#### Abstract

Based on the falling shadow theory, the concept of falling fuzzy (implicative) ideals as a generalization of that of a -fuzzy (implicative) ideal is proposed in MV-algebras. The relationships between falling fuzzy (implicative) ideals and -fuzzy (implicative) ideals are discussed, and conditions for a falling fuzzy (implicative) ideal to be a -fuzzy (implicative) ideal are provided. Some characterizations of falling fuzzy (implicative) ideals are presented by studying proprieties of them. The product and the up product operations on falling shadows and the upset of a falling shadow are established, by which -fuzzy ideals are investigated based on probability spaces.

#### 1. Introduction

Nonclassical logics take advantage of classical logics to handle information and uncertainty, and they become a formal and useful tool for dealing with fuzzy information and uncertain information in computer science. MV-algebras as the algebraic counterpart of many-valued prepositional calculus were proposed by Chang [1]. The classical two-valued logic gives rise to the study of Boolean algebras and every Boolean algebra will be an MV-algebra whereas the converse does not hold. The prototypical model of MV-algebras is based on the real interval . Motivated by the search for adequate algebraic structure for the quantum counterpart of the real interval , Giuntini [2] introduced the notion of QMV-algebras which is a nonlattice theoretic generalization of MV-algebras. From an algebraic point of view, MV-algebras and QMV-algebras share a common set of axioms, which was called supplement algebras (S-algebras). What makes an S-algebra an MV-algebra is the addition of the Łukasiewicz axiom, and it also makes MV-algebra a lattice ordered structure. MV-algebras form a category which is equivalent to the category of abelian lattice ordered groups (-groups, for short) with strong units [3]. These make the interest in MV-algebras relevant outside the realm of logic.

The ideal theory plays an important role in studying logical algebras. From the logic point of view, the sets of provable formulas in corresponding systems can be described by (fuzzy) ideals of those algebraic semantics. Some types of ideals in MV-algebras have been widely studied and many important results are obtained [4, 5]. Belluce and Di Nola [6] gave the definition of Łukasiewicz rings and derived that MV-algebras arising as the MV-algebra of ideals of a commutative ring are exactly the complete and atomic ones. Lele and Nganou [7] introduced the notion of ideals in BL-algebras as a natural generalization of that of ideals in MV-algebras and proved that quotient BL-algebras turn out to be MV-algebras. It is also proved that fuzzy Boolean ideals coincide with fuzzy implicative ideals in MV-algebras [8]. Jun and Walendziak [9] applied the fuzzy set theory to ideals of pseudo MV-algebras and introduced the notion of fuzzy (implicative) ideals; moreover, [10] extended the notions of fuzzy ideals to -fuzzy (implicative) ideals by using the concept of quasicoincidence of a fuzzy value with a fuzzy set.

Falling shadow representation theory was introduced by Goodman [11] and Wang and Sanchez [12] independently, and it directly relates probability concepts to the membership function of fuzzy sets, just as Goodman pointed out that the equivalence of a fuzzy set and a class of random sets aims to study a unified treatment of uncertainty modelled by means of combining probability and fuzzy set theory. Tan et al. [13, 14] established a theoretical approach for defining a fuzzy inference relation and fuzzy set operations based on the theory of falling shadows. Yuan and Lee [15] gave a theoretical approach of the fuzzy algebraic system based on the mathematical structure of the falling shadow theory which was formulated in [16]. The characterization of the approach is that a fuzzy subalgebraic system is considered as the falling shadow of the cloud of the subalgebraic system. The falling shadow theory was also applied to study subalgebras and ideals of BCK/BCI-algebras [17, 18] and -algebras [19]. Inspired by [15], Yu et al. investigated falling fuzzy ideals of a hemiring [20] and falling fuzzy filters of a BL-algebra [21] based on the theory of falling shadows and fuzzy sets, which provide a theoretical approach for the further studying of fuzzy ideals in MV-algebras.

The paper aims to investigate ideals of MV-algebras based on the falling shadow theory. The notion of falling fuzzy (implicative) ideals is introduced, and then some properties of them are studied. It is pointed out that a falling fuzzy (implicative) ideal is a -fuzzy (implicative) ideal and some conditions under which a falling fuzzy (implicative) ideal becomes a -fuzzy (implicative) ideal are provided. We also derive several characterizations of falling fuzzy (implicative) ideals. -fuzzy ideals are investigated based on the probability space by introducing the product and up product operations and the upset of a falling shadow.

#### 2. Preliminaries

In the section, we present some definitions and results about MV-algebras for purpose of reference.

Definition 1 (see [22]). A -norm is a binary operation on   i.e., satisfying the following conditions: (i)is commutative and associative; that is, for any , (ii) is nondecreasing in both arguments; that is, (iii) and for any .
is a continuous -norm if it is a -norm and is a continuous mapping of into .

Example 2 (see [22]). The following are some important examples of continuous -norms: (1)Łukasiewicz -norm ;(2)Gödel -norm: ;(3)product -norm: .

Lemma 3 (see [22]). Let be a continuous -norm. Then there is a unique operation satisfying the condition if and only if for all ; namely, .
The operation from Lemma 3 is called the residuum of the -norm.

Definition 4 (see [1, 23]). An MV-algebra is an algebra of type satisfying the following equations: for any , (MV1);(MV2);(MV3);(MV4);(MV5);(MV6).

Let be an MV-algebra. For any , we define , , , , , , , , , and for any ; then is a bounded distributive lattice [23].

From now on, is an MV-algebra unless otherwise mentioned, which will often be referred to by its support set . Here we summarize the necessary notions and some previous results which will be used in the sequel.

Lemma 5 (see [23, 24]). In any MV-algebra , the following properties hold: for any , (1) if and only if if and only if ;(2), , ;(3);(4);(5);(6) implies , , and ;(7), ;(8), .

If a mapping over a set is defined as follows: for any ,

then is called a fuzzy set over , where is the degree of membership of with respect to [25].

Letting be a nonempty set, denote the set of all fuzzy sets of by . For any , define : for any . implies that and hold. The fuzzy sets and over are defined by , , for any .

The theory of falling shadows is an important tool in the theoretical developments and practical applications of fuzzy sets, and some of their properties and notions are displayed in the following. For further information, the readers can be referred to [16, 26].

Given a universal set , let denote the power set of . For any , let and for any let .

An order pair is called to be a hypermeasurable structure on if is a -field in and .

Let be a probability space and a hypermeasurable structure on . If a mapping is measurable, that is, , then is called a random set on .

Let be a random set on . For any , if , then is a fuzzy set of . The fuzzy set is called a falling shadow of the random set , and is called a cloud of (see Figure 1).

Let be a probability space, where is a probability distribution of two-dimensional random variables on . There are many types of probability distributions, but only three types are the most classic ones (see Figure 2).(1)If the whole probability is concentrated and uniformly distributed on the main diagonal of the unit square , then is called a diagonal distribution.(2)If the whole probability is concentrated and uniformly distributed on the antidiagonal of the unit square , then is called an antidiagonal distribution.(3)If the whole probability is uniformly distributed on the unit square , then is called an independent distribution.

A nonempty set of is called an ideal of if it satisfies the following conditions: , implies ; and imply . For purpose of convenience, let be an ideal of in the the rest of the sections. If an ideal satisfies the following condition: and imply for any , then is called an implicative ideal of .

Proposition 6 (see [1]). Let be a nonempty set of . Then the following statements are equivalent: (1) is an ideal of ;(2); , , and imply ;(3), if and , then .

#### 3. Falling Fuzzy Ideals

In this section, we will introduce the notions of -fuzzy ideals and falling fuzzy ideals of MV-algebras. The relationships between -fuzzy ideals and falling fuzzy ideals are provided, and some characterizations of falling fuzzy ideals are displayed.

Definition 7. A fuzzy set of is called a -fuzzy ideal of , if satisfies , (1);(2) implies .

It is easy to see that a -fuzzy ideal of is a -fuzzy ideal when and a -fuzzy ideal is also called a fuzzy ideal [4]. We denote the set of all -fuzzy ideals of by TFI. The next result can be proved similar to that for -fuzzy ideals of .

Theorem 8. Let be a fuzzy set of . Then is a -fuzzy ideal of if and only if it satisfies the following conditions: , (1);(2).

Proposition 9. Let be a fuzzy set of . Then the following conditions are equivalent: (1) is a -fuzzy ideal of ;(2), ; , ;(3), is an ideal of , where .

In the following, we give some properties of -fuzzy ideals for the further discussion.

Proposition 10. Let , be fuzzy sets of . If , are -fuzzy ideals of , then is a -fuzzy ideal of .

Proof. The proof is straightforward.

As a direct consequence of Proposition 10, we have the following result.

Corollary 11. Let be fuzzy sets of , where is an index set. If is a -fuzzy ideal of , then is a -fuzzy ideal of .

Example 12. Let where , . Define the operations and on as follows:
It is clear that is an MV-algebra. Define fuzzy sets and of as follows:
Then and are fuzzy ideals of , but is not a fuzzy ideal, since .

However, the union of two -fuzzy ideals is not a -fuzzy ideal. In order to investigate the algebraic properties of the set of all -fuzzy ideals in BL-algebras, [27, 28] introduced the notion of generated fuzzy ideals in BL-algebras. At first we borrow the notion and modify it for our reasons.

Definition 13. Let be a fuzzy subset of . A -fuzzy ideal of is said to be generated by , if and implies   for any -fuzzy ideal of . The fuzzy ideal generated by will be denoted by .

The -fuzzy ideal of generated by a fuzzy subset is the least -fuzzy ideal of containing . It is also the intersection of all the -fuzzy ideals of containing .

The following theorem shows how to construct the -fuzzy ideal generated by a fuzzy subset.

Theorem 14. Let be a fuzzy subset of . If the fuzzy subset of is defined by for any , then .

For any TFI, we define . Generally, for any TFI, . Similar to Theorem  38 in [28], we can obtain the following result.

Theorem 15. TFI, is a complete modular lattice with and as the least lower bound and the largest upper bound of TFI, respectively, where and for any .

In what follows, we will introduce the notion of falling fuzzy ideals and then investigate their properties.

Definition 16. Let be a probability space and let be a random set. If is an ideal of for any , then the falling shadow of the random set , that is, , is called a falling fuzzy ideal of .

For better understanding the definition of falling fuzzy ideals, we illustrate it by the following example.

Example 17. Let , , . For any , we define operations and as follows:
It is easy to verify that is an MV-algebra.
Let , where is a Borel field on and is the usual Lebesgue measure. Denote and . The mapping is defined by and then is an ideal of for any . Thus is a falling fuzzy ideal of , where is represented as follows:

Let be a probability space and . Define the operations and on as follows: , ,

Let be defined by for any . Then it is easy to check that is an MV-algebra. Define the operation on by , for any , .

For any subset of and , let and Then .

Proposition 18. Let be a probability space, a nonempty subset of , and . If is an ideal of , then is an ideal of .

Proof. Suppose that is an ideal of . Since for any , we have that . Let be such that and . For any , we have and . Hence ; that is, . Thus is an ideal of .
Noticing that and is a random set of , we get that is a falling fuzzy ideal of , where .

Proposition 19. Let be a -fuzzy ideal of . Then is a falling fuzzy ideal of .

Proof. Consider the probability space , where is a Borel field on and is the usual Lebesgue measure. Since is a -fuzzy ideal of , then is an ideal of for any , by Proposition 9. Define the random set by for any ; then is a falling fuzzy ideal of .

However, it is important and interesting to point out that the converse of Proposition 19 is not true in general and we illustrate it by the following example.

Example 20. Let and be the power set of . Let , , and denote, respectively, the join, the complement, and the smallest element in . It is clear that is an MV-algebra.
Let , where is a Borel field on and is the usual Lebesgue measure. The mapping is defined by
Then is an ideal of for any . Thus is a falling fuzzy ideal of , where is represented as follows:
But is not a -fuzzy ideal of since .

Let be a probability space and a falling shadow of a random set . For any , let ; then . In what follows, we give a number of equivalent conditions of falling fuzzy ideals for further discussion.

Theorem 21. Let be a falling shadow of a random set . Then is a falling fuzzy ideal of if and only if , and imply for any , .

Proof. Assuming that is a falling fuzzy ideal of , then is an ideal of for any , and it follows that . Let be such that and . Supposing that , then , a contradiction, and thus is valid.
Conversely, assume that and are valid. For any , let be such that and . Supposing that , it follows that , which is a contradiction, and thus . Hence is a falling fuzzy ideal of .

Theorem 22. Let be a random set and a falling shadow of . Then is a falling fuzzy ideal of if and only if implies for any .

Proof. Let be such that . For any , we get , . Considering that is a falling fuzzy ideal of , we have that is an ideal of . By Proposition 6, ; that is, . Hence .
Conversely, assume that implies for any . Let be such that . Then there exists ; that is, . Since , then . It follows that ; that is, . Let be such that and . Then . Since , it follows that . Thus ; that is, . Hence is an ideal of , and so is a falling fuzzy ideal of .

Lemma 23. Let be a random set and a falling shadow of . If is a falling fuzzy ideal of , then the following statements hold: for any , (1);(2).

Proof. (1)Noticing that is a falling fuzzy ideal of , we have that is an ideal of for any . It follows that ; that is, . Hence .(2)For any , we get and . Since is an ideal of , then ; that is, . Thus .

Theorem 24. Let be a falling shadow of a random set . Then is a falling fuzzy ideal of if and only if, for any , (1);(2) implies .

Proof. Assuming that is a falling fuzzy ideal of , then is an ideal of for any . For any , if , , then . That is, and imply . Hence . If , that is, , it follows that by Theorem 22 and Lemma 23.
Conversely, assume that and are valid. For any , let be such that and . It follows that . Thus ; that is, . If and for any , then . Thus . Hence is an ideal of , and so is a falling fuzzy ideal of .

Proposition 25. Let be a falling shadow of a random set . Then is a falling fuzzy ideal of if and only if, for any , (1);(2).

Proof. The proof is obvious since if and only if for any .

Proposition 26. Let be a falling shadow of a random set . Then is a falling fuzzy ideal of if and only if, for any , (1);(2).

Proof. One direction is clear since . Conversely, assume that conditions and hold. By hypothesis, we have for any . Now let and . Then and . It means that implies , and so is a falling fuzzy ideal of by Theorem 24.

Theorem 27. Let be a falling shadow of a random set . If the following conditions are valid: (1);(2), for any ,then is a falling fuzzy ideal of .

Proof. Assume that conditions and are valid. For any , we have ; that is, . For any , if and , then , . It follows that . Hence ; that is, . Thus is an ideal of , and so is a falling fuzzy ideal of .

Proposition 28. Let be a falling shadow of a random set . If is a falling fuzzy ideal of , then the following relationships hold: for any , (1);(2)if , then ;(3);(4);(5);(6)if , then .

Proof. Here we only prove , and the other cases directly follow from Lemma 23 and Theorem 24. For any , we have by .

According to Proposition 28, we can provide another condition for a falling shadow to be a falling fuzzy ideal in MV-algebras.

Proposition 29. Let be a falling shadow of a random set . If the following conditions are valid: (1);(2), for any ,then is a falling fuzzy ideal of .

Proof. By hypothesis, we get , for any . It follows from Theorem 27 that is a falling fuzzy ideal of .

From Proposition 19, it is known that the notion of falling fuzzy ideals is a generalization of that of -fuzzy ideals. Under what conditions a falling fuzzy ideal becomes a -fuzzy ideal, we will give some answers to the questions in the following.

Theorem 30. Let be a falling shadow of a random set . If is a falling fuzzy ideal of , then is a -fuzzy ideal of .

Proof. Noticing that is a falling fuzzy ideal of , we have that for any , by Lemma 23. Since , thus . It follows that for any by Lemma 23; that is, . Thus   −   or . Since , then . Hence is a -fuzzy ideal of .

Proposition 31. Let be a falling shadow of a random set . If is a falling fuzzy ideal of , then the following statements hold: , (1)if or , then is a -fuzzy ideal of ;(2)if and are independent random events, then is a -fuzzy ideal of .

Proof. (1)By Theorem 30, for any . According to hypothesis and Lemma 23, we have or for any . If , then . If , then . Hence , and so is a -fuzzy ideal of .(2)It is directly obtained that for any . Since is a falling fuzzy ideal of , then , for any . It follows that . Thus is a -fuzzy ideal of .

Definition 32. Let be a probability space, and let , , and be falling shadows of random sets , , , respectively. Then the product of and is defined by
The up product between and is defined by
The upset of is defined by

From the above definition, it is easy to see that for any , and so .

If the probability distribution in Definition 32 is diagonal, antidiagonal, and independent, respectively, then and , where .

Proposition 33. Let be a probability space and a falling shadow of a random set . Then if and only if implies for any .

Proof. Suppose that . For any , if , then .
Conversely, for any , we have by hypothesis. Thus , and so .

Theorem 34. Let be a probability space and a falling shadow of a random set . If the probability distribution of two-dimensional random variables is diagonal antidiagonal or independent, resp.), then is a -fuzzy ideal of if and only if (1),(2),where .

Proof. We only consider that is diagonal, and other cases can be proved similarly. Supposing that is a -fuzzy ideal of , there exist such that for any , then we get that , and so . Thus . For any , there exists such that , then , and thus . Therefore, .
Conversely, assume that and hold. For any , we have . If , then . Therefore, is a -fuzzy ideal of .

Theorem 35. Let be a probability space and a falling shadow of a random set . If the probability distribution of two-dimensional random variables is diagonal, then is a -fuzzy ideal of if and only if .

Proof. Suppose that is a -fuzzy ideal of . For any , there exist such that , then we have , and so . Thus .
Conversely, assume that . For any , we have , and so . For any , , and then . Therefore, is a -fuzzy ideal of .

Proposition 36. Let be a probability space and a falling shadow of a random set . If and the probability distribution of two-dimensional random variables is antidiagonal or independent, then is a -fuzzy ideal or -fuzzy ideal of if and only if .

Proof. We only consider that is antidiagonal. Suppose that . For any , we have , and so . For any , ; then . Therefore is a -fuzzy ideal of .
Conversely, suppose that is a -fuzzy ideal of . The proof of is similar to that of Theorem 35.

Proposition 37. Let be a probability space, where the probability distribution of two-dimensional random variables is diagonal. Let and be falling shadows of random sets , , respectively. If and are -fuzzy ideals of , then is a -fuzzy ideal of .

Proof. Since and are -fuzzy ideals of , then we have for any .
If , then = . Therefore is a -fuzzy ideal of .

#### 4. Characterizations of Falling Fuzzy Implicative Ideals

In the section, we introduce the notion of falling fuzzy implicative ideals as a generalization of -fuzzy implicative ideals in MV-algebras and investigate some of their properties. We also give some conditions under which falling shadows (falling fuzzy ideals) become falling fuzzy implicative ideals.

Definition 38. A fuzzy set in is called a -fuzzy implicative ideal of if it satisfies , (1);(2).

Definition 39. Let be a probability space and be a random set. If is an implicative ideal of for any , then the falling shadow of , that is, , is called a falling fuzzy implicative ideal of .

For the sake of simplicity and better understanding the above definition, we give the following example.

Example 40. Let be such that , , and . Define the operations and on as follows:
Routine computations prove that is an MV-algebra.
Let , where is a Borel field on and is the usual Lebesgue measure. The mapping is defined by
Then is an implicative ideal of for any . Thus is a falling fuzzy implicative ideal of , where is represented as follows:

Proposition 41. Let be a probability space, a nonempty subset of , and . If is an implicative ideal of , then is an implicative ideal of .

Proof. It is similar to the proof of Proposition 18.

Since an implicative ideal is an ideal in MV-algebras and the converse is not true, thus we immediately get the following result.

Proposition 42. Every falling fuzzy implicative ideal of is a falling fuzzy ideal, and the converse is not valid.

The following proposition describes the relationship between falling fuzzy implicative ideals and fuzzy implicative ideals, which is similar to Proposition 19, and we omit the proof.

Proposition 43. Let be a -fuzzy implicative ideal of . Then is a falling fuzzy implicative ideal of .

According to Proposition 42 and Example 20, a falling fuzzy implicative ideal of is a falling fuzzy ideal and a falling fuzzy ideal is not a -fuzzy ideal of in general, thus the converse of Proposition 43 is not true.

In what follows, we will display some characterizations of falling fuzzy implicative ideals.

Theorem 44. Let be a falling shadow of a random set . Then is a falling fuzzy implicative ideal of if and only if , and imply for any , .

Proof. Suppose that is a falling fuzzy implicative ideal of , it follows that is an implicative ideal of for any , and thus . Let be such that and . If hold, then , which is a contradiction, and so is valid.
Conversely, let be such that and for any . If , it follows from hypothesis that , a contradiction, and so . Therefore is a falling fuzzy implicative ideal of .

Proposition 45. Let be a falling shadow of a random set . If is a falling fuzzy implicative ideal of , then we have, for any , (1)