About this Journal Submit a Manuscript Table of Contents
Advances in Fuzzy Systems
Volume 2012 (2012), Article ID 595687, 9 pages
http://dx.doi.org/10.1155/2012/595687
Research Article

Rough Fuzzy Hyperideals in Ternary Semihypergroups

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 Γ-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 Γ-semihypergroups [7], hyperrings [8], and Γ-semihyperrings [9]. Yaqoob [10] introduced the notion of rough Γ-hyperideals in left almost Γ-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 [1619]. 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: 𝑒-hyperstructures and transposition hypergroups.

Hila and Naka [2022] 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 𝐻×𝐻𝒫(𝐻) is called hyperoperation or join operation on the set 𝐻, where 𝐻 is a nonempty set and 𝒫(𝐻)=𝒫(𝐻){} denotes the set of all nonempty subsets of 𝐻.

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

Definition 1. A hypergroupoid (𝐻,), which is associative, that is, 𝑥(𝑦𝑧)=(𝑥𝑦)𝑧, for all 𝑥,𝑦,𝑧𝑆, is called a semihypergroup.
Let 𝐴 and 𝐵 be two nonempty subsets of 𝐻. Then, we define 𝐴𝐵=𝑎𝐴,𝑏𝐵𝑎𝑏,𝑎𝐴={𝑎}𝐴,𝑎𝐵={𝑎}𝐵.(1)

Definition 2. A map 𝑓𝐻×𝐻×𝐻𝒫(𝐻) is called ternary hyperoperation on the set 𝐻, where 𝐻 is a nonempty set and 𝒫(𝐻)=𝒫(𝐻){} denotes the set of all nonempty subsets of 𝐻.

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

Definition 4. A ternary hypergroupoid (𝑆,𝑓) is called a ternary semihypergroup if for all 𝑎1,𝑎2,,𝑎5𝑆, we have 𝑓𝑓𝑎1,𝑎2,𝑎3,𝑎4,𝑎5𝑎=𝑓1𝑎,𝑓2,𝑎3,𝑎4,𝑎5𝑎=𝑓1,𝑎2𝑎,𝑓3,𝑎4,𝑎5.(2)

Definition 5. Let (𝑆,𝑓) be a ternary semihypergroup. Then 𝑆 is called a ternary hypergroup if for all 𝑎,𝑏,𝑐𝑆, there exist 𝑥,𝑦,𝑧𝑆 such that 𝑐𝑓(𝑥,𝑎,𝑏)𝑓(𝑎,𝑦,𝑏)𝑓(𝑎,𝑏,𝑧).(3)

Definition 6. Let (𝑆,𝑓) be a ternary semihypergroup and 𝑇 a nonempty subset of 𝐻. Then 𝑇 is called a subsemihypergroup of 𝑆 if and only if 𝑓(𝑇,𝑇,𝑇)𝑇.

Definition 7. A nonempty subset 𝐼 of a ternary semihypergroup 𝑆 is called a left (right, lateral) hyperideal of 𝑆 if 𝑓(𝑆,𝑆,𝐼)𝐼(𝑓(𝐼,𝑆,𝑆)𝐼,𝑓(𝑆,𝐼,𝑆)𝐼).(4)

Definition 8. A subsemihypergroup 𝐵 of a ternary semihypergroup 𝑆 is called a bi-hyperideal of 𝑆 if 𝑓(𝐵,𝑆,𝐵,𝑆,𝐵)𝐵.(5)

Definition 9. Let (𝑆,𝑓) be a ternary semihypergroup and 𝑄 a subset of 𝑆. Then 𝑄 is called a quasi-hyperideal of 𝑆 if and only if 𝑓𝑓(𝑄,𝑆,𝑆)𝑓(𝑆,𝑄,𝑆)𝑓(𝑆,𝑆,𝑄)𝑄,(𝑄,𝑆,𝑆)𝑓(𝑆,𝑆,𝑄,𝑆,𝑆)𝑓(𝑆,𝑆,𝑄)𝑄.(6)

3. Rough Hyperideals in Ternary Semihypergroups

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

Given a ternary semihypergroup 𝑆, by 𝒫(𝑆) we will denote the power-set of 𝑆. Let 𝐴 and 𝐵 be two nonempty subsets of 𝑆. We define (𝐴,𝐵)𝜌 if for every 𝑎𝐴 there exists 𝑏𝐵 such that (𝑎,𝑏)𝜌 and for every 𝑑𝐵 there exists 𝑐𝐴 such that (𝑐,𝑑)𝜌. If 𝜌 is an equivalence relation on 𝑆, then, for every 𝑥𝑆, [𝑥]𝜌 stands for the equivalence class of 𝑥 with the represent 𝜌.

Definition 10. Let 𝑆 be a ternary semihypergroup. An equivalence relation 𝜌 on 𝑆 is called regular on 𝑆 if (𝑎,𝑏)𝜌implies(𝑥𝑦𝑎,𝑥𝑦𝑏)𝜌,(𝑥𝑎𝑦,𝑥𝑏𝑦)𝜌,(𝑎𝑥𝑦,𝑏𝑥𝑦)𝜌,(7) for all 𝑎,𝑏,𝑥,𝑦𝑆.
A regular relation 𝜌 on 𝑆 is called complete if [𝑎]𝜌[𝑏]𝜌[𝑐]𝜌=[𝑎𝑏𝑐]𝜌 for all 𝑎,𝑏𝑆.

Lemma 11. Let 𝑆 be a ternary semihypergroup and 𝜌 be a regular relation on 𝑆. If 𝑎,𝑏𝑆, then [𝑎]𝜌[𝑏]𝜌[𝑐]𝜌[𝑎𝑏𝑐]𝜌

Proof. Let 𝑥[𝑎]𝜌[𝑏]𝜌[𝑐]𝜌. Then there exist 𝑝[𝑎]𝜌, 𝑞[𝑏]𝜌 and 𝑟[𝑐]𝜌 such that 𝑥𝑝𝑞𝑟. Since (𝑎,𝑝)𝜌, (𝑏,𝑞)𝜌 and (𝑐,𝑟)𝜌 then by regularity of 𝜌, we have (𝑎𝑏𝑐,𝑝𝑞𝑟)𝜌.(8) So 𝑥𝑝𝑞𝑟 implies that there exists 𝑦𝑎𝑏𝑐 such that (𝑥,𝑦)𝜌, and therefore 𝑥[𝑎𝑏𝑐]𝜌.

Let 𝐴 be a nonempty subset of a ternary semihypergroup 𝑆 and 𝜌 be a regular relation on 𝑆. Then, the sets 𝐴𝑝𝑟𝜌[𝑥](𝐴)=𝑥𝑆𝜌,𝐴𝐴𝑝𝑟𝜌[𝑥](𝐴)=𝑥𝑆𝜌𝐴(9) are called 𝜌-upper and 𝜌-lower approximations of 𝐴, respectively. For a nonempty subset 𝐴 of 𝑆,𝐴𝑝𝑟𝜌(𝐴)=(𝐴𝑝𝑟𝜌(𝐴),𝐴𝑝𝑟𝜌(𝐴)) is called a rough set with respect to 𝜌 if A𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐴).

Theorem 12. Let 𝜌 and 𝜑 be regular relations on a ternary semihypergroup 𝑆. If 𝐴 and 𝐵 are nonempty subsets of 𝑆, then the following hold:(1)𝐴𝑝𝑟𝜌(𝐴)𝐴𝐴𝑝𝑟𝜌(𝐴); (2)𝐴𝑝𝑟𝜌(𝐴𝐵)=𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐵); (3)𝐴𝑝𝑟𝜌(𝐴𝐵)=𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐵);(4)𝐴𝐵 implies 𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐵);(5)𝐴𝐵 implies 𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐵);(6)𝐴𝑝𝑟𝜌(𝐴𝐵)𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐵); (7)𝐴𝑝𝑟𝜌(𝐴𝐵)𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐵); (8)𝜌𝜑 implies 𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜑(𝐴);(9)𝜌𝜑 implies 𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜑(𝐴).

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

Theorem 13. Let 𝜌 be a regular relation on a ternary semihypergroup 𝑆 and let 𝐴, 𝐵 and 𝐶 be nonempty subsets of 𝑆. Then(1)𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐵)𝐴𝑝𝑟𝜌(𝐶)𝐴𝑝𝑟𝜌(𝐴𝐵𝐶), (2)If 𝜌 is complete, then 𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐵)𝐴𝑝𝑟𝜌(𝐶)𝐴𝑝𝑟𝜌(𝐴𝐵𝐶).

Proof. (1) Let 𝑥𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐵)𝐴𝑝𝑟𝜌(𝐶). Then 𝑥𝑎𝑏𝑐 for 𝑎𝐴𝑝𝑟𝜌(𝐴), 𝑏𝐴𝑝𝑟𝜌(𝐵) and 𝑐𝐴𝑝𝑟𝜌(𝐶). There exist 𝑟,𝑠,𝑡𝑆 such that 𝑟[𝑎]𝜌𝐴, 𝑠[𝑏]𝜌𝐵 and 𝑡[𝑐]𝜌𝐶. Since 𝜌 is regular, it follows that [𝑎]𝑟𝑠𝑡𝜌[𝑏]𝜌[𝑐]𝜌[]𝑎𝑏𝑐𝜌.(10) On the other hand, since 𝑟𝑠𝑡𝐴𝐵𝐶, we have []𝑟𝑠𝑡𝑎𝑏𝑐𝜌𝐴𝐵𝐶,(11) and so 𝑥𝑎𝑏𝑐𝐴𝑝𝑟𝜌(𝐴𝐵𝐶). This shows that 𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐵)𝐴𝑝𝑟𝜌(𝐶)𝐴𝑝𝑟𝜌(𝐴𝐵𝐶).
(2) Let 𝑥𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐵)𝐴𝑝𝑟𝜌(𝐶). Then 𝑥𝑎𝑏𝑐 for 𝑎𝐴𝑝𝑟𝜌(𝐴), 𝑏𝐴𝑝𝑟𝜌(𝐵) and 𝑐𝐴𝑝𝑟𝜌(𝐶).
It follows that [𝑎]𝜌𝐴, [𝑏]𝜌𝐵 and [𝑐]𝜌𝐶. Since 𝜌 is complete, we have [𝑎]𝜌[𝑏]𝜌[𝑐]𝜌=[]𝑎𝑏𝑐𝜌𝐴𝐵𝐶,(12) and so 𝑥𝑎𝑏𝑐𝐴𝑝𝑟𝜌(𝐴𝐵𝐶). Hence 𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐵)𝐴𝑝𝑟𝜌(𝐶)𝐴𝑝𝑟𝜌(𝐴𝐵𝐶).

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

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

Theorem 15. Let 𝜌 be a regular relation on ternary semihypergroup 𝑆 and let 𝐴 be a subsemihypergroup of 𝑆. Then(1)𝐴𝑝𝑟𝜌(𝐴) is a subsemihypergroup of 𝑆,(2)if 𝜌 is complete, then 𝐴𝑝𝑟𝜌(𝐴) is, if it is nonempty, a subsemihypergroup of 𝑆.

Proof. (1) Let 𝐴 be a subsemihypergroup of 𝑆. Now by Theorem 13(1), 𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐴𝐴𝐴)𝐴𝑝𝑟𝜌(𝐴).(13) This shows that 𝐴𝑝𝑟𝜌(𝐴) is a subsemihypergroup of 𝑆, that is, 𝐴 is a 𝜌-upper rough subsemihypergroup of 𝑆.
(2) Let 𝐴 be a subsemihypergroup of 𝑆. Now by Theorem 13(2) 𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐴𝐴𝐴)𝐴𝑝𝑟𝜌(𝐴).(14) This shows that 𝐴𝑝𝑟𝜌(𝐴) is a subsemihypergroup of 𝑆, that is, 𝐴 is a 𝜌-lower rough subsemihypergroup of 𝑆.

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

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

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

Theorem 17. Let 𝜌 be a regular relation on a ternary semihypergroup 𝑆 and let 𝐴 be a left (right, lateral) hyperideal of 𝑆. Then(1)𝐴𝑝𝑟𝜌(𝐴) is a left (right, lateral) hyperideal of 𝑆,(2) if 𝜌 is complete, then 𝐴𝑝𝑟𝜌(𝐴) is, if it is nonempty, a left (right, lateral) hyperideal of 𝑆.

Proof. (1)  Let 𝐴 be a right hyperideal of 𝑆. Now by Theorem 13(1), 𝐴𝑝𝑟𝜌(𝐴)𝑆𝑆=𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝑆)𝐴𝑝𝑟𝜌(𝑆)𝐴𝑝𝑟𝜌(𝐴𝑆𝑆)𝐴𝑝𝑟𝜌(𝐴).(15) This shows that 𝐴𝑝𝑟𝜌(𝐴) is a right hyperideal of 𝑆, that is, 𝐴 is a 𝜌-upper rough right hyperideal of 𝑆.
(2) Let 𝐴 be a right hyperideal of 𝑆. Now by Theorem 13(2), 𝐴𝑝𝑟𝜌(𝐴)𝑆𝑆=𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝑆)𝐴𝑝𝑟𝜌(𝑆)𝐴𝑝𝑟𝜌(𝐴𝑆𝑆)𝐴𝑝𝑟𝜌(𝐴).(16) This shows that 𝐴𝑝𝑟𝜌(𝐴) is a right hyperideal of 𝑆, that is, 𝐴 is a 𝜌-lower rough right hyperideal of 𝑆. The case for left (lateral) hyperideal can be seen in a similar way.

Theorem 18. Let 𝜌 be a regular relation on a ternary semihypergroup 𝑆. If 𝐴, 𝐵, and 𝐶 are a right hyperideal, a lateral hyperideal, and a left hyperideal of 𝑆, respectively. Then(1)𝐴𝑝𝑟𝜌(𝐴𝐵𝐶)𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐵)𝐴𝑝𝑟𝜌(𝐶), (2)𝐴𝑝𝑟𝜌(𝐴𝐵𝐶)𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐵)𝐴𝑝𝑟𝜌(𝐶).

Proof. Since 𝐴 is a right hyperideal of 𝑆, so 𝐴𝐵𝐶𝐴𝑆𝑆𝐴. Since 𝐵 is a lateral hyperideal of 𝑆, so 𝐴𝐵𝐶𝑆𝐵𝑆𝐵, also 𝐶 is a left hyperideal of 𝑆, so 𝐴𝐵𝐶𝑆𝑆𝐶𝐶, thus 𝐴𝐵𝐶𝐴𝐵𝐶. Then by Theorem 12(7), we have 𝐴𝑝𝑟𝜌(𝐴𝐵𝐶)𝐴𝑝𝑟𝜌(𝐴𝐵𝐶)𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐵)𝐴𝑝𝑟𝜌(𝐶).(17) Also by Theorem 12(3), we have 𝐴𝑝𝑟𝜌(𝐴𝐵𝐶)𝐴𝑝𝑟𝜌(𝐴𝐵𝐶)=𝐴𝑝𝑟𝜌(𝐴)𝐴𝑝𝑟𝜌(𝐵)𝐴𝑝𝑟𝜌(𝐶).(18) This completes the proof.

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

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

Theorem 20. Let 𝜌 be a regular relation on a ternary semihypergroup 𝑆 and let 𝐵 be a bi-hyperideal of 𝑆. Then(1)𝐴𝑝𝑟𝜌(𝐵) is a bi-hyperideal of 𝑆,(2)if 𝜌 is complete, then 𝐴𝑝𝑟𝜌(𝐵) is, if it is nonempty, a bi-hyperideal of 𝑆.

Proof. (1) Let 𝐵 be a bi-hyperideal of 𝑆. Now by Theorem 13(1) 𝐴𝑝𝑟𝜌(𝐵)𝑆𝐴𝑝𝑟𝜌(𝐵)𝑆𝐴𝑝𝑟𝜌=(𝐵)𝐴𝑝𝑟𝜌(𝐵)𝐴𝑝𝑟𝜌(𝑆)𝐴𝑝𝑟𝜌(𝐵)𝐴𝑝𝑟𝜌(𝑆)𝐴𝑝𝑟𝜌(𝐵)𝐴𝑝𝑟𝜌(𝐵𝑆𝐵𝑆𝐵)𝐴𝑝𝑟𝜌(𝐵).(19) From this and Theorem 15(1), 𝐴𝑝𝑟𝜌(𝐵) is a bi-hyperideal of 𝑆, that is, 𝐴 is a 𝜌-upper rough bi-hyperideal of 𝑆.
(2) Let 𝐵 be a bi-hyperideal of 𝑆. Now by Theorem 13(2) 𝐴𝑝𝑟𝜌(𝐵)𝑆𝐴𝑝𝑟𝜌(𝐵)𝑆𝐴𝑝𝑟𝜌(𝐵)=𝐴𝑝𝑟𝜌(𝐵)𝐴𝑝𝑟𝜌(𝑆)𝐴𝑝𝑟𝜌(𝐵)𝐴𝑝𝑟𝜌(𝑆)𝐴𝑝𝑟𝜌(𝐵)𝐴𝑝𝑟𝜌(𝐵𝑆𝐵𝑆𝐵)A𝑝𝑟𝜌(𝐵).(20)
From this and Theorem 15(2), 𝐴𝑝𝑟𝜌(𝐵) is a bi-hyperideal of 𝑆, that is, 𝐴 is a 𝜌-lower rough bi-hyperideal of 𝑆.

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

Example 21. Let 𝐻={0,𝑎,𝑏,𝑐,𝑑,𝑒,𝑔} and 𝑓(𝑥,𝑦,𝑧)=(𝑥𝑦)𝑧 for all 𝑥,𝑦,𝑧𝐻, where is defined by Table 1.
Then (𝐻,𝑓) is a ternary semihypergroup. Let 𝜌 be a complete regular relation on 𝑆 such that 𝜌-regular classes are the subsets {0}, {𝑎,𝑏,𝑐,𝑑,𝑒,𝑔}. Now for 𝐴={0,𝑒,𝑔}𝑆, 𝐴𝑝𝑟𝜌(𝐴)={0,𝑎,𝑏,𝑐,𝑑,𝑒,𝑔} and 𝐴𝑝𝑟𝜌(𝐴)={0}. It is clear that 𝐴𝑝𝑟𝜌(𝐴) and 𝐴𝑝𝑟𝜌(𝐴) are bi-hyperideals of 𝑆, but the subsemihypergroup {0,𝑒,𝑔} of 𝑆 is not a bi-hyperideal of 𝑆.

tab1
Table 1

Definition 22. A subset 𝑄 of a ternary semihypergroup 𝑆 is called a quasi-hyperideal of 𝑆 if 𝑄𝑆𝑆𝑆𝑄𝑆𝑆𝑆𝑄𝑄,𝑄𝑆𝑆𝑆𝑆𝑄𝑆𝑆𝑆𝑆𝑄𝑄.(21) Let 𝜌 be a regular relation on a ternary semihypergroup 𝑆. Then a subset 𝑄 of 𝑆 is called a 𝜌-upper (𝜌-lower) rough quasi-hyperideal of 𝑆 if 𝐴𝑝𝑟𝜌(𝐴)(𝐴𝑝𝑟𝜌(𝐴)) is a quasi-hyperideal of 𝑆.

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

Proof. Let 𝑄 be a quasi-hyperideal of 𝑆. Now by Theorems 13(2) and 12(3) 𝐴𝑝𝑟𝜌(𝑄)𝑆𝑆𝑆𝐴𝑝𝑟𝜌(𝑄)𝑆𝑆𝑆𝐴𝑝𝑟𝜌(𝑄)=𝐴p𝑟𝜌(𝑄)𝐴𝑝𝑟𝜌(𝑆)𝐴𝑝𝑟𝜌(𝑆)𝐴𝑝𝑟𝜌(𝑆)𝐴𝑝𝑟𝜌(𝑄)𝐴𝑝𝑟𝜌(𝑆)𝐴𝑝𝑟𝜌(𝑆)𝐴𝑝𝑟𝜌(𝑆)𝐴𝑝𝑟𝜌(𝑄)𝐴𝑝𝑟𝜌(𝑄𝑆𝑆)𝐴𝑝𝑟𝜌(𝑆𝑄𝑆)𝐴𝑝𝑟𝜌(𝑆𝑆𝑄)=𝐴𝑝𝑟𝜌(𝑄𝑆𝑆𝑆𝑄𝑆𝑆𝑆𝑄)𝐴𝑝𝑟𝜌(𝑄).(22) Also we can show that 𝐴p𝑟𝜌(𝑄)𝑆𝑆𝑆𝑆𝐴𝑝𝑟𝜌(𝑄)𝑆𝑆𝑆𝑆𝐴𝑝𝑟𝜌(𝑄)𝐴𝑝𝑟𝜌(𝑄).(23) Hence 𝐴𝑝𝑟𝜌(𝑄) is a quasi-hyperideal of 𝑆, that is, 𝐴 is a 𝜌-lower rough quasi-hyperideal of 𝑆.

4. Rough Hyperideals in the Quotient Ternary Semihypergroups

Let 𝜌 be a regular relation on a ternary semihypergroup 𝑆. The 𝜌-upper approximation and 𝜌-lower approximation of a nonempty subset 𝐴 of 𝑆 can be presented in an equivalent form as shown below: 𝐴𝑝𝑟𝜌[𝑥](𝐴)=𝜌𝑆𝜌[𝑥]𝜌,𝐴𝐴𝑝𝑟𝜌[𝑥](𝐴)=𝜌𝑆𝜌[𝑥]𝜌,𝐴(24) respectively.

Theorem 24. Let 𝜌 be a regular relation on a ternary semihypergroup 𝑆 and let 𝐴 be a subsemihypergroup of 𝑆. Then(1)𝐴𝑝𝑟𝜌(𝐴) is a subsemihypergroup of 𝑆/𝜌,(2)if 𝜌 is complete, then 𝐴𝑝𝑟𝜌(𝐴) is, if it is nonempty, a subsemihypergroup of 𝑆/𝜌.

Proof. (1) Let [𝑎]𝜌,[𝑏]𝜌,[𝑐]𝜌𝐴𝑝𝑟𝜌(𝐴). Then [𝑎]𝜌𝐴, [𝑏]𝜌𝐴 and [𝑐]𝜌𝐴. So there exist 𝑥[𝑎]𝜌𝐴, 𝑦[𝑏]𝜌𝐴 and 𝑧[𝑐]𝜌𝐴. Since 𝐴 is a subsemihypergroup of 𝑆, we have 𝑥𝑦𝑧𝐴. By Lemma 11, we have [𝑎]𝑥𝑦𝑧𝜌[𝑏]𝜌[𝑐]𝜌[]𝑎𝑏𝑐𝜌.(25) Thus [𝑎𝑏𝑐]𝜌𝐴, which implies that [𝑎]𝜌[𝑏]𝜌[𝑐]𝜌𝐴𝑝𝑟𝜌(𝐴). Hence 𝐴𝑝𝑟𝜌(𝐴) is a subsemihypergroup of 𝑆/𝜌.
(2) Let [𝑎]𝜌,[𝑏]𝜌,[𝑐]𝜌𝐴𝑝𝑟𝜌(𝐴). Then [𝑎]𝜌𝐴, [𝑏]𝜌𝐴 and [𝑐]𝜌𝐴. Since 𝐴 is a subsemihypergroup of 𝑆, we have [𝑎]𝜌[𝑏]𝜌[𝑐]𝜌=[]𝑎𝑏𝑐𝜌𝐴𝐴𝐴𝐴.(26) This means that 𝐴𝑝𝑟𝜌(𝐴) is a subsemihypergroup of 𝑆/𝜌.

Theorem 25. Let 𝜌 be a regular relation on a ternary semihypergroup 𝑆 and let 𝐴 be a left (right, lateral) hyperideal of 𝑆. Then(1)𝐴𝑝𝑟𝜌(𝐴) is a left (right, lateral) hyperideal of 𝑆/𝜌,(2)if 𝜌 is complete, then 𝐴𝑝𝑟𝜌(𝐴) is, if it is nonempty, a left (right, lateral) hyperideal of 𝑆/𝜌.

Proof. (1) Let 𝐴 be a left hyperideal of 𝑆. Let [𝑥]𝜌𝐴𝑝𝑟𝜌(𝐴) and [𝑦]𝜌,[𝑧]𝜌𝑆/𝜌. Then [𝑥]𝜌𝐴, hence 𝑥𝐴𝑝𝑟𝜌(𝐴). Since 𝐴 is a left hyperideal of 𝑆, by Theorem 17(1), 𝐴𝑝𝑟𝜌(𝐴) is a left hyperideal of 𝑆. So, we have 𝑦𝑧𝑥𝐴𝑝𝑟𝜌(𝐴).(27) Now, for every 𝑚𝑦𝑧𝑥, we have [𝑚]𝜌𝐴. On the other hand, from 𝑚𝑦𝑧𝑥, we obtain [𝑚]𝜌[𝑦]𝜌[𝑧]𝜌[𝑥]𝜌. Therefore [𝑦]𝜌[𝑧]𝜌[𝑥]𝜌𝐴𝑝𝑟𝜌(𝐴). This means that 𝐴𝑝𝑟𝜌(𝐴) is a left hyperideal of 𝑆/𝜌.
(2) Let 𝐴 be a left hyperideal of 𝑆. Let [𝑥]𝜌𝐴𝑝𝑟𝜌(𝐴) and [𝑦]𝜌,[𝑧]𝜌𝑆/𝜌. Then, [𝑥]𝜌𝐴, which implies 𝑥𝐴𝑝𝑟𝜌(𝐴). Since 𝐴 is a left hyperideal of 𝑆, by Theorem 17(2), 𝐴𝑝𝑟𝜌(𝐴) is a left hyperideal of 𝑆. Thus, we have 𝑦𝑧𝑥𝐴𝑝𝑟𝜌(𝐴).(28) Now, for every 𝑚𝑦𝑧𝑥, we have 𝑚𝐴𝑝𝑟𝜌(𝐴), which implies that [𝑚]𝜌𝐴. Hence, [𝑚]𝜌𝐴𝑝𝑟𝜌(𝐴). On the other hand, from 𝑚𝑦𝑧𝑥, we have [𝑚]𝜌[𝑦]𝜌[𝑧]𝜌[𝑥]𝜌. Therefore [𝑦]𝜌[𝑧]𝜌[𝑥]𝜌𝐴𝑝𝑟𝜌(𝐴). This means that 𝐴𝑝𝑟𝜌(𝐴) is, if it is nonempty, a left hyperideal of 𝑆/𝜌. The other cases can be seen in a similar way.

Theorem 26. Let 𝜌 be a regular relation on a ternary semihypergroup 𝑆 and let 𝐴 be a bi-hyperideal of 𝑆. Then(1)𝐴𝑝𝑟𝜌(𝐴) is a bi-hyperideal of 𝑆/𝜌,(2)if 𝜌 is complete, then 𝐴𝑝𝑟𝜌(𝐴) is, if it is nonempty, a bi-hyperideal of 𝑆/𝜌.

Proof. (1) Let 𝐴 be a bi-hyperideal of 𝑆. Let [𝑥]𝜌,[𝑦]𝜌,[𝑧]𝜌𝐴𝑝𝑟𝜌(𝐴) and [𝑠]𝜌,[𝑡]𝜌𝑆/𝜌. Then, [𝑥]𝜌[𝑦]𝐴,𝜌[𝑧]𝐴,𝜌𝐴.(29) Hence, 𝑥𝐴𝑝𝑟𝜌(𝐴),𝑦𝐴𝑝𝑟𝜌(𝐴) and 𝑧𝐴𝑝𝑟𝜌(𝐴). By Theorem 20(1), 𝐴𝑝𝑟𝜌(𝐴) is a bi-hyperideal of 𝑆. So, we have 𝑥𝑠𝑦𝑡𝑧𝐴𝑝𝑟𝜌(𝐴).(30) Now, for every 𝑚𝑥𝑠𝑦𝑡𝑧, we obtain [𝑚]𝜌[𝑥]𝜌[𝑠]𝜌[𝑦]𝜌[𝑡]𝜌[𝑧]𝜌.(31) On the other hand, since 𝑚𝐴𝑝𝑟𝜌(𝐴), we have [𝑚]𝜌𝐴. Thus, [𝑥]𝜌[𝑠]𝜌[𝑦]𝜌[𝑡]𝜌[𝑧]𝜌𝐴𝑝𝑟𝜌(𝐴).(32) Therefore, from this and Theorem 24(1), 𝐴𝑝𝑟𝜌(𝐴) is a bi-hyperideal of 𝑆/𝜌.
(2) Let 𝐴 be a bi-hyperideal of 𝑆. Let [𝑥]𝜌,[𝑦]𝜌,[𝑧]𝜌𝐴𝑝𝑟𝜌(𝐴) and [𝑠]𝜌,[𝑡]𝜌𝑆/𝜌. Then, [𝑥]𝜌[𝑦]𝐴,𝜌[𝑧]𝐴,𝜌𝐴.(33) Hence, 𝑥𝐴𝑝𝑟𝜌(𝐴), 𝑦𝐴𝑝𝑟𝜌(𝐴), and 𝑧𝐴𝑝𝑟𝜌(𝐴). By Theorem 20(2), 𝐴𝑝𝑟𝜌(𝐴) is a bi-hyperideal of 𝑆. So, we have 𝑥𝑠𝑦𝑡𝑧𝐴𝑝𝑟𝜌(𝐴).(34) Then, for every 𝑚𝑥𝑠𝑦𝑡𝑧, we obtain [𝑚]𝜌[𝑥]𝜌[𝑠]𝜌[𝑦]𝜌[𝑡]𝜌[𝑧]𝜌.(35) On the other hand, since 𝑚𝐴𝑝𝑟𝜌(𝐴), we have [𝑚]𝜌𝐴. So, [𝑥]𝜌[𝑠]𝜌[𝑦]𝜌[𝑡]𝜌[𝑧]𝜌𝐴𝑝𝑟𝜌(𝐴).(36) Therefore from this and Theorem 24(2), 𝐴𝑝𝑟𝜌(𝐴) is, if it is nonempty, a bi-hyperideal of 𝑆/𝜌.

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 𝑆 be a ternary semihypergroup. A fuzzy subset 𝑓 of 𝑆 is called(1)a fuzzy ternary subsemihypergroup of 𝑆 if inf𝑡𝑥𝑦𝑧𝑓(𝑡)min{𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)} for all 𝑥,𝑦,𝑧𝑆,(2)a fuzzy left hyperideal of 𝑆 if inf𝑡𝑥𝑦𝑧𝑓(𝑡)𝑓(𝑧) for all 𝑥,𝑦,𝑧𝑆,(3)a fuzzy right hyperideal of 𝑆 if inf𝑡𝑥𝑦𝑧𝑓(𝑡)𝑓(𝑥) for all 𝑥,𝑦,𝑧𝑆,(4)a fuzzy lateral hyperideal of 𝑆 if inf𝑡𝑥𝑦𝑧𝑓(𝑡)𝑓(𝑦) for all 𝑥,𝑦,𝑧𝑆,(5)a fuzzy hyperideal of 𝑆 if inf𝑡𝑥𝑦𝑧𝑓(𝑡)max{𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)} for all 𝑥,𝑦,𝑧𝑆.

Theorem 28. Let 𝑆 be a ternary semihypergroup and 𝐴 a nonempty subset of 𝑆. The following statements hold true.(1)𝐴 is a ternary subsemihypergroup of 𝑆 if and only if 𝑓𝐴 is a fuzzy ternary subsemihypergroup of 𝑆.(2)𝐴 is a left hyperideal (right hyperideal, lateral hyperideal, hyperideal) of 𝑆 if and only if 𝑓𝐴 is a fuzzy left hyperideal (fuzzy right hyperideal, fuzzy lateral hyperideal, fuzzy hyperideal) of 𝑆.

Proof. (1) Let us assume that 𝐴 is a ternary subsemihypergroup of 𝑆. Let 𝑥,𝑦,𝑧𝑆.
Case 1.𝑥,𝑦,𝑧𝐴. Since 𝐴 is a ternary subsemihypergroup of 𝑆, we have 𝑥𝑦𝑧𝐴. Then inf𝑡𝑥𝑦𝑧𝑓𝐴(𝑡)=1min{𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)}.
Case 2.𝑥𝐴 or 𝑦𝐴 or 𝑧𝐴. Thus 𝑓𝐴(𝑥)=0 or 𝑓𝐴(𝑦)=0 or 𝑓𝐴(𝑧)=0. Therefore min{𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)}=0inf𝑡𝑥𝑦𝑧𝑓𝐴(𝑡).
Conversely, let 𝑥,𝑦,𝑧𝐴. We have 𝑓𝐴(𝑥)=𝑓𝐴(𝑦)=𝑓𝐴(𝑧)=1. Since 𝑓𝐴 is a fuzzy ternary subsemihypergroup of 𝑆, inf𝑡𝑥𝑦𝑧𝑓𝐴(𝑡)min{𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)}=1. Hence 𝑥𝑦𝑧𝐴.
(2) Let us assume that 𝐴 is a left hyperideal of 𝑆. Let 𝑥,𝑦,𝑧𝑆.
Case 1.𝑧𝐴. Since 𝐴 is a left hyperideal of 𝑆, then 𝑥𝑦𝑧𝐴. Then inf𝑡𝑥𝑦𝑧𝑓𝐴(𝑡)=1. Therefore inf𝑡𝑥𝑦𝑧𝑓𝐴(𝑡)𝑓𝐴(𝑧).
Case 2.𝑧𝐴. We have 𝑓𝐴(𝑧)=0. Hence inf𝑡𝑥𝑦𝑧𝑓𝐴(𝑡)𝑓𝐴(𝑧).
Conversely, let 𝑥,𝑦𝑆 and 𝑧𝐴. Since 𝑓𝐴 is a fuzzy left hyperideal of 𝑆 and 𝑧𝐴,inf𝑡𝑥𝑦𝑧𝑓𝐴(𝑡)𝑓𝐴(𝑧)=1. Thus 𝑥𝑦𝑧𝐴.
The remaing parts can be seen in similarly way.

Let 𝑆 be a ternary semihypergroup. A nonempty subset 𝑇 of 𝑆 is called prime subset of 𝑆 if for all 𝑥,𝑦,𝑧𝑆,𝑥𝑦𝑧𝑇 implies 𝑥𝑇 or 𝑦𝑇 or 𝑧𝑇. A ternary subsemihypergroup 𝑇 of 𝑆 is called prime ternary subsemihypergroup of 𝑆 if 𝑇 is a prime subset of 𝑆. Prime left hyperideals, prime right hyperideals, prime lateral hyperideals, and prime hyperideals of 𝑆 are defined analogously. A fuzzy subset 𝑓 of 𝑆 is called a prime fuzzy subset of 𝑆 if inf𝑡𝑥𝑦𝑧𝑓(𝑡)max{𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)} for all 𝑥,𝑦,𝑧𝑆. A fuzzy ternary subsemihypergroup 𝑓 of 𝑆 is called a prime fuzzy ternary subsemihypergroup of 𝑆 if 𝑓 is a prime fuzzy subset of 𝑆. Prime fuzzy left hyperideals, prime fuzzy right hyperideals, prime fuzzy lateral hyperideals, and prime fuzzy hyperideals of 𝑆 are defined analogously.

Theorem 29. Let 𝑆 be a ternary semihypergroup and 𝐴 a nonempty subset of 𝑆. The following statements hold true.(1)𝐴 is a prime subset of 𝑆 if and only if 𝑓𝐴 is a prime fuzzy subset of 𝑆.(2)𝐴 is a prime ternary subsemihypergroup (prime left hyperideal, prime right hyperideal, prime lateral hyperideal, prime hyperideal) of 𝑆 if and only if 𝑓𝐴 is a prime fuzzy ternary subsemihypergroup (prime fuzzy left hyperideal, prime fuzzy right hyperideal, prime fuzzy lateral hyperideal, prime fuzzy hyperideal) of 𝑆.

Proof. (1) Let us assume that 𝐴 is a prime subset of 𝑆. Let 𝑥,𝑦,𝑧𝑆.
Case  1.𝑥𝑦𝑧𝐴. Since 𝐴 is prime, 𝑥𝐴 or 𝑦𝐴 or 𝑧𝐴. Thus max{𝑓𝐴(𝑥),𝑓𝐴(𝑦),𝑓𝐴(𝑧)}=1inf𝑡𝑥𝑦𝑧𝑓𝐴(𝑡).
Case  2.̸𝑥𝑦𝑧𝐴. Thus inf𝑡𝑥𝑦𝑧𝑓𝐴(𝑡)=0max{𝑓𝐴(𝑥),𝑓𝐴(𝑦),𝑓𝐴(𝑧)}.
Conversely, let 𝑥,𝑦,𝑧𝑆 such that 𝑥𝑦𝑧𝐴. Thus 𝑓𝐴(𝑡)=1 for all 𝑡𝑥𝑦𝑧. Since 𝑓𝐴 is prime, max{𝑓𝐴(𝑥),𝑓𝐴(𝑦),𝑓𝐴(𝑧)}=1. This implies 𝑓𝐴(𝑥)=1 or 𝑓𝐴(𝑦)=1 or 𝑓𝐴(𝑧)=1. Hence 𝑥𝐴 or 𝑦𝐴 or 𝑧𝐴.
(2) It follows from (1) and Theorem 28.

Let 𝑓 be a fuzzy subset of a set (a ternary semihypergroup) 𝑆. For any 𝑡[0,1], the set 𝑓𝑡={𝑥𝑆𝑓(𝑥)𝑡},𝑓𝑠𝑡={𝑥𝑆𝑓(𝑥)>𝑡}(37) are called a tlevel set and a tstronglevel set of 𝑓, respectively.

Theorem 30. Let 𝑓 be a fuzzy subset of a ternary semihypergroup 𝑆. The following statements hold true:(1)𝑓 is a fuzzy ternary subsemihypergroup of 𝑆 if and only if for all 𝑡[0.1], if 𝑓𝑡, then 𝑓𝑡 is a ternary subsemihypergroup of 𝑆.(2)𝑓 is a fuzzy left hyperideal (fuzzy right hyperideal, fuzzy lateral hyperideal, fuzzy hyperideal) of 𝑆 if and only if for all 𝑡[0,1], if 𝑓𝑡, then 𝑓𝑡 is a left hyperideal (right hyperideal, lateral hyperideal, hyperideal) of 𝑆.

Proof. (1) Let us assume that 𝑓 is a fuzzy ternary subsemihypergroup of 𝑆. Let 𝑡[0,1] such that 𝑓𝑡. Let 𝑥,𝑦,𝑧𝑓𝑡. So 𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)𝑡. Thus min{𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)}𝑡. Since 𝑓 is a fuzzy ternary subsemihypergroup of 𝑆, inf𝑥𝑦𝑧𝑓()𝑡. Hence, 𝑥𝑦𝑧𝑓𝑡. Conversely, let 𝑥,𝑦,𝑧𝑆. Let we take 𝑡=min{𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)}. Then 𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)𝑡. Thus, 𝑥,𝑦,𝑧𝑓𝑡. Since 𝑓𝑡 is a ternary subsemihypergroup of 𝑆, 𝑥𝑦𝑧𝑓𝑡. Thus inf𝑥𝑦𝑧𝑓()𝑡=min{𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)}.
(2) Let us assume that 𝑓 is a fuzzy left hyperideal of 𝑆. Let 𝑡[0,1]. Let us suppose that 𝑓𝑡. Let 𝑥,𝑦𝑆 and 𝑧𝑓𝑡. Thus inf𝑥𝑦𝑧𝑓()𝑓(𝑧)𝑡. Therefore, 𝑥𝑦𝑧𝑓𝑡.
Conversely, let 𝑥,𝑦,𝑧𝑆. Let we take 𝑡=𝑓(𝑧). Thus 𝑧𝑓𝑡, this implies 𝑓𝑡. By assumption, we have 𝑓𝑡 is a left hyperideal of 𝑆. So 𝑥𝑦𝑧𝑓𝑡. Therefore, inf𝑥𝑦𝑧𝑓()𝑡. Thus inf𝑥𝑦𝑧𝑓()𝑓(𝑧).
The remain parts can be proved in a similar way.

Theorem 31. Let 𝑆 be a ternary semihypergroup and 𝑓 be a fuzzy of 𝑆. The following statements hold true:(1)𝑓 is prime fuzzy subset of 𝑆 if and only if for all 𝑡[0,1], if 𝑓𝑡, then 𝑓𝑡 is a prime subset of 𝑆.(2)𝑓 is a prime fuzzy ternary subsemihypergroup (prime fuzzy left hyperideal, prime fuzzy right hyperideal, prime fuzzy lateral hyperideal, prime fuzzy hyperideal) of 𝑆 if and only if for all 𝑡[0,1], if 𝑓𝑡, then 𝑓𝑡 is a prime ternary subsemihypergroup (prime left hyperideal, prime right hyperideal, prime lateral hyperideal, prime hyperideal) of 𝑆.

Proof. (1) Let us assume that 𝑓 is a prime fuzzy subset of 𝑆. Let 𝑡[0,1]. Let us suppose that 𝑓𝑡. Let 𝑥,𝑦,𝑧𝑆 such that 𝑥𝑦𝑧𝑓𝑡. Thus inf𝑥𝑦𝑧𝑓()𝑡. Since 𝑓 is prime, 𝑓(𝑥)𝑡 or 𝑓(𝑦)𝑡 or 𝑓(𝑧)𝑡. This implies 𝑥𝑓𝑡 or 𝑦𝑓𝑡 or 𝑧𝑓𝑡.
Conversely, let 𝑥,𝑦,𝑧𝑆. Let we take 𝑡=inf𝑥𝑦𝑧𝑓(). Then 𝑥𝑦𝑧𝑓𝑡. Since 𝑓𝑡 is prime, 𝑥𝑓𝑡 or 𝑦𝑓𝑡 or 𝑧𝑓𝑡. Then 𝑓(𝑥)𝑡 or 𝑓(𝑦)𝑡 or 𝑓(𝑧)𝑡. Hence max{𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)}𝑡=inf𝑥𝑦𝑧𝑓().
(2) It follows from (1) and Theorem 29.

Theorem 32. Let 𝑆 be a ternary semihypergroup and 𝑓 be a fuzzy subset of 𝑆. Then 𝑓 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 𝑆 if and only if for all 𝑡[0,1], if 𝑓𝑠𝑡, then 𝑓𝑠𝑡 is a prime subset (prime ternary subsemihypergroup, prime left hyperideal, prime right hyperideal, prime lateral hyperideal, prime hyperideal) of S.

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 𝑆 be a ternary semihypergroup and 𝑓 be a fuzzy subset of 𝑆. Then the sets 𝐴𝑝𝑟𝜌(𝑓)(𝑥)=sup[𝑥]𝑎𝜌𝑓(𝑎),𝐴𝑝𝑟𝜌(𝑓)(𝑥)=inf[𝑥]𝑎𝜌𝑓(𝑎)(38) are called the 𝜌-upper and 𝜌-lower approximations of a fuzzy set 𝑓, respectively.

Lemma 33. Let 𝑆 be a ternary semihypergroup, 𝜌 a regular relation on 𝑆, 𝑓 a fuzzy subset of 𝑆 and 𝑡[0,1], then(1)(𝐴𝑝𝑟𝜌(𝑓))𝑡=𝐴𝑝𝑟𝜌(𝑓𝑡),(2)(𝐴𝑝𝑟𝜌(𝑓))𝑠𝑡=𝐴𝑝𝑟𝜌(𝑓𝑠𝑡).

Proof. (1) Let 𝑥(𝐴𝑝𝑟𝜌(𝑓))𝑡. Then 𝐴𝑝𝑟𝜌(𝑓)(𝑥)𝑡. So inf𝑎[𝑥]𝜌𝑓(𝑎)𝑡. Therefore, 𝑓(𝑎)𝑡 for all 𝑎[𝑥]𝜌. This implies [𝑥]𝜌𝑓𝑡. Therefore, 𝑥𝐴𝑝𝑟𝜌(𝑓𝑡).
Conversely, let us assume that 𝑥𝐴𝑝𝑟𝜌(𝑓𝑡). Thus [𝑥]𝜌𝑓𝑡. Then 𝑓(𝑎)𝑡 for all 𝑎[𝑥]𝜌. This implies inf𝑎[𝑥]𝜌𝑓(𝑎)𝑡. Thus, 𝐴𝑝𝑟𝜌(𝑓)(𝑥)𝑡. Hence 𝑥(𝐴𝑝𝑟𝜌(𝑓))t.
(2) Let 𝑥(𝐴𝑝𝑟𝜌(𝑓))𝑠𝑡. Then 𝐴𝑝𝑟𝜌(𝑓)(𝑥)>𝑡. So sup𝑎[𝑥]𝜌𝑓(𝑎)>𝑡. Therefore, 𝑓(𝑎)>𝑡 for some 𝑎[𝑥]𝜌. This implies [𝑥]𝜌𝑓𝑠𝑡. Therefore 𝑥𝐴𝑝𝑟𝜌(𝑓𝑠𝑡).
Conversely, let us assume 𝑥𝐴𝑝𝑟𝜌(𝑓𝑠𝑡). Thus [𝑥]𝜌𝑓𝑠𝑡. Then 𝑓(𝑎)>𝑡 for some 𝑎[𝑥]𝜌. This implies inf𝑎[𝑥]𝜌𝑓(𝑎)>𝑡. Thus 𝐴𝑝𝑟𝜌(𝑓)(𝑥)>𝑡. Hence, 𝑥(𝐴𝑝𝑟𝜌(𝑓))𝑠𝑡.

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

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

7. Fuzzy Bi-Hyperideals of Ternary Semihypergroups

Let 𝑆 be a ternary semihypergroup. A fuzzy subset 𝑓 of 𝑆 is called a fuzzy bi-hyperideal of 𝑆 if inf𝑥𝑦𝑧𝑓()min{𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)} and inf𝑥𝑦𝑧𝑝𝑞𝑓()min{𝑓(𝑥),𝑓(𝑧),𝑓(𝑞)} for all 𝑥,𝑦,𝑧,𝑝,𝑞𝑆.

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

Proof. Let us assume that 𝐴 is a bi-hyperideal of 𝑆. Let 𝑎,𝑏,𝑥,𝑦,𝑧𝑆.
Case  1.𝑥,𝑦,𝑧𝐴. Since 𝐴 is a bi-hyperideal of 𝑆, then 𝑥𝑦, 𝑥𝑎𝑦𝑏𝑧𝐴. Therefore inf𝑥𝑦𝑧𝑓𝐴()=1min{𝑓𝐴(𝑥),𝑓𝐴(𝑦),𝑓𝐴(𝑧)} and inf𝑥𝑎𝑦𝑏𝑧𝑓𝐴()=1min{𝑓𝐴(𝑥),𝑓𝐴(𝑦),𝑓𝐴(𝑧)}.
Case  2.𝑥𝐴 or 𝑦𝐴 or 𝑧𝐴. Thus 𝑓𝐴(𝑥)=0 or 𝑓𝐴(𝑦)=0 or 𝑓𝐴(𝑧)=0. Hence min{𝑓𝐴(𝑥),𝑓𝐴(𝑦),𝑓𝐴(𝑧)}=0inf𝑥𝑦𝑧𝑓𝐴() and min{𝑓𝐴(𝑥),𝑓𝐴(𝑦),𝑓𝐴(𝑧)}=0inf𝑥𝑎𝑦𝑏𝑧𝑓𝐴().

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

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

Proof. It follows from Theorems 29 and 35.

Theorem 37. Let 𝑆 be a ternary semihypergroup and 𝑓 a fuzzy subset of 𝑆. Then 𝑓 is a fuzzy bi-hyperideal of 𝑆 if and only if for all 𝑡[0,1], if 𝑓𝑡, then 𝑓𝑡 is a bi-hyperideal of 𝑆.

Proof. Let us assume that 𝑓 is a fuzzy bi-hyperideal of 𝑆. Let 𝑡[0,1] such that 𝑓𝑡. Let 𝑥,𝑦,𝑧𝑓𝑡. Then 𝑓(𝑥)𝑡,𝑓(𝑦)𝑡 and 𝑓(𝑧)𝑡. Since 𝑓 is a fuzzy bi-hyperideal of 𝑆, inf𝑥𝑦𝑧𝑓()min{𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)}𝑡, and inf𝑥𝑎𝑦𝑏𝑧𝑓()min{𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)}𝑡 for all 𝑎,𝑏𝑆. Therefore, 𝑥𝑦𝑧, 𝑥𝑎𝑦𝑏𝑧𝑓𝑡. Hence 𝑓𝑡 is a bi-hyperideal of 𝑆.
Conversely, let us assume for all 𝑡[0,1], if 𝑓𝑡, then 𝑓𝑡 is a bi-hyperideal of 𝑆. Let 𝑎,𝑏,𝑥,𝑦,𝑧𝑆. Let we take 𝑡=min{𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)}. Then 𝑥,𝑦,𝑧𝑓𝑡. This implies that 𝑓𝑡. By assumption, we have 𝑓𝑡 is a bi-hyperideal of 𝑆. So 𝑥𝑦𝑧, 𝑥𝑎𝑦𝑏𝑧𝑓𝑡. Therefore, inf𝑥𝑦𝑧𝑓()𝑡 and inf𝑥𝑎𝑦𝑏𝑧𝑓()𝑡. Hence inf𝑥𝑦𝑧𝑓()min{𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)} and inf𝑥𝑎𝑦𝑏𝑧𝑓()min{𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)}.

Theorem 38. Let 𝑆 be a ternary semihypergroup and 𝑓 a fuzzy subset of 𝑆. Then 𝑓 is a prime fuzzy bi-hyperideal of 𝑆 if and only if for all 𝑡[0,1], if 𝑓𝑡, then 𝑓𝑡 is a prime bi-hyperideal of 𝑆.

Proof. It follows from Theorems 31 and 37.

Theorem 39. Let 𝑆 be a ternary semihypergroup and 𝑓 a fuzzy subset of 𝑆. Then 𝑓 is a fuzzy bi-hyperideal of 𝑆 if and only if for all 𝑡[0,1], if 𝑓𝑠𝑡, then 𝑓𝑠𝑡 is a bi-hyperideal of 𝑆.

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

Theorem 40. Let 𝑆 be a ternary semihypergroup and 𝑓 a fuzzy subset of 𝑆. Then 𝑓 is a prime bi-hyperideal of 𝑆 if and only if for all 𝑡[0,1], if 𝑓𝑠𝑡, then 𝑓𝑠𝑡 is a prime bi-hyperideal of 𝑆.

Proof. It follows from Theorems 31 and 39.

8. Rough Fuzzy Bi-Hyperideals of Ternary Semihypergroups

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

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

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

References

  1. Z. Pawlak, “Rough sets,” International Journal of Computer & Information Sciences, vol. 11, no. 5, pp. 341–356, 1982. View at Publisher · View at Google Scholar · View at Scopus
  2. M. Aslam, M. Shabir, and N. Yaqoob, “Roughness in left almost semigroups,” Journal of Advanced Research in Pure Mathematics, vol. 3, no. 3, pp. 70–88, 2011.
  3. R. Chinram, “Rough prime ideals and rough fuzzy prime ideals in gamma-semigroups,” Communications of the Korean Mathematical Society, vol. 24, no. 3, pp. 341–351, 2009. View at Publisher · View at Google Scholar · View at Scopus
  4. P. Petchkhaew and R. Chinram, “Fuzzy, rough and rough fuzzy ideals in ternary semigroups,” International Journal of Pure and Applied Mathematics, vol. 56, no. 1, pp. 21–36, 2009. View at Scopus
  5. B. Davvaz, “Roughness in rings,” Information Sciences, vol. 164, no. 1–4, pp. 147–163, 2004. View at Publisher · View at Google Scholar · View at Scopus
  6. B. Davvaz and M. Mahdavipour, “Roughness in modules,” Information Sciences, vol. 176, no. 24, pp. 3658–3674, 2006. View at Publisher · View at Google Scholar · View at Scopus
  7. S. M. Anvariyeh, S. Mirvakili, and B. Davvaz, “Pawlak's approximations in Γ-semihypergroups,” Computers and Mathematics with Applications, vol. 60, no. 1, pp. 45–53, 2010. View at Publisher · View at Google Scholar · View at Scopus
  8. B. Davvaz, “Approximations in hyperrings,” Journal of Multiple-Valued Logic and Soft Computing, vol. 15, no. 5-6, pp. 471–488, 2009. View at Scopus
  9. S. O. Dehkordi and B. Davvaz, “Γ-semihyperrings: approximations and rough ideals,” Bulletin of the Malaysian Mathematical Sciences Society. In press.
  10. N. Yaqoob, “Applications of rough sets to Γ-hyperideals in left almost Γ-semihypergroups,” Neural Computing and Applications. In press. View at Publisher · View at Google Scholar
  11. M. Aslam, S. Abdullah, B. Davvaz, and N. Yaqoob, “Rough M-hypersystems and fuzzy M-hypersystems in Γ-semihypergroups,” Neural Computing & Applications. In press. View at Publisher · View at Google Scholar
  12. N. Yaqoob, M. Aslam, and R. Chinram, “Rough prime bi-ideals and rough fuzzy prime bi-ideals in semigroups,” Annals of Fuzzy Mathematics and Informatics, vol. 3, no. 2, pp. 203–211, 2012.
  13. N. Kuroki, “Rough ideals in semigroups,” Information Sciences, vol. 100, no. 1–4, pp. 139–163, 1997. View at Scopus
  14. Y. B. Jun, “Roughness of ideals in BCK-algebras,” Scientiae Mathematicae Japonicae, vol. 57, no. 1, pp. 165–169, 2003.
  15. F. Marty, “Sur une generalization de la notion de groupe. 8iem congres,” Mathématiciens Scandinaves, Stockholm, pp. 45–49, 1934.
  16. P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Kluwer Academic Publications, 2003.
  17. P. Corsini, Prolegomena of Hypergroup Theory, Aviani Editor, 2nd edition, 1993.
  18. B. Davvaz and V. L. Fotea, Hyperring Theory and Applications, International Academic Press, New York, NY, USA, 2007.
  19. T. Vougiouklis, Hyperstructures and Their Representations, Hadronic Press, Palm Harbor, Fla, USA, 1994.
  20. K. Hila and K. Naka, “On regular ternary semihypergroups,” submitted.
  21. K. Naka and K. Hila, “On some special classes of hyperideals in ternary semihypergroups,” Utilitas Mathematica. In press.
  22. K. Naka and K. Hila, “Some properties of hyperideals in ternary semihypergroups,” Mathematica Slovaca. In press.
  23. K. Hila, B. Davvaz, and K. Naka, “On quasi-hyperideals in semihypergroups,” Communications in Algebra, vol. 39, no. 11, pp. 4183–4194, 2011.
  24. L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965. View at Scopus
  25. N. Kuroki, “Fuzzy bi-ideals in Semigroups,” Commentarii Mathematici Universitatis Sancti Pauli, vol. 28, pp. 17–21, 1979.
  26. A. Rosenfeld, “Fuzzy groups,” Journal of Mathematical Analysis and Applications, vol. 35, no. 3, pp. 512–517, 1971. View at Scopus
  27. B. Davvaz, “Fuzzy hyperideals in semihypergroups,” Italian Journal of Pure and Applied Mathematics, vol. 8, pp. 67–74, 2000.
  28. B. Davvaz, “Fuzzy hyperideals in ternary semihyperrings,” Iranian Journal of Fuzzy Systems, vol. 6, no. 4, pp. 21–36, 2009. View at Scopus
  29. S. O. Dehkordi and B. Davvaz, “A strong regular relation on Γ-semihyperrings,” Iranian Journal of Science and Technology, vol. 22, no. 3, pp. 257–266, 2011.