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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 915217, 10 pages
http://dx.doi.org/10.1155/2013/915217
Research Article

On Rough Hyperideals in Hyperlattices

1Department of Mathematics, Northwest University, Xi'an 710127, China
2Department of Mathematics, Hubei University for Nationalities, Enshi 445000, China

Received 30 March 2013; Revised 28 June 2013; Accepted 29 June 2013

Academic Editor: Hak-Keung Lam

Copyright © 2013 Pengfei He et al. 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

We introduce and study rough hyperideals in hyperlattices. First, we give some interesting examples of hyperlattices and introduce hyperideals of hyperlattices. Then, applying the notion of rough sets to hyperlattices, we introduce rough hyperideals in hyperlattices, which are extended notions of hyperideals of hyperlattices. In addition, we consider rough hyperideals in Cartesian products and quotients of hyperlattices. Finally, we investigate some properties about homomorphic images of rough hyperideals in hyperlattices.

1. Introduction

In applied mathematics, we encounter many examples of mathematical objects that can be added to each other and multiplied by scalar numbers. First of all, the real numbers themselves are such objects. Other examples are real-valued functions, the complex numbers, infinite series, vectors in -dimensional space, and vector valued functions. Sometimes the sum of two elements is not an element. There are many examples in chemistry where the sum of two elements is a set of elements. In this case we have a hyperstructure. The concept of hyperstructure was introduced in 1934 by a French mathematician, Marty [1]. Algebraic hyperstructures are suitable generalizations of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Since then, there appeared many components of hyperalgebras such as hypergroups in [2] and hyperrings in [3]. Moreover, Konstantinidou and Mittas introduced the concept of hyperlattices in [4] and superlattices in [5]; also see [68]. In particular, Rasouli and Davvaz further studied the theory of hyperlattices and obtained some interesting results [9, 10], which enriched the theory of hyperlattices.

Recently, a number of different hyperstructures are widely studied from the theoretical point of view and for their applications to many subjects of pure and applied mathematics by many mathematicians. Also, a recent book [11] contains a wealth of applications on geometry, binary relations, lattices, fuzzy sets and rough sets, automata, combinatorics, codes, artificial intelligence, and probabilistic. Another book [12] is devoted especially to the study of hyperring theory, written by Davvaz and Leoreanu-Fotea. Several kinds of hyperrings are introduced and analyzed. The volume ends with an outline of applications in chemistry and physics, analyzing several special kinds of hyperstructures: -hyperstructures and transposition hypergroups. The theory of suitable modified hyperstructures can serve as a mathematical background in the field of quantum communication systems.

The theory of rough sets was introduced by Pawlak [13] to deal with uncertain knowledge in information systems. It is an expanding research area which stimulates explorations on both real-world applications and on the theory itself. Rough set theory is an extension of set theory, in which a subset of a universe is described by a pair of ordinary sets called the lower and upper approximations. It is a natural question to ask what happens if we substitute an algebraic system with the universe set. Some authors studied algebraic properties of rough sets. Since Biswas and Nanda [14] applied the notion of rough sets to algebra and introduced the notion of rough subgroups, Davvaz et al. have been engaged in extending concepts and methods of rough set theory to various algebraic structures [1521]. With the development of the hyperstructure theory, Leoreanu-Fotea et al. attached importance to the connections among rough sets, fuzzy sets, and algebraic hyperstructures in [2230]. These not only enriched the theory of rough sets but also provided new ideas in the study of pure algebra and algebraic hyperstructures.

The combination of rough set theory and algebraic systems may provide more new interesting research topics, which have drawn attention of many mathematicians and computer scientists. One can introduce roughness into an algebraic system and investigate algebraic properties of various rough objects. In this paper, in order to broaden application fields of the theory of rough sets and hyperstructures, we introduce the rough set theory into hyperlattices. We introduce rough hyperideals in hyperlattices, which are extended notions of hyperideals in hyperlattices. And we study some properties about rough hyperideals in hyperlattices.

2. Hyperideals in Hyperlattices

In this section, we recall the notion of hyperlattices and give several new examples of it. Moreover, we will introduce hyperideals in hyperlattices and discuss some basic properties of them, which will be used in the following paragraphs.

Let be a nonempty set, and let be the set of all nonempty subsets of . A hyperoperation on is a map , which associates a nonempty subset with any pair of elements of . The couple is called a hypergroupoid.

If and are nonempty subsets of , for all , we denote    , ;    .

In what follows, let us see what a hyperlattice is.

There are several kinds of hyperlattices that can be defined on a nonempty set; see [48]. Throughout the paper, we shall consider one of general types of hyperlattices [8]; also see [4].

Definition 1 (see [8]). Let be a nonempty set endowed with two hyperoperations and . The triple is called a hyperlattice if the following conditions hold: for all ,    , ;    , ;    , ;    , .

Let be a lattice. Define hyperoperations “ ” and “ ” on as follows: for all , , , then is a hyperlattice. From this, we can see that hyperlattices are suitable generalizations of lattices.

Now, we give some new examples of hyperlattices. From these examples, we can see that hyperlattices are connected to several domains of mathematics.

Example 2. Let be a partially ordered set. Define the following hyperoperations on : for all , , . Then is a hyperlattice.

Example 3. Let be the set of all subspaces of -dimensional vectors space . Define hyperoperations and on as follows: for all , , , where represents the sum space of and . One can check that is a hyperlattice.

Example 4. Let be a lattice. Define the following hyperoperations on : for all , , . Then is a hyperlattice.

Example 5. Let be a lattice. We define two hyperoperations on : for all , , . Then is also a hyperlattice.

Example 6. Let be the set of all positive integers. We define hyperoperations and on as follows: for all , , , in which represents that divides , and . We can check that is a hyperlattice.

Definition 7 (see [8]). Let be a hyperlattice. A nonempty subset of is called a subhyperlattice of if is itself a hyperlattice.

It is easy to see that a nonempty subset of is a subhyperlattice of if and only if holds: for all , , . That is to say, is a subhyperlattice of if and only if , .

Example 8. Let be a nonempty set. Define hyperoperations on as follows: for all , , . Then is a hyperlattice. Each nonempty subset of is a subhyperlattice of .

Example 9 (see [8]). Let , and let hyperoperations and on be defined as follows:

915217.tab.001

915217.tab.002

Then is a hyperlattice, in which and are subhyperlattices.

In what follows, we introduce hyperideals of a hyperlattice.

Definition 10. Let be a hyperlattice, and let be a nonempty subset of .    is called a -hyperideal of if for all and , (i)   , (ii)   .    is called a -hyperideal of if for all and , (i)   ,(ii)   .

Obviously, a subhyperlattice of is a -hyperideal of if and only if . Similarly, a subhyperlattice of is a -hyperideal of if and only if .

Now, we present some examples of -hyperideals and -hyperideals of hyperlattices.

Example 11. Let be a lattice. Define the hyperoperations and on as follows: for all , , , then is a hyperlattice. Every ideal and filter of the lattice are -hyperideal and -hyperideal of the hyperlattice , respectively.

From the previous example, we can see that -hyperideals and -hyperideals of hyperlattices are suitable generalizations of ideals and filters of lattices, respectively.

Example 12. Let be a hyperlattice in Example 4. For any element of the lattice , denote the principal ideal generated by of the lattice by , which means that , then it is easy to check that is a -hyperideal of the hyperlattice .

Example 13. Let be a hyperlattice in Example 5. For any element of the lattice , denote the principal filter generated by of the lattice by , which means that , then is a -hyperideal of the hyperlattice .

Example 14. Let be the hyperlattice in Example 9. One can check that is a -hyperideal, but not a -hyperideal of and is a -hyperideal, but not a -hyperideal of .

Next, we discuss some basic properties of hyperideals, which will be used in the following paragraphs.

Proposition 15. Let be a hyperlattice, and let be a nonempty subset of . Then the following conditions are equivalent.    is a -hyperideal of .    and for all and .    and .
Similarly, the following conditions are equivalent.    is a -hyperideal of .    and for all and .    and .

Proof. It is obvious.

Let and be two hyperlattices. Define hyperoperations on the Cartesian product as follows: for all , , . One can check that is a hyperlattice, which is called the Cartesian product hyperlattice of and .

Proposition 16. Let and be two nonempty subsets of and , respectively.   If and are subhyperlattices of and , respectively, then is a subhyperlattice of .   If and are -hyperideals ( -hyperideals) and -hyperideals ( -hyperideals) of and , respectively, then is a -hyperideal ( -hyperideal) of .

Proof. The proof is straightforward.

Let and be two hyperlattices. A map is called a weak homomorphism if and for all . In particular, if and , then is called a homomorphism.

If such a homomorphism is surjective, injective, or bijective, then is called an epimorphism, a monomorphism, or an isomorphism from to , respectively.

Proposition 17. Let be a surjective homomorphism from a hyperlattice to a hyperlattice . If is a -hyperideal ( -hyperideal) of , then is a -hyperideal ( -hyperideal) of . If is a -hyperideal ( -hyperideal) of , then is a -hyperideal ( -hyperideal) of .

Proof. Assume that is a -hyperideal of . For all , there exist such that , . Then . It follows from that . Now, let ; notice that is surjective, then there exists such that . Hence, . By , we have . Therefore, is a -hyperideal of .
Suppose that is a -hyperideal of . For all , then . It follows that . On the other hand, let , then ; that is, . Therefore, is a -hyperideal of .

3. Rough Hyperideals in Hyperlattices

In this section, we introduce the notion of rough hyperideals in hyperlattices and discuss some properties of them.

Given a hyperlattice , by we will denote the set of all nonempty subsets of . If is an equivalence relation on , then, for every , stands for the equivalence class of with the represent . For any nonempty subset of , we denote .

For any , we denote if the following conditions hold:  for all , such that ;  for all , such that .

Now, we can introduce the notion of hypercongruences on hyperlattices in the following manner.

Definition 18. Let be a hyperlattice. An equivalence relation on is called a hypercongruence on if for all , the following implication holds: and imply and .

Obviously, an equivalence relation on is a hypercongruence if and only if for all , we have that implies and .

Lemma 19. Let be a hyperlattice, and let be a hypercongruence on . For all , then , .

Proof. Suppose that , then there exist and such that . Since , , by Definition 18, we have . So implies that there exists such that . Therefore, we have , which implies . Similarly, we can prove that .

A hypercongruence relation on is called -complete if for all . Similarly, is called -complete if for all . We call   complete if it is both -complete and -complete.

Now, we briefly recall the rough set theory in Pawlak's sense. Let be an equivalence relation on , and let be a nonempty subset of . Then, the sets and are called, respectively, the upper and lower approximations of with respect to . is called a rough set with respect to .

Proposition 20. Let be a hypercongruence on a hyperlattice . If are two nonempty subsets of , then    . In particular, if is -complete, then ,    . In particular, if is -complete, then .

Proof. Suppose that . There exist and such that . It follows that there exist such that and . Since is a hypercongruence on , we have by Lemma 19. On the other hand, since , we obtain , which implies . Therefore, .
If is -complete, let , then . Therefore, there exists , and so for some and , we have . Since is -complete, we can obtain . Thus, there exist and such that . It follows that and . Hence, and , and we have , which implies . Therefore, .
The proof is similar to that of .

Proposition 21. Let be a hypercongruence on a hyperlattice , and are two nonempty subsets of .   If and are two -hyperideals of , then .   If and are two -hyperideals of , then .

Proof. Let , then there exist and such that , which implies that there exists such that . Since is a -hyperideal of , we have . It follows that . Hence, we obtain that , which implies . In a similar way, we have . Thus, . Combining Proposition 20, we have .
The proof is similar to that of .

Proposition 22. Let be a hypercongruence relation on a hyperlattice , and let , be two nonempty subsets of .   If is -complete, then .   If is -complete, then .

Proof. Let , then there exist and such that . It follows that and . Since is -complete, we have , which implies . Therefore, .
Similar to the proof of .

The following example shows that the converses of Proposition 22 do not hold in general.

Example 23. Let be a lattice , where the partial order relation on is defined as shown in Figure 1. For all , , , then is a hyperlattice. Let be a complete hypercongruence relation on the hyperlattice with the following equivalence classes: . If , , then and . We have , . So, . If the partial order is reverse, then , . Therefore, .

915217.fig.001
Figure 1: The lattice in Example 23.

Proposition 24. Let be a hypercongruence relation on a hyperlattice . If and are -hyperideals ( -hyperideals) of , then .

Proof. Let , we have and . Then, there exist and such that and . It follows from which is a hypercongruence relation that , which implies that there exists such that . Since and are -hyperideals of , we have . So, . It follows that , which implies . Hence, . On the other hand, it is clear that . Therefore, . In a similar way, if and are -hyperideals of , we can also obtain .

Up to now, we have studied some properties of the lower and upper approximations in hyperlattices. Next, we will introduce and investigate a new algebraic structure called rough hyperideals in hyperlattices. Let us begin with introducing the following definitions.

Definition 25. Let be a hypercongruence on a hyperlattice , and let be a nonempty subset of . is called a lower (an upper) rough subhyperlattice of if ( ) is a subhyperlattice of . is called a rough subhyperlattice of if is both a lower rough subhyperlattice and an upper rough subhyperlattice of .
Similarly, is called a lower (an upper) rough -hyperideal of if ( ) is a -hyperideal of . And we call a rough -hyperideal of if is both a lower rough -hyperideal and an upper rough -hyperideal of . In a similar way, a rough -hyperideal of can be defined.

Example 26. Let and hyperoperations and on defined as follows:

915217.tab.003

915217.tab.004

Then, is a hyperlattice. Let be a complete hypercongruence relation on the hyperlattice with the following equivalence classes: , . Now for , and . It is clear that and are -hyperideals, so is a rough -hyperideal of .

Example 27. Let be the hyperlattice in Example 9. Let be a hypercongruence relation on the hyperlattice with the following equivalence classes: , . Considering , we can obtain that     and   . Notice that and are -hyperideals, so is a rough -hyperideal of . If , we have that and . From Example 14, we obtain that and are -hyperideals, so is a rough -hyperideal of .

Theorem 28. Let be a hypercongruence on a hyperlattice , and let be a nonempty subset of .   If is a subhyperlattice of , then is an upper rough subhyperlattice of .   If is a -hyperideal ( -hyperideal) of , then is an upper rough -hyperideal ( -hyperideal) of .

Proof. Suppose that , then and . It follows that there exist and . Since is a subhyperlattice of , we have . Also, by Lemma 19, we can obtain . Hence, , which implies that . In a similar way, we have . Therefore, is a subhyperlattice of ; that is, is an upper rough subhyperlattice of .
Let be a -hyperideal of ; then is a subhyperlattice of . By , we have . On the other hand, by Proposition 20, we have . Thus, is a -hyperideal of . Therefore, is an upper rough -hyperideal of . The other case can be proved in a similar way.

Theorem 29. Let be a nonempty subset of , and let be a complete hypercongruence relation on such that .   If is a subhyperlattice of , then is a lower rough subhyperlattice of .   If is a -hyperideal ( -hyperideal) of , then is a lower rough -hyperideal ( -hyperideal) of .

Proof. Let be a subhyperlattice of . Since , it follows from Proposition 22 that and . Therefore, is a subhyperlattice of ; that is, is a lower rough subhyperlattice of .
Assume that is a -hyperideal of ; then is a subhyperlattice of . Notice that is complete; by the statement of , we obtain . On the other hand, by Proposition 22, we have . Thus, is a -hyperideal of . Therefore, is a lower rough -hyperideal of . In a similar way, we can prove that is a lower rough -hyperideal of .

By the two theorems above, we have immediately the following corollary.

Corollary 30. Let be a nonempty subset of , and let be a complete hypercongruence relation on such that .   If is a subhyperlattice of , then is a rough subhyperlattice of . If is a -hyperideal ( -hyperideal) of , then is a rough -hyperideal ( -hyperideal) of .

The above corollary shows that under some conditions -hyperideals ( -hyperideals) are rough -hyperideals ( -hyperideals) in hyperlattices. The following example shows that the converse of this result does not hold in general.

Example 31. In Example 26, is a rough -hyperideal of , but is not a -hyperideal of .

Example 32. In Example 27, is a rough -hyperideal of , but is not a -hyperideal of .

Based on the discussion above, we obtain that rough hyperideals are extended notions of hyperideals in hyperlattices.

4. Rough Hyperideals in the Product Hyperlattices and Quotient Hyperlattices

In this section, we consider rough hyperideals in Cartesian products and quotients of hyperlattices. Let us begin with introducing the following proposition.

Let and be two hypercongruence relations on and , respectively. Define a relation on as follows: for all , and . It is easy to check that is a hypercongruence on the product hyperlattice . Then we can obtain the following proposition.

Proposition 33. Let and be two nonempty subsets of and , respectively. Then,    .    .

Proof. For all , such that and , . It follows that .
For all , , for all  and . We conclude that .

Theorem 34. Let and be hypercongruence relations on and , respectively. If and are two nonempty subsets of and , respectively, then    is a rough subhyperlattice of   if and only if and are rough subhyperlattices of and , respectively.    is a rough -hyperideal ( -hyperideal) of if and only if and are rough -hyperideals ( -hyperideals) and rough -hyperideals ( -hyperideals) of and , respectively.

Proof. Assume that is an upper rough subhyperlattice of . Let , it follows from Proposition 33 that . Since is a subhyperlattice of , we have , . It follows that , , , , which implies that and are subhyperlattices of and , respectively. Therefore, and are upper rough subhyperlattices of and , respectively. The case of the lower approximation can be seen in a similar way.
This follows from Propositions 16 and 33.
Assume that is an upper rough -hyperideal of . Let ; it follows from that , . Now, for all , since is a -hyperideal of . Hence, , , which implies that and are -hyperideals and -hyperideals of and , respectively. Similarly, the case of the lower approximation can be proved.
This follows from Propositions 16 and 33.

Let be a hypercongruence relation on . For all , we define and . Then, one can check is a hyperlattice, which is called the quotient hyperlattice of with respect to .

When is finite, is smaller than , and its structure is usually less complicated than that of . At the same time, simulates in many ways. In fact, we may think of a quotient hyperlattice of as a less complicated approximation of .

The lower and upper approximations can be presented in an equivalent form as shown bellow.

Let be a hypercongruence relation on , and let be a nonempty subset of . Denote , .

Theorem 35. Let be a hypercongruence relation on , and let be a nonempty subset of . Then,    is a subhyperlattice of if and only if is a subhyperlattice of .    is a -hyperideal ( -hyperideal) of if and only if is a -hyperideal ( -hyperideal) of .

Proof. Let ; then and . This implies that . Since is a subhyperlattice of , we have , . Then, for all , . Thus, . Similarly, . Therefore, is a subhyperlattice of .
Assume that is a subhyperlattice of . Let ; then and . This implies that . Since is a subhyperlattice of , we conclude that , which implies that for all , ; that is, . Hence . In a similar way, we have . Therefore, is a subhyperlattice of .
Assume that is a -hyperideal of . Let ; it follows from the necessity of that . Now, for every and , then . This implies that . Since is a -hyperideal of , we have . It follows that . Then, for every , , which implies that . Therefore, is a -hyperideal of . The other case can be seen in a similar way.
Suppose that is a -hyperideal of . Let ; it follows from the sufficiency of that . Now, let ; then and . Hence, we conclude that , which implies that for all , ; that is, . Hence . Therefore, is a -hyperideal of . In a similar way, the other case can be seen.

Combining Theorems 28 and 35, we have the following corollary.

Corollary 36. Let be a hypercongruence relation on , and let be a nonempty subset of .   If is a subhyperlattice of , then is a subhyperlattice of .   If is a -hyperideal ( -hyperideal) of , then is a -hyperideal ( -hyperideal) of .

Theorem 37. Let be a nonempty subset of , and let be a hypercongruence relation on such that .    is a subhyperlattice of if and only if is a subhyperlattice of .    is a -hyperideal ( -hyperideal) of if and only if is a -hyperideal ( -hyperideal) of .

Proof. Let , then and . That is, and . Since is a subhyperlattice of , we have , . It follows that , . Thus, , . Therefore, is a subhyperlattice of .
Assume that is a subhyperlattice of . Let ; then and . This implies that . Since is a subhyperlattice of , we infer that , which implies that for all , ; that is, . It follows that . Likewise, we have . Therefore, is a subhyperlattice of .
Assume that is a -hyperideal of . Let ; it follows from the necessity of that . Now, for every and , then . This implies that . Since is a -hyperideal of , we have . Then, for all , we have , which implies that . Hence, . On the other hand, from , we have . It follows that . Therefore, is a -hyperideal of . The other case can be seen in a similar way.
Assume that is a -hyperideal of . Let ; it follows from the sufficiency of that . Now, let ; then and . Hence, we conclude that , which implies that for all , ; that is, . Hence . Therefore, is a -hyperideal of . In a similar way, the other case can be seen.

Combining Theorems 29 and 37, we have the following corollary.

Corollary 38. Let be a nonempty subset of , and let be a complete hypercongruence relation on such that .   If is a subhyperlattice of , then is a subhyperlattice of .   If is a -hyperideal ( -hyperideal) of , then is a -hyperideal ( -hyperideal) of .

5. Homomorphic Images of Rough Hyperideals

In this section, we will discuss relations between the upper (lower) rough hyperideals of hyperlattices and the upper (lower) approximations of their homomorphic images. Finally, combining results in the previous sections, we obtain the corresponding relationships between rough hyperideals of quotient hyperlattices of two homomorphic hyperlattices.

Lemma 39. Let and be two hyperlattices, and let be a homomorphism from to . Then is a hypercongruence on , which is called the kernel of .

Proof. Clearly, is an equivalence relation on .
For all , let and ; then and . Let ; then , which implies that there exists such that . That is, there exists such that . Conversely, for any , there also exists such that . It follows that . Similarly, we can prove that . Therefore, is a hypercongruence on .

Theorem 40. Let be a homomorphism from the hyperlattice to the hyperlattice and . If is a nonempty subset of , then    .   if is one to one, .

Proof. Since , it follows that . Conversely, let ; there exists such that , so we have . Thus, there exists . So, ; that is, . Thus, . Therefore, .
It is obvious that . Let ; there exists such that . If , then . Since is one to one, we have . So , which implies . Thus, . Therefore, we obtain .

In order to discuss some relations between the upper (lower) rough hyperideals of hyperlattices and the upper (lower) approximations of their homomorphic images, we give the following lemma.

Lemma 41. Let be a surjective homomorphism from a hyperlattice to a hyperlattice , and let be a hypercongruence relation on . If is a nonempty subset of , then    is a hypercongruence relation on a hyperlattice .