# Rough Fuzzy Hyperideals in Ternary Semihypergroups

Authors: Naveed Yaqoob; Muhammad Aslam; Kostaq Hila
Journal: Advances in Fuzzy Systems
 (2012)

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## 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.


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## Body

## 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 [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: e-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∘:H×H→𝒫*(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 setH with together a (binary) hyperoperation.Definition 1.
A hypergroupoid(H,∘), which is associative, that is, x∘(y∘z)=(x∘y)∘z, for all x,y,z∈S, is called a semihypergroup.
LetA and B be two nonempty subsets of H. Then, we define
(1)A∘B=⋃a∈A,b∈B‍a∘b,a∘A={a}∘A,a∘B={a}∘B.Definition 2.
A mapf:H×H×H→𝒫*(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,…,a5∈S, we have
(2)f(f(a1,a2,a3),a4,a5)=f(a1,f(a2,a3,a4),a5)=f(a1,a2,f(a3,a4,a5)).Definition 5.
Let(S,f) be a ternary semihypergroup. Then S is called a ternary hypergroup if for all a,b,c∈S, there exist x,y,z∈S such that
(3)c∈f(x,a,b)∩f(a,y,b)∩f(a,b,z).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 subsetI of a ternary semihypergroup S is called a left (right, lateral) hyperideal of S if
(4)f(S,S,I)⊆I(f(I,S,S)⊆I,f(S,I,S)⊆I).Definition 8.
A subsemihypergroupB of a ternary semihypergroup S is called a bi-hyperideal of S if
(5)f(B,S,B,S,B)⊆B.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
(6)f(Q,S,S)∩f(S,Q,S)∩f(S,S,Q)⊆Q,f(Q,S,S)∩f(S,S,Q,S,S)∩f(S,S,Q)⊆Q.


## 3. Rough Hyperideals in Ternary Semihypergroups

In what follows, letS denote a ternary semihypergroup unless otherwise specified. In this section, for simplicity we write f(a,b,c) as a·b·c 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 semihypergroupS, 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 a∈A there exists b∈B such that (a,b)∈ρ and for every d∈B there exists c∈A such that (c,d)∈ρ. If ρ is an equivalence relation on S, then, for every x∈S, [x]ρ stands for the equivalence class of x with the represent ρ.Definition 10.
LetS be a ternary semihypergroup. An equivalence relation ρ on S is called regular on S if
(7)(a,b)∈ρimplies(x⋅y⋅a,x⋅y⋅b)∈ρ¯,(x⋅a⋅y,x⋅b⋅y)∈ρ¯,(a⋅x⋅y,b⋅x⋅y)∈ρ¯,
for all a,b,x,y∈S.
A regular relationρ on S is called complete if [a]ρ·[b]ρ·[c]ρ=[a·b·c]ρ for all a,b∈S.Lemma 11.
LetS be a ternary semihypergroup and ρ be a regular relation on S. If a,b∈S, then [a]ρ·[b]ρ·[c]ρ⊆[a·b·c]ρProof.
Letx∈[a]ρ·[b]ρ·[c]ρ. Then there exist p∈[a]ρ, q∈[b]ρ and r∈[c]ρ such that x∈p·q·r. Since (a,p)∈ρ,  (b,q)∈ρ and (c,r)∈ρ then by regularity of ρ, we have
(8)(a⋅b⋅c,p⋅q⋅r)∈ρ¯.
So x∈p·q·r implies that there exists y∈a·b·c such that (x,y)∈ρ, and therefore x∈[a·b·c]ρ.LetA be a nonempty subset of a ternary semihypergroup S and ρ be a regular relation on S. Then, the sets
(9)Apr¯ρ(A)={x∈S:[x]ρ∩A≠∅},Apr_ρ(A)={x∈S:[x]ρ⊆A}
are called ρ-upper and ρ-lower approximations of A, respectively. For a nonempty subset A of S,Aprρ(A)=(Apr_ρ(A),Apr¯ρ(A)) is called a rough set with respect to ρ if Apr_ρ(A)≠Apr¯ρ(A).Theorem 12.
Letρ and φ be regular relations on a ternary semihypergroup S. If A and B are nonempty subsets of S, then the following hold:(1)
Apr_ρ(A)⊆A⊆Apr¯ρ(A);(2)
Apr¯ρ(A∪B)=Apr¯ρ(A)∪Apr¯ρ(B);(3)
Apr_ρ(A∩B)=Apr_ρ(A)∩Apr_ρ(B);(4)
A⊆B implies Apr_ρ(A)⊆Apr_ρ(B);(5)
A⊆B implies Apr¯ρ(A)⊆Apr¯ρ(B);(6)
Apr_ρ(A∪B)⊇Apr_ρ(A)∪Apr_ρ(B);(7)
Apr¯ρ(A∩B)⊆Apr¯ρ(A)∩Apr¯ρ(B);(8)
ρ⊆φ implies Apr_ρ(A)⊇Apr_φ(A);(9)
ρ⊆φ implies Apr¯ρ(A)⊆Apr¯φ(A).Proof.
The proof of this theorem is similar to [13, Theorem  2.1].Theorem 13.
Letρ be a regular relation on a ternary semihypergroup S and let A, B and C be nonempty subsets of S. Then(1)
Apr¯ρ(A)·Apr¯ρ(B)·Apr¯ρ(C)⊆Apr¯ρ(A·B·C),(2)
Ifρ is complete, then Apr_ρ(A)·Apr_ρ(B)·Apr_ρ(C)⊆Apr_ρ(A·B·C).Proof.
(1) Letx∈Apr¯ρ(A)·Apr¯ρ(B)·Apr¯ρ(C). Then x∈a·b·c for a∈Apr¯ρ(A), b∈Apr¯ρ(B) and c∈Apr¯ρ(C). There exist r,s,t∈S such that r∈[a]ρ∩A, s∈[b]ρ∩B and t∈[c]ρ∩C. Since ρ is regular, it follows that
(10)r⋅s⋅t⊆[a]ρ⋅[b]ρ⋅[c]ρ⊆[a⋅b⋅c]ρ.
On the other hand, since r·s·t⊆A·B·C, we have
(11)r⋅s⋅t⊆[a⋅b⋅c]ρ∩A⋅B⋅C,
and so x∈a·b·c⊆Apr¯ρ(A·B·C). This shows that Apr¯ρ(A)·Apr¯ρ(B)·Apr¯ρ(C)⊆Apr¯ρ(A·B·C).
(2) Letx∈Apr_ρ(A)·Apr_ρ(B)·Apr_ρ(C). Then x∈a·b·c for a∈Apr_ρ(A), b∈Apr_ρ(B) and c∈Apr_ρ(C).
It follows that[a]ρ⊆A, [b]ρ⊆B and [c]ρ⊆C. Since ρ is complete, we have
(12)[a]ρ⋅[b]ρ⋅[c]ρ=[a⋅b⋅c]ρ⊆A⋅B⋅C,
and so x∈a·b·c⊆Apr_ρ(A·B·C). Hence Apr_ρ(A)·Apr_ρ(B)·Apr_ρ(C)⊆Apr_ρ(A·B·C).Definition 14.
LetS be a ternary semihypergroup. A nonempty subset A of S is called a subsemihypergroup of S if A·A·A⊆A.Letρ 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 Apr¯ρ(A)(Apr_ρ(A)) is a subsemihypergroup of S.Theorem 15.
Letρ be a regular relation on ternary semihypergroup S and let A be a subsemihypergroup of S. Then(1)
Apr¯ρ(A) is a subsemihypergroup of S,(2)
ifρ is complete, then Apr_ρ(A) is, if it is nonempty, a subsemihypergroup of S.Proof.
(1) LetA be a subsemihypergroup of S. Now by Theorem 13(1),
(13)Apr¯ρ(A)⋅Apr¯ρ(A)⋅Apr¯ρ(A)⊆Apr¯ρ(A⋅A⋅A)⊆Apr¯ρ(A).
This shows that Apr¯ρ(A) is a subsemihypergroup of S, that is, A is a ρ-upper rough subsemihypergroup of S.
(2) LetA be a subsemihypergroup of S. Now by Theorem 13(2)
(14)Apr_ρ(A)⋅Apr_ρ(A)⋅Apr_ρ(A)⊆Apr_ρ(A⋅A⋅A)⊆Apr_ρ(A).
This shows that Apr_ρ(A) is a subsemihypergroup of S, that is, A is a ρ-lower rough subsemihypergroup of S.Definition 16.
A nonempty subsetA of a ternary semihypergroup S is called left (right, lateral) hyperideal of S if S·S·A⊆A(A·S·S⊆A, S·A·S⊆A).A nonempty subsetA 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 A≠S.Letρ 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 Apr¯ρ(A)  (Apr_ρ(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.
Letρ be a regular relation on a ternary semihypergroup S and let A be a left (right, lateral) hyperideal of S. Then(1)
Apr¯ρ(A) is a left (right, lateral) hyperideal of S,(2)
ifρ is complete, then Apr_ρ(A) is, if it is nonempty, a left (right, lateral) hyperideal of S.Proof.
(1)  LetA be a right hyperideal of S. Now by Theorem 13(1),
(15)Apr¯ρ(A)⋅S⋅S=Apr¯ρ(A)⋅Apr¯ρ(S)⋅Apr¯ρ(S)⊆Apr¯ρ(A⋅S⋅S)⊆Apr¯ρ(A).
This shows that Apr¯ρ(A) is a right hyperideal of S, that is, A is a ρ-upper rough right hyperideal of S.
(2) LetA be a right hyperideal of S. Now by Theorem 13(2),
(16)Apr_ρ(A)⋅S⋅S=Apr_ρ(A)⋅Apr_ρ(S)⋅Apr_ρ(S)⊆Apr_ρ(A⋅S⋅S)⊆Apr_ρ(A).
This shows that Apr_ρ(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.
Letρ 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)
Apr¯ρ(A·B·C)⊆Apr¯ρ(A)∩Apr¯ρ(B)∩Apr¯ρ(C),(2)
Apr_ρ(A·B·C)⊆Apr_ρ(A)∩Apr_ρ(B)∩Apr_ρ(C).Proof.
SinceA is a right hyperideal of S, so A·B·C⊆A·S·S⊆A. Since B is a lateral hyperideal of S, so A·B·C⊆S·B·S⊆B, also C is a left hyperideal of S, so A·B·C⊆S·S·C⊆C, thus A·B·C⊆A∩B∩C. Then by Theorem 12(7), we have
(17)Apr¯ρ(A⋅B⋅C)⊆Apr¯ρ(A∩B∩C)⊆Apr¯ρ(A)∩Apr¯ρ(B)∩Apr¯ρ(C).
Also by Theorem 12(3), we have
(18)Apr_ρ(A⋅B⋅C)⊆Apr_ρ(A∩B∩C)=Apr_ρ(A)∩Apr_ρ(B)∩Apr_ρ(C).
This completes the proof.Definition 19.
A subsemihypergroupB of a ternary semihypergroup S is called a bi-hyperideal of S if B·S·B·S·B⊆B.Letρ 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 Apr¯ρ(A)(Apr_ρ(A)) is a bi-hyperideal of S.Theorem 20.
Letρ be a regular relation on a ternary semihypergroup S and let B be a bi-hyperideal of S. Then(1)
Apr¯ρ(B) is a bi-hyperideal of S,(2)
ifρ is complete, then Apr_ρ(B) is, if it is nonempty, a bi-hyperideal  of S.Proof.
(1) LetB be a bi-hyperideal of S. Now by Theorem 13(1)
(19)Apr¯ρ(B)⋅S⋅Apr¯ρ(B)⋅S⋅Apr¯ρ(B)=Apr¯ρ(B)⋅Apr¯ρ(S)⋅Apr¯ρ(B)⋅Apr¯ρ(S)⋅Apr¯ρ(B)⊆Apr¯ρ(B⋅S⋅B⋅S⋅B)⊆Apr¯ρ(B).
From this and Theorem 15(1), Apr¯ρ(B) is a bi-hyperideal of S, that is, A is a ρ-upper rough bi-hyperideal of S.
(2) LetB be a bi-hyperideal of S. Now by Theorem 13(2)
(20)Apr_ρ(B)⋅S⋅Apr_ρ(B)⋅S⋅Apr_ρ(B)=Apr_ρ(B)⋅Apr_ρ(S)⋅Apr_ρ(B)⋅Apr_ρ(S)⋅Apr_ρ(B)⊆Apr_ρ(B⋅S⋅B⋅S⋅B)⊆Apr_ρ(B).
From this and Theorem15(2), Apr_ρ(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.
LetH={0,a,b,c,d,e,g} and f(x,y,z)=(x*y)*z for all x,y,z∈H, 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, Apr¯ρ(A)={0,a,b,c,d,e,g} and Apr_ρ(A)={0}. It is clear that Apr¯ρ(A) and Apr_ρ(A) are bi-hyperideals of S, but the subsemihypergroup {0,e,g} of S is not a bi-hyperideal of S.Table 1
*
0
a
b
c
d
e
g
0
0
0
0
0
0
0
0
a
0
a
{
a
,
b
}
c
{
c
,
d
}
e
{
e
,
g
}
b
0
b
b
d
d
g
g
c
0
c
{
c
,
d
}
c
{
c
,
d
}
c
{
c
,
d
}
d
0
d
d
d
d
d
d
e
0
e
{
e
,
g
}
c
{
c
,
d
}
e
{
e
,
g
}
g
0
g
g
d
d
g
gDefinition 22.
A subsetQ of a ternary semihypergroup S is called a quasi-hyperideal of S if
(21)Q⋅S⋅S∩S⋅Q⋅S∩S⋅S⋅Q⊆Q,Q⋅S⋅S∩S⋅S⋅Q⋅S⋅S∩S⋅S⋅Q⊆Q.
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 Apr¯ρ(A)(Apr_ρ(A)) is a quasi-hyperideal of S.Theorem 23.
Letρ be a complete regular relation on a ternary semihypergroup S and let Q be a bi-hyperideal of S. Then Apr_ρ(Q) is, if it is nonempty, a quasi- hyperideal  of S.Proof.
LetQ be a quasi-hyperideal of S. Now by Theorems 13(2) and 12(3)
(22)Apr_ρ(Q)⋅S⋅S∩S⋅Apr_ρ(Q)⋅S∩S⋅S⋅Apr_ρ(Q)=Apr_ρ(Q)⋅Apr_ρ(S)⋅Apr_ρ(S)∩Apr_ρ(S)⋅Apr_ρ(Q)⋅Apr_ρ(S)∩Apr_ρ(S)⋅Apr_ρ(S)⋅Apr_ρ(Q)⊆Apr_ρ(Q⋅S⋅S)∩Apr_ρ(S⋅Q⋅S)∩Apr_ρ(S⋅S⋅Q)=Apr_ρ(Q⋅S⋅S∩S⋅Q⋅S∩S⋅S⋅Q)⊆Apr_ρ(Q).
Also we can show that
(23)Apr_ρ(Q)⋅S⋅S∩S⋅S⋅Apr_ρ(Q)⋅S⋅S∩S⋅S⋅Apr_ρ(Q)⊆Apr_ρ(Q).
Hence Apr_ρ(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

Letρ 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:
(24)Apr¯¯ρ(A)={[x]ρ∈Sρ:[x]ρ∩A≠∅},Apr__ρ(A)={[x]ρ∈Sρ:[x]ρ⊆A},
respectively.Theorem 24.
Letρ be a regular relation on a ternary semihypergroup S and let A be a subsemihypergroup of S. Then(1)
Apr¯¯ρ(A) is a subsemihypergroup of S/ρ,(2)
ifρ is complete, then Apr__ρ(A) is, if it is nonempty, a subsemihypergroup of S/ρ.Proof.
(1) Let[a]ρ,[b]ρ,[c]ρ∈Apr¯¯ρ(A). Then [a]ρ∩A≠∅, [b]ρ∩A≠∅ and [c]ρ∩A≠∅. So there exist x∈[a]ρ∩A, y∈[b]ρ∩A and z∈[c]ρ∩A. Since A is a subsemihypergroup of S, we have x·y·z⊆A. By Lemma 11, we have
(25)x⋅y⋅z⊆[a]ρ⋅[b]ρ⋅[c]ρ⊆[a⋅b⋅c]ρ.
Thus [a·b·c]ρ∩A≠∅, which implies that [a]ρ·[b]ρ·[c]ρ⊆Apr¯¯ρ(A). Hence Apr¯¯ρ(A) is a subsemihypergroup of S/ρ.
(2) Let[a]ρ,[b]ρ,[c]ρ∈Apr__ρ(A). Then [a]ρ⊆A, [b]ρ⊆A and [c]ρ⊆A. Since A is a subsemihypergroup of S, we have
(26)[a]ρ⋅[b]ρ⋅[c]ρ=[a⋅b⋅c]ρ⊆A⋅A⋅A⊆A.
This means that Apr__ρ(A) is a subsemihypergroup of S/ρ.Theorem 25.
Letρ be a regular relation on a ternary semihypergroup S and let A be a left (right, lateral) hyperideal of S. Then(1)
Apr¯¯ρ(A) is a left (right, lateral) hyperideal of S/ρ,(2)
ifρ is complete, then Apr__ρ(A) is, if it is nonempty, a left (right, lateral) hyperideal of S/ρ.Proof.
(1) LetA be a left hyperideal of S. Let [x]ρ∈Apr¯¯ρ(A) and [y]ρ,[z]ρ∈S/ρ. Then [x]ρ∩A≠∅, hence x∈Apr¯ρ(A). Since A is a left hyperideal of S, by Theorem 17(1), Apr¯ρ(A) is a left hyperideal of S. So, we have
(27)y⋅z⋅x⊆Apr¯ρ(A).
Now, for every m∈y·z·x, we have [m]ρ∩A≠∅. On the other hand, from m∈y·z·x, we obtain [m]ρ⊆[y]ρ·[z]ρ·[x]ρ. Therefore [y]ρ·[z]ρ·[x]ρ⊆Apr¯¯ρ(A). This means that Apr¯¯ρ(A) is a left hyperideal of S/ρ.
(2) LetA be a left hyperideal of S. Let [x]ρ∈Apr¯¯ρ(A) and [y]ρ,[z]ρ∈S/ρ. Then, [x]ρ⊆A, which implies x∈Apr_ρ(A). Since A is a left hyperideal of S, by Theorem 17(2), Apr_ρ(A) is a left hyperideal of S. Thus, we have
(28)y⋅z⋅x⊆Apr_ρ(A).
Now, for every m∈y·z·x, we have m∈Apr_ρ(A), which implies that [m]ρ⊆A. Hence, [m]ρ∈Apr__ρ(A). On the other hand, from m∈y·z·x, we have [m]ρ⊆[y]ρ·[z]ρ·[x]ρ. Therefore [y]ρ·[z]ρ·[x]ρ⊆Apr__ρ(A). This means that Apr__ρ(A) is, if it is nonempty, a left hyperideal of S/ρ. The other cases can be seen in a similar way.Theorem 26.
Letρ be a regular relation on a ternary semihypergroup S and let A be a bi-hyperideal of S. Then(1)
Apr¯¯ρ(A) is a bi-hyperideal of S/ρ,(2)
ifρ is complete, then Apr__ρ(A) is, if it is nonempty, a bi-hyperideal of S/ρ.Proof.
(1) LetA be a bi-hyperideal of S. Let [x]ρ,[y]ρ,[z]ρ∈Apr¯¯ρ(A) and [s]ρ,[t]ρ∈S/ρ. Then,
(29)[x]ρ∩A≠∅,[y]ρ∩A≠∅,[z]ρ∩A≠∅.
Hence, x∈Apr¯ρ(A),y∈Apr¯ρ(A) and z∈Apr¯ρ(A). By Theorem 20(1), Apr¯ρ(A) is a bi-hyperideal of S. So, we have
(30)x⋅s⋅y⋅t⋅z⊆Apr¯ρ(A).
Now, for every m∈x·s·y·t·z, we obtain
(31)[m]ρ⊆[x]ρ⋅[s]ρ⋅[y]ρ⋅[t]ρ⋅[z]ρ.
On the other hand, since m∈Apr¯ρ(A), we have [m]ρ∩A≠∅. Thus,
(32)[x]ρ⋅[s]ρ⋅[y]ρ⋅[t]ρ⋅[z]ρ⊆Apr¯¯ρ(A).
Therefore, from this and Theorem 24(1), Apr¯¯ρ(A) is a bi-hyperideal of S/ρ.
(2) LetA be a bi-hyperideal of S. Let [x]ρ,[y]ρ,[z]ρ∈Apr¯¯ρ(A) and [s]ρ,[t]ρ∈S/ρ. Then,
(33)[x]ρ⊆A,[y]ρ⊆A,[z]ρ⊆A.
Hence, x∈Apr_ρ(A), y∈Apr_ρ(A), and z∈Apr_ρ(A). By Theorem 20(2), Apr_ρ(A) is a bi-hyperideal of S. So, we have
(34)x⋅s⋅y⋅t⋅z⊆Apr_ρ(A).
Then, for every m∈x·s·y·t·z, we obtain
(35)[m]ρ∈[x]ρ⋅[s]ρ⋅[y]ρ⋅[t]ρ⋅[z]ρ.
On the other hand, since m∈Apr_ρ(A), we have [m]ρ⊆A. So,
(36)[x]ρ⋅[s]ρ⋅[y]ρ⋅[t]ρ⋅[z]ρ⊆Apr__ρ(A).
Therefore from this and Theorem 24(2), Apr__ρ(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.
LetS be a ternary semihypergroup. A fuzzy subset f of S is called(1)
a fuzzy ternary subsemihypergroup ofS if inft∈xyzf(t)≥min{f(x),f(y),f(z)} for all x,y,z∈S,(2)
a fuzzy left hyperideal ofS if inft∈xyzf(t)≥f(z) for all x,y,z∈S,(3)
a fuzzy right hyperideal ofS if inft∈xyzf(t)≥f(x) for all x,y,z∈S,(4)
a fuzzy lateral hyperideal ofS if inft∈xyzf(t)≥f(y) for all x,y,z∈S,(5)
a fuzzy hyperideal ofS if inft∈xyzf(t)≥max{f(x),f(y),f(z)} for all x,y,z∈S.Theorem 28.
LetS 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 thatA is a ternary subsemihypergroup of S. Let x,y,z∈S.
Case1.x,y,z∈A. Since A is a ternary subsemihypergroup of S, we have xyz⊆A. Then inft∈xyzfA(t)=1≥min{f(x),f(y),f(z)}.
Case2.x∉A or y∉A or z∉A. Thus fA(x)=0 or fA(y)=0 or fA(z)=0. Therefore min{f(x),f(y),f(z)}=0≤inft∈xyzfA(t).
Conversely, letx,y,z∈A. We have fA(x)=fA(y)=fA(z)=1. Since fA is a fuzzy ternary subsemihypergroup of S, inft∈xyzfA(t)≥min{f(x),f(y),f(z)}=1. Hence xyz⊆A.
(2) Let us assume thatA is a left hyperideal of S. Let x,y,z∈S.
Case1.z∈A. Since A is a left hyperideal of S, then xyz⊆A. Then inft∈xyzfA(t)=1. Therefore inft∈xyzfA(t)≥fA(z).
Case2.z∉A. We have fA(z)=0. Hence inft∈xyzfA(t)≥fA(z).
Conversely, letx,y∈S and z∈A. Since fA is a fuzzy left hyperideal of S and z∈A,inft∈xyzfA(t)≥fA(z)=1. Thus xyz⊆A.
The remaing parts can be seen in similarly way.LetS be a ternary semihypergroup. A nonempty subset T of S is called prime subset of S if for all x,y,z∈S,xyz⊆T implies x∈T or y∈T or z∈T. 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 subsetof S if inft∈xyzf(t)≤max{f(x),f(y),f(z)} for all x,y,z∈S. 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.
LetS 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 thatA is a prime subset of S. Let x,y,z∈S.
Case1.xyz⊆A. Since A is prime, x∈A or y∈A or z∈A. Thus max{fA(x),fA(y),fA(z)}=1≥inft∈xyzfA(t).
Case  2.xyz⊈A. Thus inft∈xyzfA(t)=0≤max{fA(x),fA(y),fA(z)}.
Conversely, letx,y,z∈S such that xyz⊆A. Thus fA(t)=1 for all t∈xyz. 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 x∈A or y∈A or z∈A.
(2) It follows from (1) and Theorem28.Letf be a fuzzy subset of a set (a ternary semihypergroup) S. For any t∈[0,1], the set
(37)ft={x∈S∣f(x)≥t},fts={x∈S∣f(x)>t}
are called a t—level set and a t—stronglevel set of f, respectively.Theorem 30.
Letf 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 thatf is a fuzzy ternary subsemihypergroup of S. Let t∈[0,1] such that ft≠∅. Let x,y,z∈ft. 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, infh∈xyzf(h)≥t. Hence, xyz⊆ft. Conversely, let x,y,z∈S. Let we take t=min{f(x),f(y),f(z)}. Then f(x),f(y),f(z)≥t. Thus, x,y,z∈ft. Since ft is a ternary subsemihypergroup of S, xyz⊆ft. Thus infh∈xyzf(h)≥t=min{f(x),f(y),f(z)}.
(2) Let us assume thatf is a fuzzy left hyperideal of S. Let t∈[0,1]. Let us suppose that ft≠∅. Let x,y∈S and z∈ft. Thus infh∈xyzf(h)≥f(z)≥t. Therefore, xyz⊆ft.
Conversely, letx,y,z∈S. Let we take t=f(z). Thus z∈ft, this implies ft≠∅. By assumption, we have ft is a left hyperideal of S. So xyz⊆ft. Therefore, infh∈xyzf(h)≥t. Thus infh∈xyzf(h)≥f(z).
The remain parts can be proved in a similar way.Theorem 31.
LetS 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 thatf is a prime fuzzy subset of S. Let t∈[0,1]. Let us suppose that ft≠∅. Let x,y,z∈S such that xyz⊆ft. Thus infh∈xyzf(h)≥t. Since f is prime, f(x)≥t or f(y)≥t or f(z)≥t. This implies x∈ft or y∈ft or z∈ft.
Conversely, letx,y,z∈S. Let we take t=infh∈xyzf(h). Then xyz⊆ft. Since ft is prime, x∈ft or y∈ft or z∈ft. Then f(x)≥t or f(y)≥t or f(z)≥t. Hence max{f(x),f(y),f(z)}≥t=infh∈xyzf(h).
(2) It follows from (1) and Theorem29.Theorem 32.
LetS 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 fts≠∅, then fts 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 Theorem30.


## 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.LetS be a ternary semihypergroup and f be a fuzzy subset of S. Then the sets
(38)Apr¯ρ(f)(x)=sup⁡a∈[x]ρf(a),Apr_ρ(f)(x)=inf⁡a∈[x]ρf(a)
are called the ρ-upper and ρ-lower approximations of a fuzzy set f, respectively.Lemma 33.
LetS be a ternary semihypergroup, ρ a regular relation on S, f a fuzzy subset of S and t∈[0,1], then(1)
(Apr_ρ(f))t=Apr_ρ(ft),(2)
(Apr¯ρ(f))ts=Apr¯ρ(fts).Proof.
(1) Letx∈(Apr_ρ(f))t. Then Apr_ρ(f)(x)≥t. So infa∈[x]ρf(a)≥t. Therefore, f(a)≥t for all a∈[x]ρ. This implies [x]ρ⊆ft. Therefore, x∈Apr_ρ(ft).
Conversely, let us assume thatx∈Apr_ρ(ft). Thus [x]ρ⊆ft. Then f(a)≥t for all a∈[x]ρ. This implies infa∈[x]ρf(a)≥t. Thus, Apr_ρ(f)(x)≥t. Hence x∈(Apr_ρ(f))t.
(2) Letx∈(Apr¯ρ(f))ts. Then Apr¯ρ(f)(x)>t. So supa∈[x]ρf(a)>t. Therefore, f(a)>t for some a∈[x]ρ. This implies [x]ρ∩fts≠∅. Therefore x∈Apr¯ρ(fts).
Conversely, let us assumex∈Apr¯ρ(fts). Thus [x]ρ∩fts≠∅. Then f(a)>t for some a∈[x]ρ. This implies infa∈[x]ρf(a)>t. Thus Apr¯ρ(f)(x)>t. Hence, x∈(Apr¯ρ(f))ts.Theorem 34.
LetS 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 Apr¯ρ(f) and Apr_ρ(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 Theorems29, 31, 15, and 17 and Lemma 33.


## 7. Fuzzy Bi-Hyperideals of Ternary Semihypergroups

LetS be a ternary semihypergroup. A fuzzy subset f of S is called a fuzzy bi-hyperideal of S if infh∈xyzf(h)≥min{f(x),f(y),f(z)} and infh∈xyzpqf(h)≥min{f(x),f(z),f(q)} for all x,y,z,p,q∈S.Theorem 35.
LetS 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 thatA is a bi-hyperideal of S. Let a,b,x,y,z∈S.
Case1.x,y,z∈A. Since A is a bi-hyperideal of S, then xy,  xaybz⊆A. Therefore infh∈xyzfA(h)=1≥min{fA(x),fA(y),fA(z)} and infh∈xaybzfA(h)=1≥min{fA(x),fA(y),fA(z)}.
Case  2.x∉A or y∉A or z∉A. Thus fA(x)=0 or fA(y)=0 or fA(z)=0. Hence min{fA(x),fA(y),fA(z)}=0≤infh∈xyzfA(h) and min{fA(x),fA(y),fA(z)}=0≤infh∈xaybzfA(h).LetS 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.
LetS 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 Theorems29 and 35.Theorem 37.
LetS 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 thatf is a fuzzy bi-hyperideal of S. Let t∈[0,1] such that ft≠∅. Let x,y,z∈ft. Then f(x)≥t,f(y)≥t and f(z)≥t. Since f is a fuzzy bi-hyperideal of S, infh∈xyzf(h)≥min{f(x),f(y),f(z)}≥t, and infh∈xaybzf(h)≥min{f(x),f(y),f(z)}≥t for all a,b∈S. Therefore, xyz, xaybz⊆ft. Hence ft is a bi-hyperideal of S.
Conversely, let us assume for allt∈[0,1], if ft≠∅, then ft is a bi-hyperideal of S. Let a,b,x,y,z∈S. Let we take t=min{f(x),f(y),f(z)}. Then x,y,z∈ft. This implies that ft≠∅. By assumption, we have ft is a bi-hyperideal of S. So xyz, xaybz⊆ft. Therefore, infh∈xyzf(h)≥t and infh∈xaybzf(h)≥t. Hence infh∈xyzf(h)≥min{f(x),f(y),f(z)} and infh∈xaybzf(h)≥min{f(x),f(y),f(z)}.Theorem 38.
LetS 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 Theorems31 and 37.Theorem 39.
LetS 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 fts≠∅, then fts is a bi-hyperideal of S.Proof.
The proof is similar to the proof of Theorem37.Theorem 40.
LetS 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 fts≠∅, then fts is a prime bi-hyperideal of S.Proof.
It follows from Theorems31 and 39.


## 8. Rough Fuzzy Bi-Hyperideals of Ternary Semihypergroups

Theorem 41.
LetS be a ternary semihypergroup and ρ be a complete regular relation on S. If f is a fuzzy bi-hyperideal of S, then Apr¯ρ(f) and Apr_ρ(f) are fuzzy bi-hyperideals.Proof.
This proof follows from Theorems37, 39, and 20, and Lemma 33.Note that ifApr¯ρ(f) and Apr_ρ(f) are fuzzy bi-hyperideals of a ternary semihypergroup S, in general, f need not be a fuzzy bi-hyperideal of S.



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