#### 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 [5โ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. ( ๐ฟ , ๐ ) -Fuzzy ๐ -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)* โโ, .
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. Letโโ 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. ( ๐ฟ , ๐ ) -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. ( ๐ผ , ๐ผ ) -Fuzzy ๐ -Algebras Generated by ๐ผ -Fuzzifying ๐ -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).