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Journal of Applied Mathematics

Volume 2013 (2013), Article ID 716435, 9 pages

http://dx.doi.org/10.1155/2013/716435

## Characterizations of Hemirings Based on Probability Spaces

Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei 445000, China

Received 24 March 2013; Accepted 8 May 2013

Academic Editor: Qiankun Song

Copyright © 2013 Bin Yu and Jianming Zhan. 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.

#### Abstract

The notion of falling fuzzy -ideals of a hemiring is introduced on the basis of the theory of falling shadows and fuzzy sets. Then the relations between fuzzy -ideals and falling fuzzy -ideals are described. In particular, by means of falling fuzzy -ideals, the charac-terizations of -hemiregular hemirings are investigated based on independent (prefect positive correlation) probability spaces.

#### 1. Introduction

Starting from a unified treatment of uncertainty by combining probability and fuzzy set theory [1], Goodman [2] put forward the equivalence between a fuzzy set and a class of random sets. Falling shadow representation theory was established based on the collection of Wang and Sanchez [3], which is directly related to the concept of probabilistic fuzzy set membership function. The theory shows the selection methods related to the joint degrees distributions. It provides a reasonable and convenient approach for the theoretical development and the practical applications of fuzzy sets and fuzzy logics. Utilizing the theory of falling shadows, in particular, Tan et al. [4, 5] established a theoretical approach to define a fuzzy inference relation and fuzzy set operations based on the theory of falling shadows. Yuan and Lee [6] considered a fuzzy subgroup (subring, ideal) as the falling shadow of a cloud of the subgroups (subrings, ideals). Jun and Kang [7] proposed a theoretical approach for BCK algebras.

A semiring plays an important role in studying matrices and determinants. Many aspects of the theory of matrices and determinants over semiring have been studied by Beasley and Pullman [8] and Ghosh [9]. The ideals in semiring are useful for many purposes, but they do not coincide with the usual ring ideals if is ring in general. Their use is thus somewhat limited in terms of obtaining analogues of ring theorems for semirings. In fact, many results in ring apparently have no analogues in hemirings using only ideals. LaTorre [10] investigated -ideals and -ideals in hemirings in an effort to obtain analogues of familiar ring theorems. The fuzzy theory in semirings and hemirings has been discussed by many researchers (see [11–16]). The concept of -hemiregular hemirings has been introduced by Zhan and Dudek [17] to generalize the regularity in hemirings. Further, some characterizations of -semisimple and -*intra*-hemiregular hemirings were investigated by Yin et al. [18, 19]. It is pointed out that some generalized fuzzy -ideals of hemirings were investigated by Ma et al.; for example, see [18–25].

Recently, some properties of falling fuzzy ideals of hemirings have also been investigated by Yu and Zhan [26]. As a continuation of our previous investigation of falling fuzzy ideals of hemirings, the present paper is organized as follows. In Section 2, we recall the concepts and properties of hemirings, fuzzy sets, and falling shadows. In Section 3, we introduce the concept of falling fuzzy -ideals and investigated some related properties. Finally, we investigate characterizations of -hemiregular hemirings based on independent (prefect positive correlation) probability spaces in Section 4.

#### 2. Preliminaries

A semiring is an algebraic system consisting of a nonempty set together with two binary operations on called addition and multiplication (denoted in the usual manner) such that and are semigroups and the following distributive laws: are satisfied for all .

By zero of a semiring we mean an element such that and for all . A semiring with zero and a commutative semigroup are called a hemiring. For the sake of simplicity, we shall write for .

A subset in a hemiring is called a left (right) ideal of if is closed under addition and (). Further, is called an ideal of if it is both a left ideal and a right ideal of . A left -ideal of hemiring is defined to be a left ideal of , such that, for all , and, for all , .

The -closure of in a hemiring is defined as: .

*Definition 1 (see [25]). *A fuzzy set of a hemiring is called a fuzzy left (right) -ideal if, for all , we have);
();
().

Further, is called a fuzzy -ideal of if it is both a fuzzy left -ideal and a fuzzy right -ideal of .

Note that if is a fuzzy -ideal, then for all .

For any , we denote the characteristic function of by .

Theorem 2 (see [17]). *A fuzzy set of is a fuzzy -ideal of if and only if the nonempty subset is an -ideal of for all . *

It is well known that ideals theory plays a fundamental role in the development of hemirings. Throughout this paper, is a hemiring.

We now display the basic theory on falling shadows. We refer the reader to the papers [2–5] for further information regarding 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 [0,1] and is the usual Lebesgue measure. Let be a fuzzy set in and be a -cut of . Then is a random set and is a cloud of . We shall call defined above the cut cloud of (see [2]).

#### 3. Falling Fuzzy -Ideals

In this section, we will introduce the notion of falling fuzzy -ideals of a hemiring. The relations between fuzzy -ideals and falling fuzzy -ideals are provided.

*Definition 3 (see [26]). *Let be a probability space, and let be a random set. If is a left (right) ideal of for any , then the falling shadow of the random set , that is, , is called a falling fuzzy left (right) ideal of . Further, is called a falling fuzzy ideal of if it is both a falling fuzzy left ideal and a falling fuzzy right ideal of .

Let be a probability space and , where is a hemiring.

Define two operations and on by for all .

Let be defined by , for all . Then we can check that is a hemiring.

For any subset of and , let , and then .

*Definition 4. *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 .

Proposition 5. *If is an -ideal of , then is an -ideal of . *

* Proof. *Assume that is an -ideal of and . Let be such that , and then . Since is an -ideal of , then . Thus, , and so . Let and , and then . Since is an -ideal of , then , and so , that is, . Similarly, we can prove .

Let and , and then . Hence . Since is an -ideal of , we have , that is, . Hence, is an -ideal of .

From the above proposition, we know that is a falling fuzzy ideal of , where . In fact, since we see that is a random set on . By Proposition 5, we know that is a falling fuzzy -ideal of .

*Example 6. * (1) Let be a set with an addition operation and a multiplication operation as follows:

Then is a hemiring [23].

Let and be defined by

Then is an -ideal of for all . Hence is a falling fuzzy -ideal of .

(2) The set with the following Cayley tables:
Then is a hemiring.

Let and be defined by

Then is an -ideal of for all . Hence is a falling fuzzy -ideal of .

Theorem 7. * Every fuzzy -ideal of is a falling fuzzy -ideal of .*

* Proof. *Consider the probability space , where is a Bored field on [0,1], and is the usual Lebesgue measure. Let be a fuzzy -ideal of , and then is an -ideal of for all . Let be a random set and for every . Then is a falling fuzzy -ideal of .

*Remark 8. *The following example shows that the converse of Theorem 7 is not valid.

*Example 9. *Let be a set with an addition operation and a multiplication operation as follows:
Then is a hemiring.

Let and be defined by Then is an -ideal of for all . Hence is a falling fuzzy -ideal of , and it is represented as follows: Then If , then is not an -ideal of since . Thus, it follows from Theorem 2 that is not a fuzzy -ideal of .

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

Theorem 10. * If a falling shadow of a random set is a falling fuzzy left (right) -ideal of , then, for all , we have*(1)*;
*(2)*;*(3)*.*

* Proof. *(1) Let , then . Since is a left (right) -ideal of by Definition 4, then , and so . This implies that .

(2) Let , and then . Since is a left -ideal of by Definition 4, then , and so . This implies that . Similarly, we can show that .

(3) Let and , and then . Since is a left (right) -ideal of by Definition 4, then , and so . This implies that . This completes the proof.

Theorem 11. *Let be a falling fuzzy -ideal of , and then .*

* Proof. *Since , it follows that

#### 4. Characterizations of -Hemiregular Hemirings

The concept of -hemiregularity of a hemiring was first introduced by Zhan and Dudek [17] as a generalization of the concept of regularity of a ring.

*Definition 12 (see [25]). *A hemiring is said to be -hemiregular if, for each , there exist such that .

Lemma 13 (see [17]). *If and are, respectively, a right -ideal and a left -ideal of a hemiring , then . *

Lemma 14 (see [17]). * A hemiring is hemiregular if and only if, for any right -ideal and for any left -ideal , . *

In the following sections, we divide the results into two parts. In Sections 4.1 and 4.2, we describe the characterizations of -hemiregular hemirings based on prefect positive correlation and independent probability spaces via falling fuzzy -ideals, respectively.

##### 4.1. Prefect Positive Correlation Probability Spaces

In this subsection, we describe the characterizations of -hemiregular hemirings based on prefect positive correlation probability spaces via falling fuzzy -ideals.

*Definition 15. *The probability space is called prefect positive correlation if or for all .

*Definition 16. *Let be a prefect positive correlation probability space and let and be falling fuzzy -ideals of . Then the product of and is defined by
and if cannot be expressed as .

Theorem 17. *If is a prefect positive correlation probability space and is a falling fuzzy left (right) -ideal of for all , then*(1)*;
*(2)*;
*(3)*. *

* Proof. *(1) By Definition 4, is a left -ideal of for any . Hence by Theorem 10, .

Thus, we have .

If , then . If , then , and so .

(2) By Definition 4, is a left -ideal of for any . Hence by Theorem 10, . Thus, we have

(3) By Definition 4, is a left -ideal of for any . Hence by Theorem 10, . Thus, we have .

If , then . If , then , and so .

Proposition 18 (see [5]). *If is a prefect positive correlation probability space, then .*

Proposition 19. * If is a prefect positive correlation probability space and and are two falling fuzzy left (right) -ideals of , then is a falling fuzzy left (right) -ideal of .*

* Proof. *We only consider the case of left -ideals, and the proof of right -ideals is similar:

(i)

(ii)

(iii)

Theorem 20. * If a falling fuzzy set of is a falling fuzzy left (right) -ideal of , then . *

* Proof. *We only consider the case of left -ideals, and the proof of right -ideals is similar. It is sufficient to show that the condition is satisfied. Let . If , it is clear that . Otherwise, there exist such that . Then we have

This implies that .

Theorem 21. *Let be a hemiring and . Then .*

* Proof. *Let . If , then and for some and . Thus we have
and so .

If , then . If possible, let . Then

Hence, there exist such that and , that is, , hence , and , which contradicts with . Thus we have .

In any case, we have . This completes the proof.

Theorem 22. *A hemiring is -hemiregular if and only if for any falling fuzzy right -ideal and any falling fuzzy left -ideal of we have .*

* Proof. *Let be an -hemiregular hemiring, and be any falling fuzzy right -ideal and any falling fuzzy left -ideal of , respectively. Then by Theorem 21, we have and . Thus . To show the converse inclusion, let be any element of . Since is -hemiregular, there exist such that . Then we have

This implies that . Therefore, .

Conversely, let and be any right -ideal and any left -ideal of , respectively. Then by Definition 4, and are any falling right -ideal and any falling left fuzzy -ideal of , respectively. The characteristic functions and are a fuzzy right -ideal and a fuzzy left -ideal of , respectively. Now, by Theorem 21, we have

It follows from Theorem 21 that . Thus, we have

Therefore is -hemiregular by Lemma 14.

##### 4.2. Independent Probability Spaces

In this subsection, we describe the characterizations of -hemiregular hemirings based on independent probability spaces via falling fuzzy -ideals.

*Definition 23. *The probability space is called independent if for all .

*Definition 24. *If is an independent probability space and let and be falling fuzzy -ideals of . Then the -product of and is defined by

Theorem 25. * If is an independent probability space and is a falling fuzzy left (right) -ideal of for all , then*(1)*;
*(2)*;
*(3)*. *

* Proof. *(1) By Definition 4, is a left -ideal of for any . Hence by Theorem 10, .

Thus, we have

(2) By Definition 4, is a left -ideal of for any . Hence by Theorem 10, . Thus, we have

(3) By Definition 4, is a left -ideal of for any . Hence by Theorem 10, . Thus, we have

Proposition 26 (see [5]). *If is an independent probability space, then .*

Proposition 27. *If is an independent probability space and is a falling fuzzy -ideal of , then .*

* Proof. *Since , it follows that

Proposition 28. *Let be an independent probability space and and be two falling fuzzy left (right) -ideals of , and then is a falling fuzzy left (right) -ideal of . *

* Proof. *We only consider the case of left -ideals, and the proof of right -ideals is similar:

(i)

(ii)

(iii)

Theorem 29. *If a falling fuzzy set of is a falling fuzzy left (right) -ideal of , then . *

* Proof. *We only consider the case of left -ideals, and the proof of right -ideals is similar. It is sufficient to show that the condition is satisfied. Let . If , it is clear that . Otherwise, there exist such that . Then we have

This implies that .

Theorem 30. *Let be a hemiring and , and then . *

* Proof. *Let . If , then and for some , and . Thus we have
and so .

If , then . If possible, let . Then

Hence, there exist such that and , that is, ; hence, , and , which contradicts with . Thus we have .

In any case, we have . This completes the proof.

Theorem 31. *A hemiring is -hemiregular if and only if for any falling fuzzy right -ideal and any falling fuzzy left -ideal of , . *

* Proof. *(1) Let be an -hemiregular hemiring, any falling fuzzy right -ideal, and any falling fuzzy left -ideal of , respectively. Then by Theorem 29, we have and . Thus . To show the converse inclusion, let be any element of . Since is -hemiregular, there exist such that . Then we have

This implies that . Therefore, we have .

Conversely, let and be any right -ideal and any left -ideal of , respectively. Then by Definition 4, and are a falling right -ideal and a falling left fuzzy -ideal of , respectively. The characteristic functions and of and are a fuzzy right -ideal and a fuzzy left -ideal of , respectively. Now, by Theorem 30, we have

It follows from Theorem 30 that . So .

Therefore, is -hemiregular by Lemma 14.

#### 5. Conclusions

In this paper, we introduce the notion of falling fuzzy -ideals of a hemiring. Then we investigate some characteristics of -hemiregular by means of falling fuzzy -ideals based on independent (prefect positive correlation) probability spaces. In future work, one can consider -hemiregular using falling fuzzy --ideals and falling fuzzy -*quasi*-ideals. One also can apply fuzzy inference relations to hemirings. Further, one can investigate this theory to information sciences and intelligent and fuzzy systems.

#### Acknowledgments

The authors are extremely grateful to the referees and Editor: Professor Qiankun Song for giving them many valuable comments and helpful suggestions which helped them improve the presentation of this paper. This research is partially supported by a Grant of National Natural Science Foundation of China, (61175055), Innovation Term of Higher Education of Hubei Province, China, (T201109), Natural Science Foundation of Hubei Province (2012FFB01101), and Natural Science Foundation of Education Committee of Hubei Province (D20131903).

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