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Advances in Fuzzy Systems

Volume 2012 (2012), Article ID 925948, 6 pages

http://dx.doi.org/10.1155/2012/925948

## -Fuzzy Connectedness in Stratified Order-Preserving Operator Spaces

^{1}Department of Mathematics, Hunan University of Science and Engineering, Yongzhong 425100, China^{2}Department of English Teaching, Hunan University of Science and Engineering, Yongzhong 425100, China^{3}Department of Physics and Mathematics, WuYi University, Jangmen 529020, China

Received 6 August 2011; Revised 8 December 2011; Accepted 8 December 2011

Academic Editor: Ping Feng Pai

Copyright © 2012 Xiu-Yun Wu et al. 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 concepts of stratified order-preserving operators and stratified continuity are introduced in -fuzzy topological spaces. Their basic properties are discussed, and their characteristic properties are observed. The relationship between induced stratified order-preserving topological spaces and general order-preserving operator topological spaces is studied. Finally, stratified connectedness is introduced, and its properties are studied systematically.

#### 1. Introduction

Since the concept of -fuzzy order-preserving operator was introduced by Professor Chen in 2002 (see [1]), a lot of papers have being devoted to it (see [2–5]). In general topology, similar researches were established (see [5]). However, it has not yet been found in stratified space. As we know, it is quite different between -fuzzy stratified topological spaces and -fuzzy topological spaces. Thus, it is important and meaningful to introduce the concept.

#### 2. Preliminaries

In this paper, a lattice is called a completely distributive lattice with an order-reserving involution′, if the following conditions hold.(1) is a completely lattice. That is, the largest element , and smallest elements . Besides, , , .(2)Distributive law holds in . That is, , , , then (3)There is a mapping satisfies: , then , and , implies (see [6]).

In this paper, 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 [6]). If , , then is called a molecule in . The set of all molecules in is denoted by (see [7]). If , , take . If , the complement of , denoted by , and (see [8]).

The following are the concepts of -fuzzy order-preserving operator in -fuzzy topological space and order-preserving operator in general topological space.

Let be an nonempty set. An operator is called an -fuzzy order preserving operator in , if it satisfies: (1) , (2) 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, -). 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 , . 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 (see [1]).

Let be an nonempty set, be the family of all subsets of . An operator is called an order preserving operator in , if it satisfies: (1) , (2) 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 on (briefly, ). Let , is called an -remote neighborhood of , if there is , such that . The set of all -remote neighborhood of is denoted by . Let , is called an -adherent point of , if , . The union of all -adherent points of is called the -closure of , denoted by . An set is called -closed, if . The set of all -closed sets in is denoted by . is finite union and infinite intersection preserving (see [5]).

Let be an , be an fuzzy lattice, . A fuzzy set is called a lower continuous function, if . Then, the set of all the lower continuous functions, denoted by consists an -fuzzy topology in . The space is called the induced - by (see [5]).

Considering the above definitions, if we say there is an - or an , we mean there is an order-preserving operator on , or on . The two spaces are generated by them, respectively.

Let and be two -fuzzy topological spaces, be induced by simple mapping . If and its reverse mapping satisfies:(1), ;(2), .

Then is called an -fuzzy mapping (see [6]).

An -fuzzy mapping is called homomorphism, if it satisfies:(1) is union preserving. That is, for , ;(2) is reserving involution preserving. That is, for , (see [6]).

#### 3. -Closed Set

*Definition 1. *Let be an -, . An operator is defined by: ,

Since the operator is related to , such kind of operator is generally called stratified order-preserving operator. Clearly, if is a closure operator, then is (see [9]).

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

*Proof. *(1), (2) are easy.

(3) by (1), we have . On the other hand,

(4) , we have
On the other hand, as , then by Definition 1, we have . Hence, (4) holds.

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

By Definition 3, if is the closure operator, then an -closed set is a -closed set (see [9]).

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

*Proof. *(1) As , it is easy to check . Thus, .

(2) By Theorem 2, if . We have . Then
Hence, (2) holds.

(3) If , so

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

Corollary 6. *Let be an -, . If there is , such that . Then consists an -fuzzy co-topology on .*

*Proof. *By Theorem 4, we only need to prove . In fact, as there is , such that . Then
Hence, .

*Remark 7. *In fact, the smallest fuzzy set is -closed. That is, . Otherwise, , . It means is an -adherent point of . However, for all , . This is a contradiction with the hypothesis that is an -adherent point of .

Taking the fact above, we can conclude that given an -fopos , is always an -fuzzy co-topology.

Corollary 8. *Let be an -. is an -fuzzy co-topology on . Then is a co-topology on . So is an on . Moreover, one has , .*

*Proof. *We only prove . By Theorem 2 (1), . Again, by Theorem 2 (4), we have . So .

Conversely, if there is , but . Since, there is , such that and . So we have . This is a contradiction, because . Therefore, .

Lemma 9. *Let be an , be an - induced by . . Then if and only if .*

* Proof. *For any . If , then . If , then . Thus, by the definition of lower continuous function, we have if and only if .

Theorem 10. *Let be an , be an - induced by . , . Then if and only if .*

*Proof. *For any , by Lemma 9, we have iff . Let . Then
So .

Conversely, if , by Lemma 9, we have . And , so
Hence, . This implies .

Theorem 11. *Let be an , is an - induced by . Then iff , .*

*Proof. *By Theorem 10, we have

*Definition 12. *Let , be -, -, respectively. An -fuzzy homomorphism is called -continuous, if , then .

Theorem 13. *Let , be -, -, respectively. is an -fuzzy mapping. Then the following statements are equivalent.*(1)* is -continuous;*(2)*, ;*(3)*, .*

*Proof. *(1)(2) , so , then . Therefore, .

On the other hand, by Theorem 2, we get that . And by hypothesis, is -continuous, then, . Hence, . Therefore, we have

(2)(3) , so . By (2), we get . So,
And obviously, . Thus, (2) holds.

(3)(1). . By (3), we have
So, .

On the other hand, obviously, . Therefore, . This shows that .

#### 4. -Connectedness

*Definition 14. *Let be an -, . are called -separated, if . Otherwise, are called -connected.

Clearly, if is the closure operator in , then -connected is -connected [8].

*Definition 15. *Let be an -, . is called an -connected set, if there not exist -separated sets , such that, , and . Specially, if is -connected, then, is called -connected.

Theorem 16. *Let be an -, . Then is not -connected if and only if there are , such that, , and , .*

*Proof. *Necessity. Let be not -connected. That is, is not -connected. By Definition 15, there are two -separated sets , such that, , and .

Since, are -separated, we have

Therefore, . Hence, the necessity holds.

Sufficiency. Take satisfies the conditions stated in the theorem. So . As , and , we have . Similarly, . Hence, are -separated. By Definition 15, we have is not -connected.

Theorem 17. *Let is an -, . be -connected, then implies is -connected.*

*Proof. *Since is -connected. is connected in . By Theorem 20 in reference [10] (page 54) and Corollary 8, is connected. So is connected. Therefore, is -connected.

Theorem 18. *Let be an -. , is -connected, and there is , such that for any , and are -connected. Then is -connected.*

*Proof. *Since is -connected, is connected in . and are collected for each . Then by Theorem 21 in reference [10] (page 54) and Corollary 8, is connected. Therefore, is -connected.

Theorem 19. *Let be an , and be the induced -. Then is -connected if and only if is connected.*

*Proof. *Necessity. Let be -connected. Suppose that is not connected, then there are nonempty , such that , and . By Theorem 10, , . So
This means are -separated sets, and . A contradiction.

Sufficiency. Suppose is not -connected. Then There are -separated sets , such that , and . Since , by Theorem 10, we have . Thus, . Therefore,
Similarly, we get that . This means is not connected. A contradiction.

Theorem 20. *Let be two -s, be -continuous, be -connected. Then is -connected.*

*Proof. *Let , . Since . Take
we have
Besides, by
we get

Denote , . Then , and

Since is -connected, then , or . Take the former, for example, . Thus,
Thus,

Theorem 21. *Let be an -, , for all, . Then is -connected if and only if for all , for all , there are finite molecular , such that .*

*Proof. *Necessity. Suppose the result does not holds. That is, there is , . And there is a mapping
such that for any finite , does not holds.

For the convenient sake, , and are called joined, if there are finite , such that , denoted by . Otherwise, and are called disjoined, denoted by . Let
and . Obviously, , so . and , so . Hence, . Since , , or . So . Next, let us prove are not -connected.

Suppose that , then there is . By , we have . So there is , such that , and , hence . This means that , and .

On the other hand, by , then , so there is , such that . Therefore, . And by , we have . This means and . This is a contradiction with . Therefore . Similarly, we get that .

Sufficiency. Suppose is not -connected. Then there are , such that are -separated, and , . Define a mapping
Then , we have , .

Take . Then for any finite , , or . Thus, , or , . However, since , there is , such that . Therefore, . A contradiction.

#### Acknowledgment

The work is supported by Youth Science Foundation of Hunan University of Science and Engineering (10XKYTB038).

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