Advances in Fuzzy Systems

Volume 2012 (2012), Article ID 595687, 9 pages

doi:10.1155/2012/595687

Research Article

Rough Fuzzy Hyperideals in Ternary Semihypergroups

Naveed Yaqoob,¹ Muhammad Aslam,¹ and Kostaq Hila²

¹Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

²Department of Mathematics and Computer Science, University of Gjirokastra, Gjirokastra, Albania

Received 29 April 2012; Accepted 25 June 2012

Academic Editor: M. Efe

Copyright © 2012 Naveed Yaqoob 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.

The relations between rough sets and algebraic systems have been already considered by many mathematicians, and rough sets have been studied in various kinds of algebraic systems. This paper concerns a relationship between rough sets and ternary semihypergroups. We introduce the notion of rough hyperideals and rough bi-hyperideals in ternary semihypergroups. We also study fuzzy, rough, and rough fuzzy ternary subsemihypergroups (left hyperideals, right hyperideals, lateral hyperideals, hyperideals, and bi-hyperideals) of ternary semihypergroups.

1. Introduction

The notion of a rough set was proposed by Pawlak [1] as a formal tool for modeling and processing incomplete information in information systems. Since then the subject has been investigated in many papers. The theory of rough sets is an extension of set theory, in which a subset of an universe is described by a pair of ordinary sets called the lower and upper approximations. A key notion in the Pawlak rough set model is the equivalence relation. The equivalence classes are the building blocks for the construction of the lower and upper approximations. The lower approximation of a given set is the union of all equivalence classes which are subsets of the set, and the upper approximation is the union of all equivalence classes which have a nonempty intersection with the set. It is a natural question to ask what happens if we substitute the universe set with an algebraic system. Some authors have studied the algebraic properties of rough sets. Aslam et al. [2] introduced the notion of roughness in left almost semigroups. Chinram [3], introduced rough prime ideals and rough fuzzy prime ideals in -semigroups. Petchkhaew and Chinram [4], introduced the notion of rough fuzzy ideals in ternary semigroups. In [5], Davvaz considered the relationship between rough sets and ring theory, considered a ring as a universal set, and introduced the notion of rough ideals and rough subrings with respect to the ideal of a ring. Also, rough modules have been investigated by Davvaz and Mahdavipour [6]. Davvaz et al. applied rough theory to -semihypergroups [7], hyperrings [8], and -semihyperrings [9]. Yaqoob [10] introduced the notion of rough -hyperideals in left almost -semihypergroups, also see [11, 12]. Kuroki, in [13], introduced the notion of a rough ideal in a semigroup. Jun applied the rough set theory to BCK-algebras [14].

Hyperstructure theory was introduced in 1934, when Marty [15] defined hypergroups, began to analyze their properties and applied them to groups. In the following decades and nowadays, 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. Nowadays, hyperstructures have a lot of applications to several domains of mathematics and computer science and they are studied in many countries of the world. 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. A lot of papers and several books have been written on hyperstructure theory, see [16–19]. A recent book on hyperstructures [16] points out on their applications in rough set theory, cryptography, codes, automata, probability, geometry, lattices, binary relations, graphs, and hypergraphs. Another book [18] is devoted especially to the study of hyperring theory. 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.

Hila and Naka [20–22] worked out on ternary semihypergroups and introduced some properties of hyperideals in ternary semihypergroups, also see [23].

The concept of a fuzzy set, introduced by Zadeh in his classic paper [24], provides a natural framework for generalizing some of the notions of classical algebraic structures. Fuzzy semigroups have been first considered by Kuroki [25]. After the introduction of the concept of fuzzy sets by Zadeh, several researches conducted the researches on the generalizations of the notions of fuzzy sets with huge applications in computer, logics, and many branches of pure and applied mathematics. Fuzzy set theory has been shown to be an useful tool to describe situations in which the data are imprecise or vague. Fuzzy sets handle such situations by attributing a degree to which a certain object belongs to a set. In 1971, Rosenfeld [26] defined the concept of fuzzy group. Since then many papers have been published in the field of fuzzy algebra. Recently fuzzy set theory has been well developed in the context of hyperalgebraic structure theory. A recent book [16] contains an wealth of applications. In [27], Davvaz introduced the concept of fuzzy hyperideals in a semihypergroup, also see [28, 29]. A several papers are written on fuzzy sets in several algebraic hyperstructures. The relationships between the fuzzy sets and algebraic hyperstructures have been considered by Corsini, Davvaz, Leoreanu, Zhan, Zahedi, Ameri, Cristea, and many other researchers. The notion of a rough set has often been compared to that of a fuzzy set, sometimes with a view to prove that one is more general, or, more useful than the other. Several researchers were conducted on the generalizations of the notion of fuzzy sets and rough sets.

In this paper, the notion of rough subsemihypergroup (rough hyperideal, rough bi-hyperideal resp.) in ternary semihypergroups has been introduced which is a generalization of subsemihypergroup (hyperideal, bi-hyperideal resp.). We also study fuzzy, rough and rough fuzzy ternary subsemihypergroups (left hyperideal, right hyperideals, lateral hyperideals, hyperideals, and bi-hyperideals) of ternary semihypergroups.

2. Ternary Semihypergroups

In this section we will present some basic definitions of ternary semihypergroups.

A map is called hyperoperation or join operation on the set , where is a nonempty set and denotes the set of all nonempty subsets of .

A hypergroupoid is a set with together a (binary) hyperoperation.

Definition 1. A hypergroupoid , which is associative, that is, , for all , is called a semihypergroup.

Let and be two nonempty subsets of . Then, we define

Definition 2. A map is called ternary hyperoperation on the set , where is a nonempty set and denotes the set of all nonempty subsets of .

Definition 3. A ternary hypergroupoid is called the pair , where is a ternary hyperoperation on the set .

Definition 4. A ternary hypergroupoid is called a ternary semihypergroup if for all , we have

Definition 5. Let be a ternary semihypergroup. Then is called a ternary hypergroup if for all , there exist such that

Definition 6. Let be a ternary semihypergroup and a nonempty subset of . Then is called a subsemihypergroup of if and only if .

Definition 7. A nonempty subset of a ternary semihypergroup is called a left (right, lateral) hyperideal of if

Definition 8. A subsemihypergroup of a ternary semihypergroup is called a bi-hyperideal of if

Definition 9. Let be a ternary semihypergroup and a subset of . Then is called a quasi-hyperideal of if and only if

3. Rough Hyperideals in Ternary Semihypergroups

In what follows, let denote a ternary semihypergroup unless otherwise specified. In this section, for simplicity we write as and consider the ternary hyperoperation as “”. Suppose that is a ternary semihypergroup. A partition or classification of is a family of nonempty subsets of such that each element of is contained in exactly one element of .

Given a ternary semihypergroup , by we will denote the power-set of . Let and be two nonempty subsets of . We define if for every there exists such that and for every there exists such that . If is an equivalence relation on , then, for every , stands for the equivalence class of with the represent .

Definition 10. Let be a ternary semihypergroup. An equivalence relation on is called regular on if

for all .

A regular relation on is called complete if for all .

Lemma 11. Let be a ternary semihypergroup and be a regular relation on . If , then

Proof. Let . Then there exist , and such that . Since , and then by regularity of , we have

So implies that there exists such that , and therefore .

Let be a nonempty subset of a ternary semihypergroup and be a regular relation on . Then, the sets

are called -upper and -lower approximations of , respectively. For a nonempty subset of is called a rough set with respect to if .

Theorem 12. Let and be regular relations on a ternary semihypergroup . If and are nonempty subsets of , then the following hold:

(1); (2); (3)(4) implies ;(5) implies ;(6); (7); (8) implies ;(9) implies .

Proof. The proof of this theorem is similar to [13, Theorem  2.1].

Theorem 13. Let be a regular relation on a ternary semihypergroup and let , and be nonempty subsets of . Then

(1) , (2)If is complete, then .

Proof. (1) Let . Then for , and . There exist such that , and . Since is regular, it follows that

On the other hand, since , we have

and so . This shows that .

(2) Let . Then for , and .

It follows that , and . Since is complete, we have

and so . Hence .

Definition 14. Let be a ternary semihypergroup. A nonempty subset of is called a subsemihypergroup of if .

Let be a regular relation on a ternary semihypergroup . Then a nonempty subset of is called a -upper (-lower) rough subsemihypergroup of if is a subsemihypergroup of .

Theorem 15. Let be a regular relation on ternary semihypergroup and let be a subsemihypergroup of . Then

(1) is a subsemihypergroup of ,(2)if is complete, then is, if it is nonempty, a subsemihypergroup of .

Proof. (1) Let be a subsemihypergroup of . Now by Theorem 13(1),

This shows that is a subsemihypergroup of , that is, is a -upper rough subsemihypergroup of .

(2) Let be a subsemihypergroup of . Now by Theorem 13(2)

This shows that is a subsemihypergroup of , that is, is a -lower rough subsemihypergroup of .

Definition 16. A nonempty subset of a ternary semihypergroup is called left (right, lateral) hyperideal of if , .

A nonempty subset of a ternary semihypergroup is called a hyperideal of if it is a left, right and lateral hyperideal of . A nonempty subset of a ternary semihypergroup is called two-sided hyperideal of if it is a left and right hyperideal of . A lateral hyperideal of a ternary semihypergroup is called a proper lateral hyperideal of if .

Let be a regular relation on a ternary semihypergroup . Then a nonempty subset of is called a -upper (-lower) rough left hyperideal of if    is a left hyperideal of . Similarly -upper (-lower) rough right and rough lateral hyperideals of can be defined.

Theorem 17. Let be a regular relation on a ternary semihypergroup and let be a left (right, lateral) hyperideal of . Then

(1) is a left (right, lateral) hyperideal of ,(2) if is complete, then is, if it is nonempty, a left (right, lateral) hyperideal of .

Proof. (1)  Let be a right hyperideal of . Now by Theorem 13(1),

This shows that is a right hyperideal of , that is, is a -upper rough right hyperideal of .

(2) Let be a right hyperideal of . Now by Theorem 13(2),

This shows that is a right hyperideal of , that is, is a -lower rough right hyperideal of . The case for left (lateral) hyperideal can be seen in a similar way.

Theorem 18. Let be a regular relation on a ternary semihypergroup . If , , and are a right hyperideal, a lateral hyperideal, and a left hyperideal of , respectively. Then

(1), (2).

Proof. Since is a right hyperideal of , so . Since is a lateral hyperideal of , so , also is a left hyperideal of , so , thus . Then by Theorem 12(7), we have

Also by Theorem 12(3), we have

This completes the proof.

Definition 19. A subsemihypergroup of a ternary semihypergroup is called a bi-hyperideal of if .

Let be a regular relation on a ternary semihypergroup . Then a subsemihypergroup of is called a -upper (-lower) rough bi-hyperideal of if is a bi-hyperideal of .

Theorem 20. Let be a regular relation on a ternary semihypergroup and let be a bi-hyperideal of . Then

(1) is a bi-hyperideal of ,(2)if is complete, then is, if it is nonempty, a bi-hyperideal of .

Proof. (1) Let be a bi-hyperideal of . Now by Theorem 13(1)

From this and Theorem 15(1), is a bi-hyperideal of , that is, is a -upper rough bi-hyperideal of .

(2) Let be a bi-hyperideal of . Now by Theorem 13(2)

From this and Theorem 15(2), is a bi-hyperideal of , that is, is a -lower rough bi-hyperideal of .

The following example shows that the converse of above theorem does not hold.

Example 21. Let and for all , where is defined by Table 1.

Then is a ternary semihypergroup. Let be a complete regular relation on such that -regular classes are the subsets , . Now for , and . It is clear that and are bi-hyperideals of , but the subsemihypergroup of is not a bi-hyperideal of .

Table 1

0
0
0
0
0
0
0
0

0

0

0

0

0

0

Definition 22. A subset of a ternary semihypergroup is called a quasi-hyperideal of if

Let be a regular relation on a ternary semihypergroup . Then a subset of is called a -upper (-lower) rough quasi-hyperideal of if is a quasi-hyperideal of .

Theorem 23. Let be a complete regular relation on a ternary semihypergroup and let be a bi-hyperideal of . Then is, if it is nonempty, a quasi- hyperideal of .

Proof. Let be a quasi-hyperideal of . Now by Theorems 13(2) and 12(3)

Also we can show that

Hence is a quasi-hyperideal of , that is, is a -lower rough quasi-hyperideal of .

4. Rough Hyperideals in the Quotient Ternary Semihypergroups

Let be a regular relation on a ternary semihypergroup . The -upper approximation and -lower approximation of a nonempty subset of can be presented in an equivalent form as shown below:

respectively.

Theorem 24. Let be a regular relation on a ternary semihypergroup and let be a subsemihypergroup of . Then

(1) is a subsemihypergroup of ,(2)if is complete, then is, if it is nonempty, a subsemihypergroup of .

Proof. (1) Let . Then , and . So there exist , and . Since is a subsemihypergroup of , we have . By Lemma 11, we have

Thus , which implies that . Hence is a subsemihypergroup of .

(2) Let . Then , and . Since is a subsemihypergroup of , we have

This means that is a subsemihypergroup of .

Theorem 25. Let be a regular relation on a ternary semihypergroup and let be a left (right, lateral) hyperideal of . Then

(1) is a left (right, lateral) hyperideal of ,(2)if is complete, then is, if it is nonempty, a left (right, lateral) hyperideal of .

Proof. (1) Let be a left hyperideal of . Let and . Then , hence . Since is a left hyperideal of , by Theorem 17(1), is a left hyperideal of . So, we have

Now, for every , we have . On the other hand, from , we obtain . Therefore . This means that is a left hyperideal of .

(2) Let be a left hyperideal of . Let and . Then, , which implies . Since is a left hyperideal of , by Theorem 17(2), is a left hyperideal of . Thus, we have

Now, for every , we have , which implies that . Hence, . On the other hand, from , we have . Therefore . This means that is, if it is nonempty, a left hyperideal of . The other cases can be seen in a similar way.

Theorem 26. Let be a regular relation on a ternary semihypergroup and let be a bi-hyperideal of . Then

(1) is a bi-hyperideal of ,(2)if is complete, then is, if it is nonempty, a bi-hyperideal of .

Proof. (1) Let be a bi-hyperideal of . Let and . Then,

Hence, , and . By Theorem 20(1), is a bi-hyperideal of . So, we have

Now, for every , we obtain

On the other hand, since , we have . Thus,

Therefore, from this and Theorem 24(1), is a bi-hyperideal of .

(2) Let be a bi-hyperideal of . Let and . Then,

Hence, , , and . By Theorem 20(2), is a bi-hyperideal of . So, we have

Then, for every , we obtain

On the other hand, since , we have . So,

Therefore from this and Theorem 24(2), is, if it is nonempty, a bi-hyperideal of .

5. Fuzzy Hyperideals of Ternary Semihypergroups

In this section we introduce and study fuzzy ternary subsemihypergroups, fuzzy left hyperideals, fuzzy right hyperideals, fuzzy lateral hyperideals, and fuzzy hyperideals of ternary semihypergroups.

Definition 27. Let be a ternary semihypergroup. A fuzzy subset of is called

(1)a fuzzy ternary subsemihypergroup of if for all ,(2)a fuzzy left hyperideal of if for all ,(3)a fuzzy right hyperideal of if for all ,(4)a fuzzy lateral hyperideal of if for all ,(5)a fuzzy hyperideal of if for all .

Theorem 28. Let be a ternary semihypergroup and a nonempty subset of . The following statements hold true.

(1) is a ternary subsemihypergroup of if and only if is a fuzzy ternary subsemihypergroup of .(2) is a left hyperideal (right hyperideal, lateral hyperideal, hyperideal) of if and only if is a fuzzy left hyperideal (fuzzy right hyperideal, fuzzy lateral hyperideal, fuzzy hyperideal) of .

Proof. (1) Let us assume that is a ternary subsemihypergroup of . Let .

Case 1.. Since is a ternary subsemihypergroup of , we have . Then .

Case 2. or or . Thus or or . Therefore .

Conversely, let . We have . Since is a fuzzy ternary subsemihypergroup of , . Hence .

(2) Let us assume that is a left hyperideal of . Let .

Case 1.. Since is a left hyperideal of , then . Then . Therefore .

Case 2.. We have . Hence .

Conversely, let and . Since is a fuzzy left hyperideal of and . Thus .

The remaing parts can be seen in similarly way.

Let be a ternary semihypergroup. A nonempty subset of is called prime subset of if for all implies or or . A ternary subsemihypergroup of is called prime ternary subsemihypergroup of if is a prime subset of . Prime left hyperideals, prime right hyperideals, prime lateral hyperideals, and prime hyperideals of are defined analogously. A fuzzy subset of is called a prime fuzzy subset of if for all . A fuzzy ternary subsemihypergroup of is called a prime fuzzy ternary subsemihypergroup of if is a prime fuzzy subset of . Prime fuzzy left hyperideals, prime fuzzy right hyperideals, prime fuzzy lateral hyperideals, and prime fuzzy hyperideals of are defined analogously.

Theorem 29. Let be a ternary semihypergroup and a nonempty subset of . The following statements hold true.

(1) is a prime subset of if and only if is a prime fuzzy subset of .(2) is a prime ternary subsemihypergroup (prime left hyperideal, prime right hyperideal, prime lateral hyperideal, prime hyperideal) of if and only if is a prime fuzzy ternary subsemihypergroup (prime fuzzy left hyperideal, prime fuzzy right hyperideal, prime fuzzy lateral hyperideal, prime fuzzy hyperideal) of .

Proof. (1) Let us assume that is a prime subset of . Let .

Case  1.. Since is prime, or or . Thus .

Case  2.. Thus .

Conversely, let such that . Thus for all . Since is prime, . This implies or or . Hence or or .

(2) It follows from (1) and Theorem 28.

Let be a fuzzy subset of a set (a ternary semihypergroup) . For any , the set

are called a t—level set and a t—stronglevel set of , respectively.

Theorem 30. Let be a fuzzy subset of a ternary semihypergroup . The following statements hold true:

(1) is a fuzzy ternary subsemihypergroup of if and only if for all , if , then is a ternary subsemihypergroup of .(2) is a fuzzy left hyperideal (fuzzy right hyperideal, fuzzy lateral hyperideal, fuzzy hyperideal) of if and only if for all , if , then is a left hyperideal (right hyperideal, lateral hyperideal, hyperideal) of .

Proof. (1) Let us assume that is a fuzzy ternary subsemihypergroup of . Let such that . Let . So . Thus . Since is a fuzzy ternary subsemihypergroup of , . Hence, . Conversely, let . Let we take . Then . Thus, . Since is a ternary subsemihypergroup of , . Thus .

(2) Let us assume that is a fuzzy left hyperideal of . Let . Let us suppose that . Let and . Thus . Therefore, .

Conversely, let . Let we take . Thus , this implies . By assumption, we have is a left hyperideal of . So . Therefore, . Thus .

The remain parts can be proved in a similar way.

Theorem 31. Let be a ternary semihypergroup and be a fuzzy of . The following statements hold true:

(1) is prime fuzzy subset of if and only if for all , if , then is a prime subset of .(2) is a prime fuzzy ternary subsemihypergroup (prime fuzzy left hyperideal, prime fuzzy right hyperideal, prime fuzzy lateral hyperideal, prime fuzzy hyperideal) of if and only if for all , if , then is a prime ternary subsemihypergroup (prime left hyperideal, prime right hyperideal, prime lateral hyperideal, prime hyperideal) of .

Proof. (1) Let us assume that is a prime fuzzy subset of . Let . Let us suppose that . Let such that . Thus . Since is prime, or or . This implies or or .

Conversely, let . Let we take . Then . Since is prime, or or . Then or or . Hence .

(2) It follows from (1) and Theorem 29.

Theorem 32. Let be a ternary semihypergroup and be a fuzzy subset of . Then is a prime fuzzy subset (prime fuzzy ternary subsemihypergroup, prime fuzzy left hyperideal, prime fuzzy right hyperideal, prime fuzzy lateral hyperideal, prime fuzzy hyperideal) of if and only if for all , if , then is a prime subset (prime ternary subsemihypergroup, prime left hyperideal, prime right hyperideal, prime lateral hyperideal, prime hyperideal) of .

Proof. The proof is similar to the proof of Theorem 30.

6. Rough Fuzzy Hyperideals of Ternary Semihypergroups

In this section we study rough fuzzy ternary semihypergroups, left hyperideals, right hyperideals, lateral hyperideals and hyperideals of ternary semihypergroups.

Let be a ternary semihypergroup and be a fuzzy subset of . Then the sets

are called the -upper and -lower approximations of a fuzzy set , respectively.

Lemma 33. Let be a ternary semihypergroup, a regular relation on , a fuzzy subset of and , then

(1),(2).

Proof. (1) Let . Then . So . Therefore, for all . This implies . Therefore, .

Conversely, let us assume that . Thus . Then for all . This implies . Thus, . Hence .

(2) Let . Then . So . Therefore, for some . This implies . Therefore .

Conversely, let us assume . Thus . Then for some . This implies . Thus . Hence, .

Theorem 34. Let be a ternary semihypergroup and be a regular relation on . If is a fuzzy ternary subsemihypergroup (fuzzy left hyperideal, fuzzy right hyperideal, fuzzy lateral hyperideal, fuzzy hyperideal) of , then and are fuzzy ternary subsemihypergroups (fuzzy left hyperideals, fuzzy right hyperideals, fuzzy lateral hyperideals, fuzzy hyperideals) of .

Proof. It can be obtained easily by Theorems 29, 31, 15, and 17 and Lemma 33.

7. Fuzzy Bi-Hyperideals of Ternary Semihypergroups

Let be a ternary semihypergroup. A fuzzy subset of is called a fuzzy bi-hyperideal of if and for all .

Theorem 35. Let be a ternary semihypergroup and a nonempty subset of . Then is a bi-hyperideal of if and only if is a fuzzy bi-hyperideal of .

Proof. Let us assume that is a bi-hyperideal of . Let .

Case  1.. Since is a bi-hyperideal of , then , . Therefore and .

Case  2. or or . Thus or or . Hence and .

Let be a ternary semihypergroup. A bi-hyperideal of is called a prime bi-hyperideal of if is a prime subset of . A fuzzy bi-hyperideal of is called a prime fuzzy bi-hyperideal of if is a prime fuzzy subset of .

Theorem 36. Let be a ternary semihypergroup and a nonempty subset of . Then is a prime bi-hyperideal of if and only if is a prime fuzzy bi-hyperideal of .

Proof. It follows from Theorems 29 and 35.

Theorem 37. Let be a ternary semihypergroup and a fuzzy subset of . Then is a fuzzy bi-hyperideal of if and only if for all , if , then is a bi-hyperideal of .

Proof. Let us assume that is a fuzzy bi-hyperideal of . Let such that . Let . Then and . Since is a fuzzy bi-hyperideal of , , and for all . Therefore, , . Hence is a bi-hyperideal of .

Conversely, let us assume for all , if , then is a bi-hyperideal of . Let . Let we take . Then . This implies that . By assumption, we have is a bi-hyperideal of . So , . Therefore, and . Hence and .

Theorem 38. Let be a ternary semihypergroup and a fuzzy subset of . Then is a prime fuzzy bi-hyperideal of if and only if for all , if , then is a prime bi-hyperideal of .

Proof. It follows from Theorems 31 and 37.

Theorem 39. Let be a ternary semihypergroup and a fuzzy subset of . Then is a fuzzy bi-hyperideal of if and only if for all , if , then is a bi-hyperideal of .

Proof. The proof is similar to the proof of Theorem 37.

Theorem 40. Let be a ternary semihypergroup and a fuzzy subset of . Then is a prime bi-hyperideal of if and only if for all , if , then is a prime bi-hyperideal of .

Proof. It follows from Theorems 31 and 39.

8. Rough Fuzzy Bi-Hyperideals of Ternary Semihypergroups

Theorem 41. Let be a ternary semihypergroup and be a complete regular relation on . If is a fuzzy bi-hyperideal of , then and are fuzzy bi-hyperideals.

Proof. This proof follows from Theorems 37, 39, and 20, and Lemma 33.

Note that if and are fuzzy bi-hyperideals of a ternary semihypergroup , in general, need not be a fuzzy bi-hyperideal of .

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