Ideal Convergence in Generalized Topological Molecular Lattices

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 GT2 separation axiom.


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.

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 a L ∈ is said to be a molecule (some time called co-prime or join-irreducible The set of all molecules in L is denoted by ( ) The greatest minimal family of a is denoted by ( ) Throughout this paper, the entry ( ) L M 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 , L M be a molecular lattice.A subfamily L η ⊂ is said to be a generalized closed topology, or briefly, generalized co-topology, if (T 1 ) η is closed under arbitrary intersections; , L M η is called a generalized topological molecular lattice, or briefly, GTML.
∈ , and a F ≤ / .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 ( ) Recall that according to the definition of ideals, the family ( ) a η is not necessary be an ideal in GTMLs while the family ∈ , the intersection of all η-elements containing A is called the generalized closure of A and denoted by A − .that is, { } , , L M η be GTMLs and 2) continuous at a molecule frequently true, and denoted by S a ∞ .The join of all cluster points of S will be denoted by cluS .i.e, { }

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 ( ) ⊆ , denoted by I a → .In this case, we say that I converges to a.
The join of all limit points of I will be denoted by limI .
2) a is said to be a cluster point of I if ( ) and A I ∀ ∈ , we have F A ∨ ≠ , denoted by I a ∞ .In this case, we say that I accumulates to a.
The join of all cluster points of I will be denoted by cluI .
As a consequence, we obtain the following proposition: , L M η be a GTML, I and J be ideals of L with I J ⊂ and , a b M ∈ .Then we have: and hence ( ) ( ),  ( ) , L M η be GTMLs, be a GOH, and I be an ideal in , , L M η be GTMLs, , , L M η be GTMLs, . By the definition of ( ) Then, ( ) ( ) Therefore, f is continuous GOH.□

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 is a directed set with respect to the relation " ≤ "defined as 2) ( ) Proof. 1) Case I: Let ( ) eventually, i.e; there exists ( ) ( ) , we have ( )( ) Therefore, ( ) frequently and ( ) 2) Let ( ) . By the definition of ( )

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 GT 2 separation axiom.
, i.e; F I ∈ .Since A I ∈ / , then A F ≤ / .So, by Definition 2.3, a is an adherence point of A and hence a A − ≤ net in ( ) L M called the molecular net generated by the ideal I. in L called the ideal generated by S. in L and S be a molecular net in L, then we have1)

,Theorem 6 ,
we have ( )( ), S I b B b B = ≤ / .But B ≥ A, hence ( )( ) , S I b B b A = ≤ / , i.e;( ) one can get directly the following result: be a GTML, I be an ideal in L and S be a molecular net in L, then the following statements hold: be a GTML, I be an ideal in L and S be a molecular net in L, then we have be GT 2 , I be an ideal in L. Assume that , the definition of I.Therefore, limI contains no disjoint molecules.ThenI is an ideal in L with a limI ≤ and b limI ≤.Hence, limI contains two disjoint molecules , a b M ∈ which contradicts the assumption.Therefore, be a GTML, then the following statements are equivalents:

that a is an adherence point of A if and only if a A − ≤ . Definition 2.4 [1] Let 1
Since a A − ≤, then a is an adherence point of A, i.e; , L M η be a GTML, I be an ideal of L and a M ⇒ GOH at . so, ( ) S , we have S F Yang [9]introduced the concepts of maximal ideals and universal nets.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 ( ) conclude this section by studying the relation between the ideal convergence and the GT 2 separation axiom in GTMLs.
Let S be a universal net and x M ∈ , then by the definition, there exists a maximal ideal I such that S is a subnet of ( ) S I .By (ii), we have I x → and hence ( ) Let I be a maximal ideal, then ( ) S I is a universal net, by (iii), x M , L M η be a GTML, then it is GT 2 , if and only if for every ideal I in L, limI contains no disjoint molecules.Proof.( ) ⇒ Let