Abstract

Based on the theory of a falling shadow which was first formulated by Wang (1985), a theoretical approach of the ideal structure in -algebras is established. The notions of a falling -subalgebra, a falling -ideal, a falling -ideal, and a falling -ideal of a -algebra are introduced. Some fundamental properties are investigated. Relations among a falling -subalgebra, a falling -ideal, a falling -ideal, and a falling -ideal are stated. Characterizations of falling -ideals and falling -ideals are discussed. A relation between a fuzzy -subalgebra and a falling -subalgebra is provided.

1. Introduction

Iséki and Tanaka introduced two classes of abstract algebras -algebras and -algebras [1, 2]. It is known that the class of -algebras is a proper subclass of the class of -algebras. -algebras have several connections with other areas of investigation, such as: lattice ordered groups, -algebras, Wajsberg algebras, and implicative commutative semigroups. Font et al. [3] have discussed Wajsberg algebras which are term-equivalent to -algebras. Mundici [4] proved that -algebras are categorically equivalent to bounded commutative -algebras. Meng [5] proved that implicative commutative semigroups are equivalent to a class of -algebras. Neggers and Kim [6] introduced the notion of -algebras which is another useful generalization of -algebras. They investigated several relations between -algebras and -algebras as well as several other relations between -algebras and oriented digraphs. After that, some further aspects were studied in [7, 8]. Neggers et al. [9] introduced the concept of -fuzzy function which generalizes the concept of fuzzy subalgebra to a much larger class of functions in a natural way. In addition, they discussed a method of fuzzification of a wide class of algebraic systems onto along with some consequences.

In the study of a unified treatment of uncertainty modelled by means of combining probability and fuzzy set theory, Goodman [10] pointed out the equivalence of a fuzzy set and a class of random sets. Wang and Sanchez [11] introduced the theory of falling shadows which directly relates probability concepts with the membership function of fuzzy sets. Falling shadow representation theory shows us the way of selection relaid on the joint degrees distributions. It is reasonable and convenient approach for the theoretical development and the practical applications of fuzzy sets and fuzzy logics. The mathematical structure of the theory of falling shadows is formulated in [12]. Tan et al. [13, 14] established a theoretical approach to define a fuzzy inference relation and fuzzy set operations based on the theory of falling shadows. Jun and Kang [15] established a theoretical approach to define a fuzzy positive implicative ideal in a -algebra based on the theory of falling shadows. They provided relations between falling fuzzy positive implicative ideals and falling fuzzy ideals. They also considered relations between fuzzy positive implicative ideals and falling fuzzy positive implicative ideals. Jun and Kang [16] considered the fuzzification of generalized Tarski filters of generalized Tarski algebras and investigated related properties. They established characterizations of a fuzzy-generalized Tarski filter and introduced the notion of falling fuzzy-generalized Tarski filters in generalized Tarski algebras based on the theory of falling shadows. They provided relations between fuzzy-generalized Tarski filters and falling fuzzy-generalized Tarski filters and established a characterization of a falling fuzzy-generalized Tarski filter.

In this paper, we establish a theoretical approach to define a falling -subalgebra, a falling -ideal, a falling -ideal, and a falling -ideal in -algebras based on the theory of falling shadows which was first formulated by Wang [12]. We provide relations among a falling -subalgebra, a falling -ideal, a falling -ideal, and a falling -ideal. We consider characterizations of falling -ideals and falling -ideals and discuss a relation between a fuzzy -subalgebra and a falling -subalgebra.

2. Preliminaries

A -algebra is a nonempty set with a constant 0 and a binary operation “” satisfying the following axioms: (i), (ii), (iii) and imply ,

for all .

A -algebra is a -algebra satisfying the following additional axioms: (iv), (v),

for all .

Any -algebra satisfies the following conditions: (a1), (a2).

A subset of a -algebra is called a -ideal of if it satisfies (b1), (b2).

We now display the basic theory on falling shadows. We refer the reader to the papers [10–14] for further information regarding the theory of falling shadows.

Given a universe of discourse , let denote the power set of . For each , let and for each , let An ordered pair is said to be a hypermeasurable structure on if is a -field in and . Given a probability space and a hypermeasurable structure on , a random set on is defined to be a mapping which is - measurable, that is, Suppose that is a random set on . Let Then is a kind of fuzzy set in . We call a falling shadow of the random set , and is called a cloud of .

For example, , where is a Borel field on and is the usual Lebesgue measure. Let be a fuzzy set in and let be a -cut of . Then is a random set and is a cloud of . We will call defined above as the cut-cloud of (see [10]).

3. Falling -Subalgebras/Ideals

In what follows let denote a -algebra unless otherwise specified.

A nonempty subset of is called a -subalgebra of (see [8]) if whenever and .

A subset of is called a -ideal of (see [8]) if it satisfies conditions (b1) and (b2).

A subset of is called a -ideal of (see [8]) if it satisfies conditions (b2) and (b3) .

Definition 3.1. Let be a probability space, and let be a random set. If is a -subalgebra (resp., -ideal and -ideal) of for any with , then the falling shadow of the random set , that is, is called a falling -subalgebra (resp., falling -ideal and falling -ideal) of .

Example 3.2. Let be a probability space and let Define an operation on by for all . Let be defined by for all . It is routine to check that is a -algebra. For any -subalgebra (resp., -ideal and -ideal) of and , let Then and is a -subalgebra (resp., -ideal and -ideal) of . Since is a random set of . Hence the falling shadow on is a falling -subalgebra (resp., falling -ideal and falling -ideal) of .

Example 3.3. Let be a -algebra which is not a -algebra with the following Cayley table: (3.7) Let and define a random set as follows: Then the falling shadow of is a falling -subalgebra of .

Example 3.4. Let be a -algebra which is not a -algebra with the Cayley table as follows: (3.9) Let and define a random set as follows: Then the falling shadow of is a falling -ideal of .

Example 3.5. Let be a -algebra which is not a -algebra with the Cayley table as follows: (3.11) Let and define a random set as follows: Then the falling shadow of is a falling -ideal of .

Note that the falling shadow of in Example 3.4 is not a falling -subalgebra of because if we take , then is not a -subalgebra of . This shows that, in a -algebra, a falling -ideal need not be a falling -subalgebra.

The following example shows that a falling -subalgebra need not be a falling -ideal in -algebras.

Example 3.6. Consider the -algebra which is given in Example 3.4. Let and define a random set Then the falling shadow of is a falling -subalgebra of , but it is not a falling -ideal of since is not a -ideal of for .

Theorem 3.7. Every falling -ideal is a falling -subalgebra.

Proof. It is clear, and we omit the proof.

The following example shows that the converse of Theorem 3.7 is not true.

Example 3.8. Let be a -algebra which is not a -algebra with the Cayley table as follows: (3.14) Let and define a random set Then the falling shadow of is a falling -subalgebra of , but not a falling -ideal of , since is not a -ideal of for .

Let be a probability space and a falling shadow of a random set . For any , let Then .

Lemma 3.9. If is a falling -subalgebra of , then

Proof. If , then it is clear. Assume that and let be such that . Then , and so since is a -subalgebra of . Hence , and therefore for all .

Combining Theorem 3.7 and Lemma 3.9, we have the following corollary.

Corollary 3.10. If is a falling -ideal of , then (3.17) is valid.

We provide a characterization of a falling -ideal.

Theorem 3.11. Let be a falling shadow of a random set on . Then is a falling -ideal of if and only if the following conditions are valid: (a), (b).

Proof. Assume that is a falling -ideal of . For any , if then and . Since is a -ideal of , it follows from (b2) that so that . Hence for all . Now let and be such that . Then and so by (b3). Thus , and therefore for all .
Conversely, suppose that two conditions (a) and (b) are valid. Let and be such that and . Then and . It follows from (a) that so that . Now, assume that for every and . Then for all , and so . Therefore is a -ideal of for all . Hence is a falling -ideal of .

Proposition 3.12. For a falling shadow of a random set on , if is a falling -ideal of , then (a), (b), (c).

Proof. (a) Let and be such that and . Then and by (b1). It follows from (b2) that so that . Hence for all with .
(b) Let and be such that . Then and . Since is a -ideal of , it follows from (b2) that so that . Hence for all .
(c) It follows from (ii) and (a).

We give conditions for a falling shadow to be a falling -ideal.

Theorem 3.13. For a falling shadow of a random set on , assume that the following conditions are satisfied: (a), (b). Then is a falling -ideal of .

Proof. Using (a), we have for all . Let and be such that and . Then by (b), and so . Therefore is a -ideal of for all . Hence is a falling -ideal of .

Proposition 3.14. If is a falling -ideal of , then

Proof. Let be such that . Let . Then and by Corollary 3.10. Hence . Since is a -ideal of , it follows from (b2) that . Therefore (3.19) holds.

A -ideal of is called a -ideal of (see [8]) if, for arbitrary , (b4) whenever and .

Definition 3.15. Let be a probability space, and let be a random set. If is a -ideal of for any with , then the falling shadow of the random set is called a falling -ideal of .

Example 3.16. Let be a -algebra as in Example 3.8. Let and define a random set Then the falling shadow of is a falling -ideal of , and it is represented as follows:

Theorem 3.17. Every falling -ideal is a falling -ideal.

Proof. Straightforward.

We provide an example to show that the converse of Theorem 3.17 is not true.

Example 3.18. Consider the falling -ideal of which is given in Example 3.5. For is not a -ideal of since ,, but . Hence is not a falling -ideal of .

In the above discussion, we can see the following relations:(3.23)

In this diagram, the reverse implications are not true, and we need additional conditions for considering the reverse implications.

A -algebra is called a -algebra (see [8]) if it satisfies the identity for all .

Theorem 3.19. In a -algebra, every falling -ideal is a falling -ideal.

Proof. Let be a falling -ideal of a -algebra . Then for all by Proposition 3.12. Let and . Then . Since is a -algebra, we have and so by (b2). Hence , which shows that for all . Using Theorem 3.11, we conclude that is a falling -ideal of .

Corollary 3.20. In a -algebra, every falling -ideal is a falling -subalgebra.

Proof. It follows from Theorems 3.7 and 3.19.

The following example shows that, in a -algebra, any falling -subalgebra is neither a falling -ideal nor a falling -ideal.

Example 3.21. Let be a -algebra which is not a -algebra with the following Cayley table: (3.24) Let and define a random set as follows: Then the falling shadow of is a falling -subalgebra of , but it is neither falling -ideal nor a falling -ideal of since is neither a -ideal nor a -ideal of for .

Hence, in a -algebra, we have the following relations among falling -ideals, falling -subalgebras, and falling -ideals:(3.26)

We now establish a characterization of a falling -ideal.

Theorem 3.22. For a falling shadow of a random set on , the followings are equivalent. (a) is a falling -ideal of .(b) is a falling -ideal of that satisfies the following inclusion:

Proof. Assume that is a falling -ideal of . Then is a falling -ideal of . Let and be such that . Then and , and so since is a -ideal of . Hence , and therefore for all .
Conversely, let be a falling -ideal of satisfying the condition (3.27). Then is a -ideal of . Let and be such that and . Then by (3.27), and thus . Hence is a falling -ideal of .