-Fuzzy -Algebras

Abstract

The notion of ()-fuzzy -algebras is introduced in the lattice value fuzzy set theory. It is a generalization of Klement's fuzzy -algebras. In our definition of ()-fuzzy -algebras, each -fuzzy subset can be regarded as an -measurable set to some degree.

1. Introduction and Preliminaries

In 1980, Klement established an axiomatic theory of fuzzy -algebras in [1] in order to prepare a measure theory for fuzzy sets. In the definition of Klement's fuzzy -algebra , was defined as a crisp family of fuzzy subsets of a set satisfying certain set of axioms. In 1991, Biacino and Lettieri generalized Klement's fuzzy -algebras to -fuzzy setting [2].

In this paper, when both and are complete lattices, we define an -fuzzy -algebra on a nonempty set by means of a mapping satisfying three axioms. Thus each -fuzzy subset of can be regarded as an -measurable set to some degree.

When is an -fuzzy -algebra on , is called an -fuzzy measurable space. An -fuzzy -algebra is also called an --algebra. A Klement -algebra can be viewed as a stratified --algebra. A Biacino-Lettieri --algebra can be viewed as a stratified --algebra. A -fuzzy -algebra is also called an -fuzzifying -algebra. A crisp -algebra can be regarded as a -fuzzy -algebra.

Throughout this paper, both and denote complete lattices, and has an order-reversing involution. is a nonempty set. is the set of all -fuzzy sets (or -sets for short) on . We often do not distinguish a crisp subset of and its character function . The smallest element and the largest element in are denoted by and , respectively.

The binary relation in is defined as follows: for , if and only if for every subset , the relation always implies the existence of with [3]. is called the greatest minimal family of in the sense of [4], denoted by . Moreover, for , we define . In a completely distributive lattice , there exist and for each , and (see [4]).

In [4], Wang thought that and . In fact, it should be that and .

For a complete lattice , and , we use the following notation:

If is completely distributive, then we can define

Some properties of these cut sets can be found in [5–10].

Theorem 1.1 (see [4]). Let be a completely distributive lattice and . Then (1), that is, is an map;(2), that is, is a union-preserving map. For and , we define two -fuzzy sets and as follows: Then for each -fuzzy set in , it follows that

Theorem 1.2 (see [5, 7, 10]). If is completely distributive, then for each -fuzzy set in , we have (1)(2), (3), For a family of -fuzzy sets in , it is easy to see that If is completely distributive, then it follows [7] that

Definition 1.3. Let be a nonempty set. A subset of is called a Klement fuzzy -algebra if it satisfies the following three conditions:(1)for any constant fuzzy set , ;(2)for any , ;(3)for any , .The fuzzy sets in are called fuzzy measurable sets, and the pair a fuzzy measurable space.

Definition 1.4. Let be a complete lattice with an order-reversing involution and a nonempty set. A subset of is called an --algebra if it satisfies the following three conditions:(1)for any , constant -fuzzy set ;(2)for any , ;(3)for any , .The -fuzzy sets in are called -measurable sets, and the pair an -measurable space.

2. -Fuzzy -Algebras

L. Biacino and A. Lettieri defined that an --algebra is a crisp subset of . Now we consider an -fuzzy subset of .

Definition. Let be a nonempty set. A mapping is called an -fuzzy -algebra if it satisfies the following three conditions: (LMS1);(LMS2) for any , ;(LMS3) for any , .An -fuzzy -algebra is said to be stratified if and only if it satisfies the following condition:
(LMS1)*   , .
If is an -fuzzy -algebra, then is called an -fuzzy measurable space.
An -fuzzy -algebra is also called an --algebra, and an -fuzzy measurable space is also called an -measurable space.
A -fuzzy -algebra is also called an -fuzzifying -algebra, and a -fuzzy measurable space is also called an -fuzzifying measurable space.
Obviously a crisp measurable space can be regarded as a -fuzzy measurable space.
If is an -fuzzy -algebra, then can be regarded as the degree to which is an -measurable set.

Remark 2.2. If a subset of is regarded as a mapping , then is an --algebra if and only if it satisfies the following conditions:(LS1);(LS2);(LS3) for any , . Thus we easily see that a Klement -algebra is exactly a stratified --algebra, and a Biacino-Lettieri --algebra is exactly a stratified --algebra.
Moreover, when , a mapping is an -fuzzifying -algebra if and only if it satisfies the following conditions:
(MS1);(MS2) for any , ;(MS3) for any , .

Example 2.3. Let   be a crisp measurable space. Define by Then it is easy to prove that is a -fuzzifying measurable space.

Example 2.4. Let be a nonempty set and a mapping defined by Then it is easy to prove that is a -fuzzifying measurable space. If with , then 0.5 is the degree to which is measurable.

Example 2.5. Let be a nonempty set and a mapping defined by Then it is easy to prove that is a -fuzzy measurable space. If with , then 0.5 is the degree to which is -measurable.

Proposition 2.6. Let be an -fuzzy measurable spaces. Then for any , .

Proof. This can be proved from the following fact: The next two theorems give characterizations of an -fuzzy -algebra.

Theorem 2.7. A mapping is an -fuzzy -algebra if and only if for each , is an --algebra.

Proof. The proof is obvious and is omitted.

Corollary 2.8. A mapping is an -fuzzifying -algebra if and only if for each , is a -algebra.

Theorem 2.9. If is completely distributive, then a mapping is an -fuzzy -algebra if and only if for each , is an --algebra.

Proof. Necessity.. Suppose that is an -fuzzy -algebra and . Now we prove that is an --algebra.(LS1) By and , we know that this implies that .(LS2) If , then this shows that .(LS3) If , then , . Hence . By , we know that This shows that . Therefore, . The proof is completed.

Corollary 2.10. If is completely distributive, then a mapping is an -fuzzifying -algebra if and only if for each , is a -algebra.

Now we consider the conditions that a family of --algebras forms an -fuzzy -algebra. By Theorem 1.2, we can obtain the following result.

Corollary 2.11. If is completely distributive, and is an -fuzzy -algebra, then (1) for any with ;(2) for any with .

Theorem 2.12. Let be completely distributive, and let be a family of --algebras. If for all , then there exists an -fuzzy -algebra such that .

Proof. Suppose that for all . Define by By Theorem 1.2, we can obtain that .

Corollary 2.13. Let be completely distributive, and let be a family of -algebras. If for all , then there exists an -fuzzifying -algebra such that .

Theorem 2.14. Let be completely distributive, and let be a family of --algebra. If for all , then there exists an -fuzzy -algebra such that .

Proof. Suppose that for all . Define by By Theorem 1.2, we can obtain .

Corollary 2.15. Let be completely distributive, and let be a family of -algebra. If for all , then there exists an -fuzzifying -algebra such that .

Theorem 2.16. Let be a family of -fuzzy -algebra on . Then is an -fuzzy -algebra on , where is defined by

Proof. This is straightforward.

3. -Fuzzy Measurable Functions

In this section, we will generalize the notion of measurable functions to fuzzy setting.

Theorem 3.1. Let be an -fuzzy measurable space and a mapping. Define a mapping by , Then is an -fuzzy measurable space.

Proof. (LMS1) holds from the following equality:
(LMS2) can be shown from the following fact: ,
(LMS3) for any , by

Definition 3.2. Let and be -fuzzy measurable spaces. A mapping is called -fuzzy measurable if for all .
An -fuzzy measurable mapping is called an -measurable mapping, and a -fuzzy measurable mapping is called an -fuzzifying measurable mapping.

Obviously a Klement fuzzy measurable mapping can be viewed as an -measurable mapping.

The following theorem gives a characterization of -fuzzy measurable mappings.

Theorem 3.3. Let and be two -fuzzy measurable spaces. A mapping is -fuzzy measurable if and only if for all .

Proof. Necessity.. If is -fuzzy measurable, then for all . Hence for all , we haveSufficiency.. If for all , then for all ; this shows that is -fuzzy measurable.

The next three theorems are trivial.

Theorem 3.4. If and are -fuzzy measurable, then is -fuzzy measurable.

Theorem 3.5. Let and be -fuzzy measurable spaces. Then a mapping is -fuzzy measurable if and only if is -measurable for any .

Theorem 3.6. Let be completely distributive, and let and be -fuzzy measurable spaces. Then a mapping is -fuzzy measurable if and only if is -measurable for any .

Corollary 3.7. Let and be -fuzzifying measurable spaces. Then a mapping is -fuzzifying measurable if and only if is measurable for any .

Corollary 3.8. Let be completely distributive, and let and be -fuzzifying measurable spaces. Then a mapping is -fuzzifying measurable if and only if is measurable for any .

4. -Fuzzy -Algebras Generated by -Fuzzifying -Algebras

In this section, will be used to denote the -algebra of Borel subsets of .

Theorem 4.1. Let be an -fuzzifying measurable space. Define a mapping by Then is a stratified -fuzzy -algebra, which is said to be the -fuzzy -algebra generated by .

Proof. (LMS1) For any and for any , if , then ; if , then . However, we have that . This shows that .
(LMS2) and , we have
(LMS3) for any and , by
we obtain .

Corollary 4.2. Let be a measurable space. Define a subset can be viewed as a mapping by Then is a stratified --algebra.

From Corollary 4.2, we see that the functor in Theorem 4.1 is a generalization of Klement functor .

Theorem 4.3. Let and be two -fuzzifying measurable spaces, and is a map. Then is -fuzzifying measurable if and only if is -fuzzy measurable.

Proof. Necessity. Suppose that is -fuzzifying measurable. Then for any . In order to prove that is -fuzzy measurable, we need to prove that for any .
In fact, for any , by we can prove the necessity.
Sufficiency. Suppose that is -fuzzy measurable. Then for any . In particular, it follows that for any . In order to prove that is -fuzzifying measurable, we need to prove that for any . In fact, for any and for any , if , then ; if , then ; if only one of 0 and 1 is in , then or . However, we have
This shows that is -fuzzifying measurable.

Corollary 4.4. Let and be two measurable spaces, and is a mapping. Then is measurable if and only if is -measurable.

Acknowledgments

The author would like to thank the referees for their valuable comments and suggestions. The project is supported by the National Natural Science Foundation of China (10971242).

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Copyright

Copyright © 2010 Fu-Gui Shi. 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.