– algebra of Sets and Some of its Properties

The main objective of this work is to generalize the concept of fuzzy – algebra by introducing the notion of fuzzy – algebra. Characterization and examples of the proposed generalization are presented, as well as several different properties of fuzzy – algebra are proven. Furthermore, the relationship between fuzzy – algebra and fuzzy algebra is studied, where it is shown that the fuzzy – algebra is a generalization of fuzzy algebra too. In addition, the notion of restriction, as an important property in the study of measure theory, is studied as well. Many properties of restriction of a nonempty family of fuzzy subsets of fuzzy power set are investigated and it is shown that the restriction of fuzzy – algebra is fuzzy – algebra too


Introduction and Basic concepts
Wang [1] in 2009 studied some types of collections of sets which are generalizations of -algebra such as ring, -ring and proved some important results of these concepts, where a nonempty class is called a ring iff E,F , then E-F and E F . In 2019 Ahmed and Ebrahim [2] introduced some generalizations of -algebra and -ring. Many other authors are interested in studying -algebra andring, for example, see [3], [4] and [5]. Zadeh [6] in 1965 first introduced the concept of the fuzzy set where is a nonempty set, then a fuzzy set in is defined as a set of ordered pairs { )) : } where : is a function such that for every , ) represents the degree of membership of in . Brown [7] studied some types of fuzzy sets such as fuzzy power set, empty fuzzy set, universal fuzzy set, the complement of a fuzzy set, the union of two fuzzy sets and intersection of two fuzzy. Ahmed et al. [8][9][10][11] first introduced the concept of fuzzy σ algebra and fuzzy algebra where is a nonempty set and ) . A nonempty collection ) is said to be a fuzzy -algebra of sets over a fuzzy set { ) : }, if the following conditions are satisfied: If condition 3 is satisfied only for finite sets, then is said to be a fuzzy algebra over a fuzzy set . Another generalization of the fuzzy -algebra introduced in this paper, which is a fuzzy -algebra. The main aim of this chapter is to study this generalization and introduce some of its basic properties, examples and some characterizations of them.

Definition (1.4) [9]
Let be a nonempty set. Then the union of the two fuzzy sets in with respective membership functions ) ) is a fuzzy set in whose membership function is related to those of by ) { ) )}, In symbols:

Definition (1.5) [10]
Let be a nonempty set. Then the intersection of two fuzzy sets in with respective membership functions ) ) is a fuzzy set in whose membership function is related to those of by ) { ) )}, In symbols: Definition (1.6) [6] Let be a nonempty set and is a fuzzy sets in . Then the complement of a fuzzy set is denoted by and defined as: Every fuzzy -algebra is a fuzzy algebra .

The main results:
In this section, we introduce the concept of fuzzy -algebra which is a generalization for the concept of the fuzzy -algebra and fuzzy algebra. Also, we present many properties of fuzzy -algebra.

Definition (2.1):
Let be a nonempty set. A nonempty collection ) is said to be a fuzzy -algebra of sets ( -field ) over a fuzzy set , if the following conditions are satisfied:

Definition(2.2):
Let be a nonempty set and ) be a fuzzy -algebra ( -field ) over a fuzzy set . Then the pair ( , ) is said to be fuzzy measurable space relatively to fuzzy -algebra.

Example(2.3):
Then the pair ( , ) is said to be fuzzy measurable space relatively to fuzzy -algebra.

Proposition(2.5):
Let be an infinite set and ={ , , all s.t is finite}. Then ( , ) is a fuzzy measurable space relatively to fuzzy -algebra.

Proof:
From the definition of , we get , . Let . Then is finite for all k =1, 2, …, . Hence, ⋂ is finite. Now, since . Therefore, is a fuzzy and ( , ) is a fuzzy measurable space relatively to fuzzy -algebra.

Proof:
Since be an infinite set, then each of , be an infinite fuzzy set, but and , then by definition of we have, . Let . Then is an infinite fuzzy set for every k=1,2,…, and hence is an infinite fuzzy set. Therefore, and is a -.

Proof:
Since is a -, , then .
Since is a i , then , . Hence, ⋂ , therefore ⋂ is a -

Definition(2.11):
Let be a nonempty set and ). Then the intersection of all fuzzy -algebra over a fuzzy set , which includes is said to be the fuzzy -algebra over a fuzzy set that is generated by and denoted by ), that is,

Proposition (2.12):
Let be a nonempty set and ). Then ) is the smallest fuzzy -algebra over a fuzzy set that includes .

Proof:
From the definition of ), we have: -algebra over a fuzzy set . To prove ) . let be a fuzzy -algebra over a fuzzy set that includes . Then ⋂ , thus ). Now, let be a fuzzy -algebra over a fuzzy set that includes . Then . Therefore, ) is the smallest fuzzy -algebra over a fuzzy set that includes .

Proposition (2.14):
Let be a nonempty set and ). Then ) if and only if is fuzzy -algebra over a fuzzy set .

Proof:
Let ) and let ) . Since ) is a fuzzy -algebra over a fuzzy set , then is fuzzy -algebra over a fuzzy set . Conversely, suppose that is a fuzzy -algebra over a fuzzy set . Since ) is a fuzzy -algebra over a fuzzy set which includes , then ) . But is fuzzy -algebra over a fuzzy set such that and ) is the smallest fuzzy -algebra over a fuzzy set that includes , then ) and hence ) .

Proposition (2.15):
Every fuzzyis a fuzzy -. . Therefore, is a fuzzy -. In general, the converse of Proposition (2.15) is not true as shown in the following example:

Proposition (2.17):
Let be a nonempty set and ). Then ) ), where ) is the smallest fuzzy -algebra over a fuzzy set that includes .

Proof:
Let be a nonempty set and ). Then ) is a fuzzy -algebra over a fuzzy set that includes . Since every fuzzy -algebra over a fuzzy set is a fuzzy -algebra over a fuzzy set , then ) is a fuzzy -algebra over a fuzzy set that includes . But ) is the smallest fuzzy -algebra over a fuzzy set that includes this implies that ) ).

Proof:
The result follows from the definition of fuzzy In general, the converse of the previous Remark is not true as shown in the following example:

Proposition (2.20):
Let be a nonempty set and ). Then ) ), where ) is the smallest fuzzy algebra over a fuzzy set that includes .

Proof:
It is clear, so that it is omitted.

Definition(2.21) :
Let be a nonempty collection of fuzzy subsets of fuzzy power set ) of a nonempty set that is ) and let be a nonempty fuzzy subset of a fuzzy set that is . Then the restriction of on is denoted by which is defined as follows: = { : = ⋂ , for some }.
)} This implies that for any there is such that = ⋂ . Therefore, is the restriction of on .

Proposition (2.23):
Let be a fuzzy -algebra over a fuzzy set and let be a nonempty fuzzy subset of a fuzzy set . Then is a fuzzy -algebra over a fuzzy set .

Proof:
Since is a fuzzy -algebra over a fuzzy set , then . Since , then ) ) for all and hence Let . Then there are such that = ⋂ for all =1,2,…, which implies that = ⋂ )= ) ⋂ . Since is a fuzzy -algebra over a fuzzy set , then and hence by definition of we get . Therefore, is a fuzzy -algebra over a fuzzy set .

Proposition(2.24):
Let be a fuzzy -algebra over a fuzzy set and . If , then .

Proof:
Let be a fuzzy -algebra over a fuzzy set and let . Suppose that , since , then ) ) . So, we have

Proposition (2.25):
Let be a fuzzy -algebra over a fuzzy set and let be a nonempty fuzzy subset of a fuzzy set such that .

Proposition(2.27):
Let be a nonempty set and ) and let be a nonempty fuzzy subset of a fuzzy set . Then ) is a fuzzy -algebra over a fuzzy set .

Proposition(2.28):
Let be a nonempty set and ) and let be a nonempty fuzzy subset of a fuzzy set . Let be a nonempty set and ) and let be a nonempty fuzzy subset of a fuzzy set . Then ( ) ) .