generalized topological molecular lattice, or briefly, GTML.
Example 1. Let
be a generalized topological space [6] . Then it is clear that
is a molecular lattice and
is a GTML, where
.
Example 2. Let
be an L-generalized topological space [5] . Then we have that
is a GTML, where
is a molecular lattice and
.
Definition 2.2. [7] Let
be a GTML,
, and
. Then F is said to be a generalized remote neighborhood of a. The set of all generalized remote neighborhoods of a will be denoted by
.
In a GTML, if
and
such that
, then we get
. However, for
,
need not to be in
, since it does not necessary be closed element because
is not necessary be closed under finite joins.
Definition 2.3. [8] Let L be a complete lattice. A non empty subset I of L is said to be an ideal, if it satisfies the following conditions:
(i) For
and
.
(ii) For all
.
(iii)
.
Generally, one can get that
is not necessary be an ideal in GTMLs. So, let us define the following:
![]()
Then
is an ideal in GTMLs.
Definition 2.4. Let
be a GTML and
. The intersection of all h-elements containing A will be called the generalized closure of A denoted by
. i.e.,
.
By the definition of
, one can obtain that A is a closed element if and only if
.
Proposition 4 Let
be a GTML, then the following statements are hold:
(1)
, if
, then
;
(2)
;
(3)
.
Proof.
(1) For
, with
, we have
![]()
i.e.,
.
(2) For all
. obvious;
(3) For all
, we have
.
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
, and
be a generalized topology on X. The class
is a generalized co-topology on
. So for
and
, we have that
,
and
. But
implies that
which means that
.
Definition 2.5. Let
be a GTML,
, then a is said to be an adherence point of A, if for all
, we have
.
Since
is a GTML with L equipped with an order reversing involution, we can define the generalized interior by
![]()
Proposition 5. Let
be a GTML, then the following statements are hold:
(1)
.
(2)
, if
, then ![]()
(3)
.
Proof.
(1) For all
. obvious;
(2) For
, with
, we have
![]()
i.e.,
.
(3) For all
,
.
The relation
which is true in TMLs, it is not true in GTMLs as shown in the following example:
Example 4. For
and the generalized co-topology
on
as given in Example 3. Let
and
, we have that
,
and
. But
implies that
which means that
.
Definition 2.6. Let
and
be GTMLs. An GOH
is called:
(1) continuous GOH, if for every
, we have
.
(2) continuous at a molecule
, if for every
, we have
.
It is clear that the generalized topological molecular lattices GTMLs and the continuous GOHs form a category denoted by GTML.
Theorem 6. Let
and
be GTMLs,
be a GOH, then the following statements are equivalent:
(i) f is a continuous GOH.
(ii)
.
(iii)
.
Proof. The proof the same as given for Theorem 5.2 [1] .
For an L-generalized continuous mapping
, it is well-known that
induced by an ordinary mapping
, and satisfied many useful properties ([9] [10] ). Hence, the continuous GOHs can be regarded as a generalization of L-generalized continuous mappings.
Definition 2.7. Let
be an isomorphism and f and
be continuous. A GOH
is said to be a homeomorphism.
Definition 2.8. Let
and
be GTMLs. A GOH
is said to be:
(1) closed, if for every
, we have ![]()
(2) open, if for every
and every
such that
, there exists
such that
and
.
Remark 1. In the case
and
are equipped with order reversing involutions, we can say that a GOH is open if it maps open elements in
into open elements in
. Clearly, every L-generalized closed (resp. open) mapping is a closed GOH (resp. an open GOH).
As given in [1] , we have the following easily established result.
Proposition 7 The compositions of closed (resp.,open) GOHs are closed (resp.,open) GOHs.
Definition 2.9. [1] Let
be a molecular lattice,
and D is a directed set, then the mapping
is called a molecular net and denoted by
. S is said to be in A, if
.
Definition 2.10. [1] Let
be a molecular lattice,
and
be two molecular nets, then T is said to be a subnet of S, if there exists a mapping
such that
(i)
.
(ii)
such that
.
Definition 2.11. 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
.
In symbol,
.
(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
.
In symbol,
.
Corollary 1 [1] Suppose that
(resp.,
) and
. Then
(resp.,
).
From the Definition 1.1, similarly to the case of TMLs, the following proposition is hold:
Proposition 8. Let
be a GTML,
be a molecular net and
, then:
(1)
if and only if
.
(2)
if and only if
.
Proposition 9. Let
be a GTML,
be a molecular net and
, then:
(1)
if and only if
.
(2)
if and only if
.
Proof.
(1) We only prove the sufficiency. Suppose that
and
is a standard minimal family of a. Since
. Then for all
, there exists a limit point x of S such that
. By Corollary 1,
and therefore
.
(2) The proof is similar to that of (1) and is omitted.
Remark 2. Let
be a molecular lattice, S be a molecular net and
, then
(1) If
is a constant net, i.e.,
, then
.
(2) If
and T be a subnet of S, then
.
Theorem 10 [8] Let
be a GTML,
and
, then:
(1) If
, then there exists a molecular net
in A, such that
.
(2) If
is a molecular net in A, such that
, then
.
Corollary 2 [8] Let
be a GTML,
and
, then
if and only if there exists a molecular net
in A, such that
.
3. Separation Axioms in GTMLs
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
be a GTML, then
(1)
is called
, if
, there exists
such that
.
(2)
is called
, if
, there exists
such that
or there exists
such that
.
(3)
is called
, if
, there exists
such that
.
(4)
is called
, if
, there exists
and
such that
.
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
does not imply
. The next example [1] is clear.
Example 5. Take
and
, then clearly that
is not
and hence it is not
. But there are no disjoint points, so
is
.
Definition 3.2. [1] Let
be a molecular lattice,
with
, then m is called a component of A, if
and
with
and
imply that
.
Lemma 1. [1] Let
be a molecular lattice,
with
and
, then A has at least one component m such that
.
Theorem 11. Let
be a GTML, then it is
, if and only if
, we have a is a component of
.
Proof.
Assume that there exists
such that a is not a component of
, then by the preceding lemma, we can choose a component b of
such that
. Since
is a
, then
such that
. Hence
and so
. a contradiction.
Let
with
, then by the assumption, we have a is a component of
and hence
. Then
and
.
Therefore,
is
.
Theorem 12. Let
be a GTML, then it is
, if and only if
, we have
or
.
Proof.
Let
be
, then
, there exists
such that
or there exists
such that
. Hence, a is not an adherence point of b or b is not an adherence point of a which implies that
or
.
Let
with
, then we have
or
. Hence, we get
with
or
with
which complete the proof.
Theorem 13. Let
be a GTML, then it is
, if and only if
, a is closed.
Proof.
Let
be
and
, if
, then there exists
such that
. Hence, a is not an adherence point of b. Thus, b contains all its adherence points and hence, b is closed.
Let
. By the assumption, b is closed and then
with
. Therefore,
is
.
Theorem 14. Let
be a GTML, then it is
, if and only if for every molecular net S,
contains no disjoint molecules.
Proof.
Let
be a molecular net such that
with
. Let
and
, since
, then
eventually, i.e.;
such that
we have
. Similarly, we have
, but D is a directed set, hence
such that
and
. Then
. Therefore,
, then
is not
.
Assume that
is not
, then
with
and
, we have
. Thus, we can choose a molecule
such that
. Put
![]()
Then S is a molecular net with both a and b are limit points of S. Hence,
contains at least two disjoint molecules.
Theorem 15. Let
and
be GTMLs. If
is a
and
be a homeomorphism, then so is
.
Proof. We only show the case of
and the others are similar. Let
be a
and
with
. Since f is bijective, then there exist
such that
and
. Then there exist
and
such that
. But f is isomorphic GOH, so
and
.
Thus,
.
Therefore,
is also a
.
Similarly, one can check the other cases.
Analogously to [4] , we give the next definitions:
Definition 3.3. Let
be a GTML, then
is said to be
, if and only if
with
, there exists
and
such that
.
Clearly, if
is a
, then it is
. Furthermore, we have the following:
Definition 3.4. Let
be a GTML, then
is called an interior additive if
, we have
.
Theorem 16. Let
be a GTML, if it is both
and an interior additive, then
is a
.
Proof. Let
with
. Since
is
, then there exists
and
such that
. Hence,
.
But
is an interior additive, then
.
Therefore,
is a
.
4. Conclusion
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.