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Advances in Fuzzy Systems

VolumeΒ 2012Β (2012), Article IDΒ 582105, 8 pages

http://dx.doi.org/10.1155/2012/582105

## On -Layer Ideal Topologies

^{1}Department of Mathematics, Institute of Computational Mathematics, Hunan University of Science and Engineering, Yongzhong 425100, China^{2}Department of English Teaching, Hunan University of Science and Engineering, Yongzhong 425100, China

Received 8 April 2012; Accepted 22 June 2012

Academic Editor: KatsuhiroΒ Honda

Copyright Β© 2012 Xiu-Yun Wu and Li-Li Xie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The purpose of this paper is to study theory of two different kinds of -layer order-preserving operator space, namely, -*opos* and -*opos*. The former kind of space is formed by -layer function in *L*-fuzzy order-preserving operator space. The later kind of space is derived by local -remote neighborhood function, which is related with -*opos* and -ideal. We study characteristic properties of the two kinds of spaces, respectively, and give some applications to show the intimate relations under two different -*oposs*.

#### 1. Introduction

In general topology, Vaidyanathaswamy firstly defined concepts of local function and its derived ideal topology from initial topology and ideal [1]. Some interesting extensive works were done by JankoviΔ and Hamlett [2]. After M. E. Abd El-Monsef introduced the concept of -open set, many researchers were devoted to research on local semitopology. There are many local semiopen sets, such as --open set, strong --open set, and --open set [3β5]. All of these local semiopen sets are given by comparing interior and closure operators in the initial topology and its ideal derived topology.

By utilizing *q*-neighborhood which is mentioned in [6], Sarkar generalized the concepts of local function and derived topology into fuzzy topology in 1997 [7].

As there is a layer structure in -fuzzy topology, fuzzy local functions and their derived fuzzy ideal topology must be more complex. Hence, in this paper, we will analyze the ideal topological properties in terms of layer structure of -fuzzy topology and reveal the inner relations between -layer ideal space and fuzzy ideal topological space.

In the first part of the paper, we establish the theory of -*opos*. We introduce the concept of -layer function in -fuzzy order-preserving operator space. Then, based on its basic properties, we form -*opos*. We also prove it preserves many good properties. In the second part of the paper, we establish the theory of -*opos*. It is a -layer space with local topological properties. We introduce the concept of local -remote neighborhood function via an -*opos* and an -ideal. On the basis of its basic properties, we form -*opos*. It is finer than the old one. We observe the structures of local -remote neighborhood functions under different -ideals and -*opos* as well as the relations of the correspondent -*opos*. We also obtain some equivalent conditions of -*opos* under compatibility of -ideal and -*opos*. Finally, as an application, we define four kinds of connectivity and reveal their inner relations.

#### 2. Preliminaries

In a general topological space , is an ideal, . The concept of local function of with respect to and is given by , in which is the neighborhood system of [1].

In this paper, an lattice is called a completely distributive lattice with an order reserving involutionβ². will always denote nonempty crisp sets, A mapping is called an -fuzzy set. is the set of all -fuzzy sets on . An element is called an irreducible element in , if implies or , where . The set of all nonzero irreducible elements in will be denoted by (see [1]). If , , then is called a molecule in . The set of all molecules in is denoted by . If , , take . If , the complement of , denoted by , and [8].

An -fuzzy order-preserving operator space and some related conceptions are given in the following.

Let be an nonempty set. An operator is called a -fuzzy order preserving operator in , if it satisfies (1) , (2) for all and implies . A set is called an -set, if . The set of all -sets in is denoted by . And is called an order-preserving operator space (briefly, -*opos*). A molecule , is called an -remote neighborhood of , if . The set of all -remote neighborhood of is denoted by . Let , is called an -adherent point of , if for all , . The union of all -adherent points of is called the -closure of , denoted by . A set is called -closed, if . The set of all -closed sets in is denoted by . is finite union and infinite intersection preserving [9].

Similar concepts in a general topology are defined as follows.

Let be a nonempty set, and let be the family of all subsets of . An operator is called an order-preserving operator in , if it satisfies (1) , (2) for all implies . A set is called a -set, if . The set of all -sets in is denoted by . is called an order-preserving operator space on (briefly, *opos*). Let , is called a -remote neighborhood of , if there is , such that . The set of all -remote neighborhood of is denoted by . Let , is called a -adherent point of , if for all , . The union of all -adherent points of is called the -closure of , denoted by . A set is called -closed, if . The set of all -closed sets in is denoted by . is finite union and infinite intersection preserving.

Let be an *opos*, and let be an fuzzy lattice, for all . A fuzzy set is called a lower continuous function, if . Then, the set of all the lower continuous functions, denoted by , consists an -fuzzy cotopology in . The space is called the induced -*opos* by .

A nonempty subfamily of is called an -ideal, if satisfies the following conditions:(1), and implies ,(2) implies .

It is easy to check if , , such that , then . Moreover, of is an -idea if and only if is an ideal on . If , , such that , then . If are two -ideals, then and are -ideals too.

In this paper, if , we denote and .

#### 3. -Closed Set and -*opos *

In this section, we list out the main results in [10], which we will use in the following sections. Proofs of the theorems in the following can be found in [10] as well.

*Definition 1. *Let be an -*opos*, . An operator is defined by, for all ,

Theorem 2. *Let be an -opos, . , then the following statements hold:*(1)* implies ;*(2)*;*(3)*;*(4)*.*

*Definition 3. *Let be an -*opos*, . A set is called -closed, if . The set of all -closed sets in is denoted by . is called an -order-preserving operator space (briefly, -*opos*).

Theorem 4. *Let be an -opos, . Then,*(1)*;*(2)*if , then ;*(3)*if , then .*

*Remark 5. *The theorem shows that is finite union and infinite intersection preserving.

Corollary 6. *Let be an -opos, . Then, consists an -fuzzy cotopology on .*

Lemma 7. *Let be an opos, be an -opos induced by and let . Then, if and only if .*

Theorem 8. *Let be an opos, an -opos induced by . , . Then, if and only if .*

Theorem 9. *Let be an opos, is an -opos induced by . Then, if and only if for all , .*

*Definition 10. *Let , be -*opos*, -*opos*, respectively. An -fuzzy homomorphism is called -continuous, if for all , then .

Theorem 11. *Let , be -opos, -opos, respectively. is an -fuzzy mapping. Then, the following statements are equivalent:*(1)* is -continuous;*(2)*for all , ;*(3)*for all , .*

*Definition 12. *Let be an -*opos*, . is called an -closed remote neighborhood of , if . The set of all -closed remote neighborhood of will be denoted by . An fuzzy point is called -adherent point of , if for every , .

*Remark 13. *, where is the neighborhood system of in . Hence, .

Theorem 14. *Let be an -opos, , then*(1)* is an -adherent point of if and only if ,*(2)* is the union of all -adherent points of .*

#### 4. --Closed Sets

Let be an -*opos*. be an -ideal. , be the -remote family of . For any , take
Then, is called the local -remote function of with respect to and , simply denoted by or .

*Remark 15. *If , is the closure operator, then is the local function of in , and is the fuzzy local function of in [7]. Furthermore, we have .

Theorem 16. * Let be an -opos. be two -ideals. . Then, the following statements hold.*(1)*, .*(2)*.*(3)*.*(4)*.*(5)*.*(6)*.*(7)*.*(8)*.*(9)*.*(10)*.*

*Proof. *(1) Suppose , by Theorem 14, we have

If , then there must be , such that . Therefore, .

(2), (3), and (6) Easy.

(4) implies . By , we have . So . Thus, . Besides, implies for all , . Take , so and for all , . Particularly, let . So . This means . Hence, . Therefore, .

(5) By (4), .

(7) By (2), . Conversely, if and only if , then for all , , and . Hence, . This implies . So .

(8) For each , we may remark . If not, there are , such that . Thus, . Since , there are , . This shows
Consequently, . But this contradicts with . Therefore, . The reverse inclusion is obvious. So .

(9) Obviously, . And . Conversely, , there are , . So . Thus, . Then, (9) holds.

(10) Suppose there is . So . By for each , we have , a contradiction.

Theorem 17. * Let be an -opos. Let be an ideal. . Take , then*(1)*, if and only if ,*(2)*,
*(3)*,*(4)*.*

*Proof. * According to (5), (7) in Theorem 16, the proof is trivial.

By Theorem 17, we know if and only if we take , then (simply denoted by ) consists an -fuzzy cotopology on . So it is called -cotopology formed by the ideal and . The pair , simply denoted by , is called -. By Theorem 16 (1), we have , thus , and , thus . But in general, , , consequently, . Moreover, if and only if , then .

*Example 18. *Let . An -fuzzy set satisfying will be denoted by . Let be a cotopology on . . Put . Then, consists an -cotopology, and . Totally, there are four kinds of -ideals in , namely, , , and . and are the trivial -ideals. Let us study and .

Denote . Then, . So , But . In addition, . However, , which means . It is easy to check .

In this example, we see . However, . Thus, . Furthermore, is not an -cotopology.

Theorem 19. *Let be an -opos. . is an -ideal. , then*(1)*,*(2)*,*(3)*.*

Theorem 20. *Let be an -, an -ideal . Then, .*

*Proof. * Obviously, . If , that is, . Take . Let us prove . In fact, if , there are , such that . So

It means . On the other hand, as , we have . Then,
Notice that , we get , a contradiction. Hence, . So , which implies . Therefore, . The proof is completed.

Theorem 21. *Let be an -opos, and let be an -ideal. Take as the supremum - cotopology generated by . Then, .*

*Proof. *Put . It is easy to prove is a base of .

Lemma 22. *Let be an -opos. Let be two ideals. , . Then, .*

*Proof. *By Theorem 21,

Theorem 23. *Let be an -opos. Let be two -ideals. , then*(1)*,*(2)*.*

*Proof. *(1) Suppose , there are , and , such that , and . Thus, . As , and , it is clear that . So . The reverse inclusion is obvious according to (3) in Theorem 16. Therefore, (1) holds.

(2) Suppose . There are , and , such that . So , . Hence, by Lemma 22, we have , and . This shows that . Therefore, .

Conversely. suppose . Then, there exists , and , such that . Because of , there is , such that . Hence, . This shows . Therefore, . Similarly, we can prove . So . The proof is completed.

By Theorem 20, we get two important results.

Corollary 24. *Let be an -opos. Let be an -ideal, then , and as a result .*

*Proof. * By (2) in Theorem 23, take , clearly.

Corollary 25. *Let be an -opos. Let be two -ideals. Then,*(1)*,*(2)*,*(3)*.*

*Proof. *(1) By (2) in Theorem 23, easy.

(2) Since for every -ideal , by (1) and Theorem 20, we have .

(3) Clearly, . Conversely, . Then, , and . So by (1) in Theorem 23, . Therefore, .

Theorem 26. *Let be an -opos, and let be an -ideal. . Then, if and only if .*

*Proof. **Sufficiency*. according to (4) in Theorem 16, . On the other hand, if , then and . So there are , , such that . Here, we may assume , since is an -ideal. Therefore, . This implies . .*Necessary*. if , then . Since , .

Since is an *opos* on . is an ideal on . We have the following results.

Theorem 27. * Let be an -, and let be an -ideal. Then, for every , , and therefore .*

*Proof. * Since
then , .

Corollary 28. *Let be an -opos, and let be an -ideal. Then, for every , .*

*Proof. *. So . Besides, by Theorem 27, the later equation is easy.

Corollary 29. * Let be an opos on , an ideal. The induced -opos and -ideal are denoted, respectively, by and . Then, .*

*Proof. *By Theorem 27 and Corollary 28, . On other hand, again by Corollary 28, for every , .

#### 5. Compatibility of with

*Definition 30. *Let be an -*opos*. is called an --remote neighborhood family of (briefly, --RF of , if for all , there is , such that ).

*Definition 31. *Let be an -*opos*. Let be an -ideal. is said to be compatible with , denoted by , if for any and , there are and , such that , then .

*Example 32. * In Example 18, we know , . Take , , , .

It is easy to check are all -ideals, and . Take , then and , such that . But , therefore . Furthermore, for all , if or , then . If or , there does not exist , such that . Therefore, .

Theorem 33. *Let be an -opos, an ideal, . Then, .*

*Proof. *For every , . Put . If , then . So there are and , such that . Thus, . Since , we get . Besides, by (4) in Theorem 16, . Notice that . Thus, , and . The reverse inclusion is obvious. Therefore, .

Lemma 34. * Let be an -opos, an -ideal. Then, for every , .*

*Proof. *If , so . That is, there are , such that . As , we have . Therefore, the conclusion holds.

Theorem 35. *Let be an -opos, an ideal. Then, the following statements are equivalent.*(1)*.*(2)*If has an --RF , satisfying for all , there is , such that , then .*(3)*For all , .*(4)*For all , .*(5)*For all , .*(6)*For all , if , , then .*

*Proof. * (1) (2) If has a --RF , satisfying the condition in (2), then for every , there are and such that and . As , we have .

(2) (1) By Definitions 30 and 31, clearly.

(1) (3) For all , . If , then . If , then , there are and , such that . Because , we have .

(3) (1) if satisfying, for every , there are and , such that . Then, . By (3), we have . Hence, .

(3) (4) By Lemma 34, obviously.

(4) (5) Directly.

(5) (1) If for every and , there are and , such that , this means . Put , then . So . This means . By (5), we have . Since
we draw a conclusion: for every , . Therefore, .

(4) (6) For any , . So . By (4), , so . Thus . Since , we have . Here by the assumption in (6), . This implies . Again by (4), .

(6) (4) By Lemma 34, for any , . Let's prove satisfies the condition in (6), In fact, if not, there is , , then . We have . A contradiction. Therefore, .

Theorem 36. * Let be an -opos, be an -ideal. Then the following statements are equivalent, and implied by .*(1)*For every , .*(2)*For every , .*(3)*For every , .*

*Proof. *. By (3) in Theorem 35, If . Thus .

By Lemma 34, . Then by (1), we have .

. By (8) in Theorem 16, . According to (2), . Thus, .

Straightforward.

Theorem 37. *Let be an -opos, be an -ideal. . Then for every if and only if there are and , such that .*

*Proof. **Necessity.* suppose . So and . By (4) in Theorem 2, and (5) in Theorem 17, we have and .*Sufficiency.* suppose , and . Then . Therefore, .

#### 6. -Connectivity and -Connectivity

In this section, we shall introduce four kinds of connectedness and discuss their relations.

Concepts of -separated sets and -connected are defined as:

*Definition 38. * Let and be -*opos* and -*opos*, respectively.(1) are called -separated sets (resp. -separated set), if , (resp. ).(2) is called an -connected set (resp. -connected set), if there not exist -separated sets , such that, , and , (resp. there not exist -separated sets , such that, , and ).

Theorem 39. * Let be an -opos. is -connected and . Then, is -connected.*

*Proof. * Let , and are -separated. Take
Then, , and
Similarly, we have . Thus,