(Γ, N )-SEMIHYPERGROUPS AND T -FUNCTOR

doi:10.7151/dmgaa.1231

SEMIGROUPS DERIVED FROM

(Γ, N )-SEMIHYPERGROUPS AND T -FUNCTOR

S. Ostadhadi-Dehkordi

Department of Mathematics, Hormozgan University Bandar Abbas, Iran

e-mail: Ostadhadi@hormozgan.ac.ir

Abstract

The main purpose of this paper is to introduce the concept of (Γ, n)- semihypergroups as a generalization of hypergroups, as a generalization of n- ary hypergroups and obtain an exact covariant functor between the category (Γ, n)-semihypergrous and the category semigroups. Moreover, we introduce and study complete part. Finally, we obtain some new results and some fundamental theorems in this respect.

Keywords: (Γ, n)-semihypergroup, Θ-relation, T -fuctor, fundamental semi- group.

2010 Mathematics Subject Classification: 20N15.

1. Introduction

Algebraic hyperstructures are a suitable generalization of classical algebraic struc- tures. 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. The hypergroup notion was introduced in 1934 by the French mathe- matician F. Marty [16], at the 8

Congress of Scandinavian Mathematicians. He published some notes on hypergroups, using them in different contexts: algebraic functions, rational fractions, non commutative groups. Since then, hundreds of papers and several books have been written on this topic and several kinds of hy- pergroups have been intensively studied, such as: regular hypergroups, reversible regular hypergroups, canonical hypergroups, cogroups, cyclic hypergroups, asso- ciativity hypergroups.

A recent book on hyperstructures [4] points out on their applications in fuzzy

and rough set theory, cryptography, codes, automata, probability, geometry, lat-

tices, binary relations, graphs and hypergraphs. Hypergraph theory is a useful

toll for discrete optimization problems. A comprehensive review of the theory of hypergraph appears in [2].

Let H be a nonempty set and ◦ : H ×H −→ ℘

(H), be a map such that ℘

(H) be the set of all nonempty subset of H. The couple (H, ◦) is called hypergroupoid.

If A and B are nonempty subset of H, then we define A ◦ B = [

a∈A,b∈B

a ◦ b, x ◦ A = {x} ◦ A, A ◦ x = A ◦ {x}.

A hypergroupoid (H, ◦) is called a semihypergroup if for all x, y, z ∈ H, we have x ◦ (y ◦ z) = (x ◦ y) ◦ z. A semihypergroup (H, ◦) is called hypergroup if for every x ∈ H, we have x ◦ H = H ◦ x = H. Several books have been written on hyperstructure theory, see [3, 4, 7]. A regular hypergroup (H, ◦) is a hypergroup which has at least an identity and any element of H has at least an inverse. In other words, there exists e ∈ H, such that for all x ∈ H, we have x ∈ x ◦ e ∩ e ◦ x and there exists x

∈ H such that e ∈ x ◦ x

∩ x

◦ x.

An n-ary structure generalizations of algebraic structures is the most natu- ral way for further development and deeper understanding of their fundamental properties. The notion of n-ary group, which is a generalization of the notion of a group, was introduced by W. D¨ ornte in 1928 [10]. Since then many papers concerning various n-ary algebra have appeared in the literature [8, 9, 11, 12].

The notion of Γ-semigroups was introduced by Sen in [17, 18]. Let G and Γ be two nonempty sets. Then, G is called a Γ -semigroup if there exists a mapping G × Γ × G −→ G, written (a, α, b) by aαb, such that it satisfies the identities aα(bβc) = (aαb)βc, for all a, b, c ∈ G and α, β ∈ Γ. The concept of Γ-semihypergroups was introduced by Davvaz et al. [13]. Let G and Γ be two nonempty sets. Then, G is called a Γ-semihypergroup if each α ∈ Γ be a hyperoperation on G, i.e., aαb ⊆ G, and for every a, b, c ∈ G, and for every α, β ∈ Γ we have the associative property that is aα(bβc) = (aαb)βc. Let G

be a Γ

-semihypergroup and G

be a Γ

-semigroup. If there exists a map ϕ : G

−→

G

and a bijection f : Γ

−→ Γ

such that ϕ(xαy) ⊆ ϕ(x)f (α)ϕ(y), for every x, y ∈ G

and α ∈ Γ

, then ϕ a homomorphism between G

and G

.

In 1964, Nobusawa introduced Γ-rings as a generalization of rings. Barnes [1] weakened slightly the conditions in the definition of Γ-ring in the sense of Nobusawa. Barnes [1], Luh [15] and Kyuno [14] studied the structure of Γ-rings and obtained various generalization analogous to corresponding parts in ring theory. After that, Dehkordi et. al. [5, 6] investigated the ideals, rough ideals, homomorphisms and regular relations of Γ-semihyperrings.

The aim of this research work is to define a new class of n-ary multialgebras

that we call (Γ, n)-semihypergroups that is a generalization of n-ary semihyper-

groups, a generalization of Γ-semihypergroups, a generalization of semihyper-

group and a generalization of semigroups. Also, we define complete part and

regular relation. Moreover, we introduce an exact covariant functor between the category (Γ, n)-semihypergroups and the category semigroups.

2. (Γ,n)-Semihypergroup

In this section, we present some definitions and results concerning. First of all, let us introduced (Γ, n)-semihypergroup. Let G, Γ be nonempty sets and n ∈ N, n ≥ 2. A map α : G

−→ ℘

(G) is called n-ary hyperoperation on G, where

℘

(G) is the set of all nonempty subsets of G and α ∈ Γ. Then, (G, Γ) is called (Γ, n)-hypergroupoid. If G

, G

, . . . , G

are subsets of G, then we define

α(G

, G

, . . . , G

) = [

{α(x

, x

, . . . , x

) : x

∈ G

, 1 ≤ i ≤ n}, Γ(G

, G

, . . . , G

) = [

{α(x

, x

, . . . , x

) : x

∈ G

, α ∈ Γ, 1 ≤ i ≤ n}.

The sequence x

, x

, . . . , x

, will be denoted by x

. For j ≤ i, x

is empty. In the case when x

= · · · = x

= x will be written be written in the form x

.

A (Γ, n)-hypergroupoid is called (Γ, n)-semihypergroup if for every α, β ∈ Γ and x

, x

, . . . , x

∈ G

α

 x

, β



x

 , x



= β

 x

, α

 x

 , x

 . A (Γ, n)-hypergroupoid (G, Γ) in which for every α ∈ Γ the equation

y ∈ α y

, x

, y

,

has the solution x

∈ G for every y

, y

, y ∈ G is called (Γ, n)- quasihyper- group. A (Γ, n)-hypergroup is both a (Γ, n)-semihypergroup and (Γ, n)- quasihy- pergroup. A (Γ, n)-hypergroup G is commutative if for every x

of G and any permutation δ of {1, 2, . . . , n} and for all α ∈ Γ we have

α(x

) = α(x

, x

, . . . , x

).

An element e of a (Γ, n)-hypergroup G is called an n-ary identity or a neutral element, if there exist α ∈ Γ such that

x = α(e

, x, e

).

Let G be a (Γ, n)-semihypergroup and α ∈ Γ be a fixed element. We define

f (a

, a

, . . . , a

) = α(a

, a

, . . . , a

). It is easy to see that (G, f ) is an n-ary

semihypergroup and when n = 2, (G, f ) is a semihypergroup. We denote this

n-ary semihypergroup by G[α].

Proposition 2.1. Let G be a (Γ, n)-semihypergroup and for every α ∈ Γ, the element e ∈ G be neutral element. Then, for every α

, α

∈ Γ and x

∈ G, we have α

(x

) = α

(x

).

Proof. Suppose that e is a neutral element for every α ∈ Γ. Then for x

∈ G, we have x

= α(x

, e

) and x

= β(x, e

). Hence

α(x

, x

) = α(β(x

, e

), x

) = β(x

, α(x

, e

), x

) = β(x

, x

, x

).

This completes the proof.

By Proposition 2.1, if for every α, β ∈ Γ, e is a neutral element, then G[α] = G[β].

This implies that (Γ, n)-semihypergroup G is an n-ary hypergroup.

Definition 2.2. Let (G, Γ) be a (Γ, n)-hypergroup and H be a nonempty subset of G. We say that H is a (Γ, n)-subhypergroup of G if following conditions hold:

1. For every α ∈ Γ, H is closed under the n-ary hyperoperation α,

2. For all x

, x

, . . . , x

∈ H, α ∈ Γ and fixed i ∈ {1, 2, . . . , n} there exists x ∈ H such that x

∈ α(x

, x, x

).

Definition 2.3. A nonempty subset I of a (Γ, n)-semihypergroup is said to be a k-ideal of G if

1. I is a (Γ, n)-subsemihypergroup of G, 2. Γ(G

, I, G

) ⊆ I.

If for every 1 ≤ k ≤ n, I is a k-ideal, then we say that I is an ideal.

Definition 2.4. Let G be a semigroup and I be a nonempty subset of G. We say that I is a left ideal if I is a subsemigroup of G and GI ⊆ I. In the same way can define right ideal.

Definition 2.5. Let G

and G

be (Γ

, n) and (Γ

, n)-semihypergroup, respec- tively. A map (ϕ, f ) : G

× Γ

−→ G

× Γ

is called a homomorphism if for every x

∈ G

ϕ(α(x

)) = f (α)(ϕ(x

)).

Also, if ϕ and f are onto, then (ϕ, f ) is called an epimorphism.

Example 1. Let G be a group and Γ = {α

: n ∈ N}. Then, for every x

∈ G, we define

α

(x

) = G.

Then, G is a (Γ, n)-hypergroup.

Example 2. Let X be a totally ordered set and Γ be a nonempty subset of X.

We define

α(x

) = {x ∈ G : x ≥ max{x

, α}},

for every α ∈ Γ and x

∈ X. Then, (X, Γ) is a (Γ, n)-semihypergroup.

Example 3. Let H be a semigroup and {X

}

be a collection of disjoint sets.

Consider G = S

h∈H

X

and Γ = Z(H). For every g ∈ G there exist h ∈ H such that g ∈ X

. We define

α(x

) = X

,

where x

∈ X

, for 1 ≤ i ≤ n. Then G is a (Γ, n)-hypergroup and is called (Z(H), n)-hypergroup.

Example 4. Let A

= [n, n+1), Γ

= 2Z, Γ

= 2Z+1 and G

, G

be (Z(2Z), n), (Z(2Z + 1), n)-semihyperring, respectively. Then, (ϕ, f ) is a homomorphism.

ϕ : G

−→ G

, ϕ(x) = x + 1 f : γ

−→ Γ

, f (α) = α.

Example 5. Let (H, ◦) be a hypergroup and Γ ⊆ H be a nonempty set. We define for every x

∈ H and α ∈ Γ

α(x

) = α ◦ x

◦ · · · ◦ x

. Then H is a (Γ, n)-semihypergroup.

Example 6. Let G be a group and H

be a normal subgroups of G such that H

⊆ H

. We define n-ary hyperoperation on G as follows:

α

(x

) = H

◦ x

◦ x

, . . . , ◦x

. Then G is a (Γ, n)-hypergroup.

3. Fundamental relation and complete part

By using a certain type of equivalence relations, we can connect (Γ, n)-semi-

hypergroup to semigroups and (Γ, n)-hypergroups to groups. These equivalence

relations are called strong regular relations. More exactly, starting with a (Γ, n)-

semihypergroup (hypergroup) and using a strong regular relation, we can con-

struct semigroup (group). In this section, we introduce a strong regular relation

β

and complete part such that has an important role in the study of (Γ, n)-

semihypergroups.

Let G be a (Γ, n)-hypergroup. We define α

= {α

(x

) : x

∈ G, 1 ≤ i ≤ n}

α

= α

x

, α

, x

, x

∈ G, 2 ≤ i ≤ n α

= α

x

, α

, x

, x

∈ G, 2 ≤ i ≤ n

.. .

α

= α

x

, α

, x

, x

∈ G, 2 ≤ i ≤ n , for every α

, α

, . . . , α

∈ Γ. Let U = S

k≥1,α∈Γ

U

. We define xβ

y ⇔ ∃α

∈ U , such that {x, y} ⊆ α

. We have β = S

n≥1

β

is reflexive and symmetric. Let β

be the transitive closure of β. This relation is called fundamental relation.

Let G be a (Γ, n)-semihypergroup and ρ be an equivalence relation on G. If A and B are nonempty subset of G, then

AρB ⇐⇒ ∀a ∈ A, ∃b ∈ B such that aρb

∀b ∈ B, ∃a ∈ A such that aρb.

and

AρB ⇐⇒ ∀a ∈ A, and b ∈, B, aρb.

The equivalence relation ρ is called k-regular if from aρb, it follows that α(x

, a, x

) ρ α(x

, b, x

),

for every α ∈ Γ and is called k-strongly regular if from aρb, α(x

, a, x

) ρ α(x

, b, x

).

for every α ∈ Γ. ρ is called regular (strongly regular) if it is k-regular (strongly regular) for every 1 ≤ k ≤ n.

Proposition 3.1. Let G be a (Γ, n)-semihypergroup and β

be a fundamenteal relation on G. Then, β

is the smallest strongly regular relation on G.

Proof. Suppose that aβ

b and x is an arbitrary element of G. It follows that

thee exists x

= a, x

, . . . , x

= b such that for very i ∈ {0, 1, . . . , n − 1} such

that x

βx

. Let u

∈ α(a, y

) and u

∈ α(b, y

). It follows that there exist

ξ

such that {x

, y

} ⊆ ξ

. Hence α(x

, y

) ⊆ α(ξ

, y

) and α(x

, y

)

⊆ α(ξ

, y

). Thus α(x

, y

)βα(x

, y

). This implies that for every i ∈ {0, 1, 2, . . . , n − 1} and for all t

∈ α(x

, y

), we have t

βt

. If we consider t

= u

and t

= u

, then we obtain u

β

u

. Then β

is 1-strongly regular and similarly, it is j-strongly regular for 2 ≤ j ≤ n.

Let R be a strongly regular relation on G. Since R is reflexive, we have β ⊆ R. Suppose that β

⊆ R. If aβ

b, then {a, b} ⊆ α

. Since α

= α y

, ξ

, y

there exists u, v ∈ ξ

such that a ∈ α

y

, u, y

and b ∈ α

y

, v, y

and according to the hypothesis since uβ

v, we have uRv. Since R is strongly regular it follows that aRb and by induction, it follows that β ⊆ R, whence β

⊆ R.

Proposition 3.2. Let G be a (Γ, n)-semihypergroup and ρ be an equivalence relation on G. Then, ρ is regular if and only if [G : ρ] is a (b Γ, n)- semihypergroup with respect the following operation:

α(ρ(a b

), ρ

(a

), . . . , ρ(a

)) = {ρ(a) : a ∈ α(a

, a

, . . . , a

)}.

Proof. First we check that the hyperoperation α is well defined. Let ρ(a b

) = ρ(b

), for 1 ≤ i ≤ n. Then, we have a

ρb

. Since ρ is regular, it follows that

α(a

, a

, . . . , a

) ρ α(b

, a

, . . . , a

), α(b

, a

, . . . , a

) ρ α(b b

, b

, . . . , a

),

.. .

α(b

, b

, · · ·

, a

) ρ α(b

, b

, . . . , b

).

Hence for every u

∈ α(a

, a

, . . . , a

) there exists u

∈ α(b

, b

, . . . , b

) such that ρ(u

) = ρ(u

). It follows that

α(ρ(a b

), ρ

(a

), . . . , ρ(a

)) ⊆ α(ρ(b b

), ρ

(b

), . . . , ρ(b

)),

and similarly we obtain the converse inclusion. Now, we check the associativity of n-ary hyperoperation α. Let

ρ(u) ∈ α b 

ρ(x

)

, b β(ρ(y

))

, ρ(x

)

)  .

This means that there exists ρ(v) ∈ b β(ρ(y

))

such that ρ(u) ∈ α b



ρ(x

)

, ρ(v), ρ(x

)



.

Hence there exist u

∈ α x

, v, x

such that ρ(u) = ρ(u

) and there exist v

∈ β(y

)

such that ρ(v) = ρ(v

). Since ρ is regular there exist

u

∈ α 

x

, v

, x



⊆ α 

x

, β(y

)

, x



= β



x

, α(y

)

, x

 , such that ρ(u

) = ρ(u). Hence we obtain that there exists u

∈ α(y

)

such that u

∈ β x

, u

, x

. We have

ρ(u) = ρ(u

) ∈ b β 

ρ(x

)

, α(ρ(y b

))

, ρ(x

)

)  . It follows that

α b 

ρ(x

)

, b β(ρ(y

))

, ρ(x

)

) 

⊆ b β 

ρ(x

)

, α(ρ(y b

))

, ρ(x

)

)  . Similarly, we obtain the converse inclusion.

Let [G : ρ] be a (b Γ, n)-semihypergroup, aρb and x

∈ G, for 1 ≤ i ≤ n − 1.

Since ρ is well-defined. If u ∈ α x

, a, x

, then ρ(u) ∈ α b 

ρ(x

), ρ(a), ρ(x

) 

= α b 

ρ(x

), ρ(b), ρ(x

) 

= n

ρ(v) : v ∈ α



x

, b, x

o .

Hence there exists v ∈ α



x

, b, x



such that uρv, whence

α



x

, a, x

 ρ α



x

, b, x

 . This completes the proof.

Definition 3.3. Let G be a (Γ, n)-semihypergroup and C be a nonempty subset of G. We say that C is an α-complete part of G if for any nonzero number n, the following implication holds:

C ∩ α

6= ∅ =⇒ α

⊆ C.

If for every α ∈ Γ, C is an α-complete part, then C is complete part.

Example 7. Let A

= [n, n + 1), Γ = Z. Then, R is a (Z, n)-semihypergroup

by n-ary hyperoperation defined in the Example 3. For every n ∈ Z, A

is a

complete part but C = N is not complete part.

Proposition 3.4. Let G be a (Γ, n)-semihypergroup and ρ is a strongly regular relation on G, then for every a ∈ G, the equivalence class ρ(a) is a complete part of G.

Proof. Suppose that for n ∈ N, ρ(a) ∩ α

6= ∅. This implies that there exists b ∈ α

such that ρ(a) = ρ(b). Let π : G −→ [G : ρ] be a natural homomorphism.

Then, we have

π(a) = π(α

) = π α x

, β

, x

= α π(x b

, π(β

), π(x

)

. This meanies that α

⊆ π(a) = ρ(a).

Definition 3.5. Let A be a subset of G. Then, the smallest complete part of G that contain A denoted by by C(A).

Denote K

(A) = A and for every n ≥ 1 denote

K

(A) = x ∈ G : ∃m ∈ N, x ∈ α

, K

(A) ∩ α

6= ∅ , and K

(A) = S

n≥1

K

(A). Let K(A) = S

α∈Γ

K

(A).

Proposition 3.6. Let G be a (Γ, n)-hypergroup. Then (1) For every n ≥ 2, we have K

(K

(x)) = K

(x), (2) the following relation is equivalence

x ∼ y ⇐⇒ ∃n ≥ 1, x ∈ K

(y).

Proof. Suppose that n = 2. We have

K

(K

(x)) = y ∈ G : ∃n ∈ N, y ∈ α

, K

(x) ∩ α

6= ∅

= K

(x).

Let K

(K

(x)) = K

(x). Then,

K

(K

(x)) = y ∈ G : ∃n ∈ N, y ∈ α

, K

(K

(x)) ∩ α

6= ∅

= K

(x).

(2) Suppose that n = 2 and x ∈ K

(y). Then,

x ∈ K

(y) = {z ∈ G : ∃m ∈ N, z ∈ α

, K

(y) ∩ α

6= ∅}.

Hence {x, y} ⊆ α

which implies that y ∈ K

(x). Let x ∈ K

(y) ⇐⇒ y ∈ K

(x),

and x ∈ K

(y). Then, there exists n ∈ N such that x ∈ α

and b ∈ α

∩ K

(y). This implies that b ∈ K

(x) and y ∈ K

(b). Hence y ∈ K

(K

(x)) = K

(x). In the same way, the converse implication holds.

Proposition 3.7. Let G be a (Γ, n)-hypergroup and A be a nonempty subset of G. Then, C(A) = K(A).

Proof. Suppose that A is a nonempty subset of G and K(A) ∩ α

6= ∅. Then there exist m ≥ 1 and α ∈ Γ such that K

(A) ∩ α

6= ∅. Hence α

⊆ K

(A), which means that α

⊆ K(A).

Let A ⊆ B and B is a complete part of G, then we show that K(A) ⊆ B.

We have K

(A) ⊆ B and suppose that K

(A) ⊆ B. We check that K

(A) ⊆ B. Let b ∈ K

(A). Then there exists m ∈ N, such that b ∈ α

and K

(A) ∩ α

6= ∅. Hence B ∩ α

6= ∅ and we obtain α

⊆ B. Therefore, b ∈ B and K(A) = C(A).

Proposition 3.8. Let G be a (Γ, n)-hypergroup. Then, the relation ∼ and β

are coincide.

Proof. Suppose that xβy. Then there exists α ∈ Γ such that x ∈ K

(y) ⊆ K(y).

This implies that x ∼ y. Now, if we have x ∼ y, then there exists α ∈ Γ and n ≥ 1 such that xK

y which implies that α

∩ K

(y) 6= ∅. Let a ∈ α

∩ K

(y).

Hence aβx. Since a

∈ K

(y), it follows that there exists α

such that α

∩ K

(y) 6= ∅. Let a

∈ α

∩ K

(y). Hence a

βa

and a

∈ K

(y). After finite number of steps, we obtain that there exists a

and a

such that a

βa

and a

∈ K

(y) = {y}. Therefore, xβ

y.

4. Θ relation and T Functor

The category CΓH in which the objects are (Γ, n)-semihypergroups. For (Γ, n)- semihypergroups G

and G

M or(G

, G

), are epimorphism from G

to G

and CG is the category of all semigroups. The purpose of this section is to introduce the concept of T - functor. First we shall present the fundamental definitions.

We denote the equivalence class of element x ∈ G by β

(x). Let

[G : Γ] = (β

(x

))

, α) .

We define the relation Θ as follows:



(β

(x

))

, α), ((β

(y

))

, β) 

∈ Θ if an only if

α (β b

(x

))

, β

(x)
= β b 

(β

(y

))

, β

(x)  ,

for every β

(x) ∈ [G : β

]. Obviously, Θ is an equivalence relation. Let Θ (β

(x

))

, α)

denote the equivalence class contain (β

(x

))

, α). Let

∆[G] be the set of all equivalence classes on [G : Γ]. We define operation as follows:

Θ (β

(x

))

, α)
◦ Θ (β

(y

))

, β)

= Θ α β b

(x

))

, β

(y

)
, β

(y

)

, β
,

for every β

(x

), β

(y

) ∈ [G : β

], 1 ≤ i ≤ n − 1 and α, β ∈ Γ. This operation is well-defined. Indeed,

Θ (β

(x

))

, α

= Θ (β

(a

))

, α

, Θ (β

(y

))

, β

= Θ (β

(b

))

, β

. Hence

α b

(β

(x

))

, β

(x)) = α b

(β

(a

))

, β

(x)), β b

(β

(y

))

, β

(y)) = b β

(β

(b

))

, β

(y)), for every β

(x), β

(y) ∈ [G : β

]. We have

β b

α b

(β

(a

)

, β

(b

)), β

(b

)

, β

(y))

= α b



β

(a

)

, b β

(β

(b

)

, β

(y))



= α b



β

(a

)

, b β

(β

(y

))

, β

(y))



= b β

α b

(β

(a

)

, β

(y

)), β

(y

)

, β

(y)

= b β

α b

(β

(x

)

, β

(y

)), β

(y

)

, β

(y)
. This implies that

Θ ( α b

(β

(a

)

, β

(b

)), β

(b

)

, β

= Θ ( α b

(β

(x

)

, β

(y

)), β

(y

)

, β

.

Hence

Θ (β

(x

))

, α

◦ Θ (β

(y

))

, β

= Θ ( α b

(β

(a

)

, β

(b

)), β

(b

)

, β

= Θ ( α b

(β

(x

)

, β

(y

)), β

(y

)

, β

= Θ (β

(a

))

, α

◦ Θ (β

(b

))

, β

. Thus ◦ is well-defined.

Moreover, the function ◦ is associative. Indeed,

Θ(β

(x

))

, α) ◦ Θ(β

(y

))

, β) ◦ Θ(β

(z

)

, γ)

= Θ (β

(x

))

, α)
◦ Θ( β(β b

(y

)

, β

(z

)), β

(z

), . . . , β

(z

), γ))

= Θ



α(β b

(x

)

, b β(β

(y

)

, β

(z

)), β

(z

)

)

 , γ

 . On the other hand,

Θ(β

(x

)

, α) ◦ Θ(β

(y

))

, β)
◦ Θ(β

(z

)

, γ)

=

 Θ



α(β b

(x

)

, β

(y

)), β

(y

)

, β



◦ Θ(β

(z

)

, γ)

= Θ



β( b α(β b

(x

)

, β

(y

)), β

(y

)

, β

(z

)), β

(z

)

, γ

 . Hence (∆[G], ◦) is a semigroup.

Let G be a (Γ, n)-semihypergroup. Then, for ∆

⊆ ∆ and G

⊆ G we define [∆

] = β

(x) ∈ [G : β

] : Θ β

(x), β

(y)

α
∈ ∆

, ∀α ∈ Γ, β

(y) ∈ [G : β

] , [[G

]] = Θ(β

(x

)

, α

) ∈ ∆ : α b

(β

(x

)

, β

(x)) ∈ G

, ∀β

(x) ∈ [G : β

] . Proposition 4.1. Let G be a commutative (Γ, n)-semihypergroup. Then, the following statements are true:

1. If ∆

⊆ ∆[G] is an ideal, then [∆

] is a (Γ, n)-ideal of [G : β

].

2. If G

is a (Γ, n)-ideal of [G : β

], then [[G

]] is an ideal of ∆[G].

Proof. (i) Suppose that ∆

is an ideal of ∆[G] and β

(x) ∈ ∆

. This implies

that Θ(β

(x), β

(y))

, α) ∈ ∆

, for every α ∈ Γ and β

(y) ∈ [G : β

]. Let

Θ (β

(y

))

, β) ∈ ∆[G]. Since ∆

is an ideal of ∆[G], thus

Θ β

(y

))

, β
◦ Θ β

(x), β

(y))

, α
∈ ∆

=⇒ Θ



β(β b

(y

))

, β

(x)), β

(y))

), α



∈ ∆

.

So for every α, β ∈ Γ and β

(y

)

∈ [G : β

], we have b β(β

(y

))

, β

(x)) ∈ ∆

. Therefore, ∆

is an ideal of [G : β

].

(ii) Let Θ(β

(x

)

, α) ∈ [[G

]] and Θ(β

(y

)

, β) ∈ ∆[G]. Hence for every β

(x) ∈ [G : β

],

α(β b

(x

), β

(x

), . . . , β

(x

), β

(x)) ∈ G

. On the other hand

Θ(β

(x

)

, α) ◦ Θ(β

(y

)

, β) = Θ α(β b

(x

)

, β

(y

)), β

(y

)

, β
. Since G

is an ideal of [G : β

], this implies that

Θ α(β b

(x

)

, β

(y

)), β

(y

)

), β
∈ G

. Therefore, [[G

]] is a right ideal of ∆[G]. This completes the proof.

Let G be a (Γ, n)-semihypergroup and e is a natural element of G. It is easy to see that Θ(β

(e)

, α) is a left unity of ∆[G].

Proposition 4.2. Let G be a (Γ, n)-semihypergroup with natural element and I be an ideal of G. [[[I]]] = I.

Proof. Suppose that β

(x) ∈ [[[I]]]. Hence Θ(β

(x), β

(y

)

, α) ∈ [[I]], for every α ∈ Γ and β

(y

)

∈ [G : β

]. So α(β b

(x), β(e)

) ⊆ I. Since G has a left unity, thus β

(x) ∈ I. Therefore, I = [[I]].

Theorem 4.3. Let G

and G

be (Γ

, n) and (Γ

, n)-semihypergroups and (ϕ, f ) : G

× Γ

−→ G

× Γ

be an epimorphism. Then, there exists a homomorphism (ϕ, f ) : ∆[G \

] −→ ∆[G

]. Moreover, if (ϕ, f ) is an isomorphism then, \ (ϕ, f ) is isomorphism.

Proof. We define

(ϕ, f ) Θ(β \

(x

)

, α)
= Θ β

(ϕ(x

))

, f (α)
.

First we prove that \ (ϕ, f ) is well-defined. Let

Θ(β

(x

)

, α) = Θ(β

(y

)

, β).

This implies that

(ϕ, f ) α(β b

(x

)

, β

(x))
= (ϕ, f ) 

β(β b

(y

)

, β

(x))  , for every β

(x) ∈ [G : β

]. Hence

f (α) β [

(ϕ(x

))

, β

(ϕ(x))
= d f (β) β

(ϕ(y

))

, β

(ϕ(x))
. Since ϕ is onto,

f (α) β [

(ϕ(x

))

, β

(y)
= d f (β) β

(ϕ(y

))

, β

(y)
,

for every β

(y) ∈ [G

: β

]. Hence the function \ (ϕ, f ) is well defined. Let Θ β

(x

)

, α
, Θ(β

y

)

, β
∈ ∆[G

]. Then,

(ϕ, f ) Θ(β \

(x

)

, α) ◦ Θ(β

(y

)

, β)

= \ (ϕ, f ) Θ( α(β b

(x

)

, β

(y

)), β

(y

)

, β

= Θ([ f (α)(β

(ϕ(x

))

, β

(ϕ(y

)), β

(ϕ(y

))

, f (β))

= Θ(β

(ϕ(x

))

, f (α)) ◦ Θ(β

ϕ(y

))

, f (β)

= \ (ϕ, f ) Θ(β

(x

)

, α)
◦ \ (ϕ, f ) Θ(β

(y

)

, β)
. Hence \ (ϕ, f ) is homomorphism.

Let (ϕ, f ) be one to one and \ (ϕ, f ) Θ(β

(x

)

, α)
= \ (ϕ, f ) Θ(β

(y

)

, β)
.

Then, we have

Θ(β

ϕ(x

))

, f (α)
= Θ β

(ϕ(y

))

, f (β)

=⇒ [ f (α)(β

ϕ(x

))

, β

(y)
= d f (β) β

(ϕ(y

))

, β

(y)

=⇒ [ f (α) β

(ϕ(x

))

, β

(ϕ(x)
= d f (β)(β

ϕ(y

))

, β

(ϕ(x)

=⇒ (ϕ, f ) Θ(β

(x

)

, α)
= (ϕ, f ) Θ β

(y

)

, α

=⇒ Θ(β

(x

)

, α) = Θ β

(y

)

, α
,

where y = ϕ(x). One can see that if (ϕ, f ) is an onto, then \ (ϕ, f ) is an onto.

This completes the proof.

Proposition 4.4. Let G

and G

be (Γ

, n) and (Γ

, n)-semihypergroups, respec- tively. Then,

∆[G

× G

] ∼ = ∆[G

] × ∆[G

].

Proof. Suppose that β

, β

and β

be fundamental relations on G

× G

, G

and G

, respectively. It is easy to see that

[G

× G

: β

] ∼ = [G

: β

] × [G

: β

].

We define

ψ : ∆[G

× G

] −→ ∆[G

] × ∆[G

]

Θ β

((x

, y

)

), (α

, α

)
−→ Θ(β

(x

)

, α

), Θ(β

(y

)

, α

)
. Obviously, this function is well-defined. We proof ψ is a homomorphism.

ψ Θ β

(x

, y

)

, (α

, α

)
Θ β

(z

, w

)

, (β

, β

)

= ψ Θ α

(β

(x

)

, β

(z

)), β

(z

)

),

Θ(α

(β

(y

)

, β

(w

)), β

(w

)

, (β

, β

)

= Θ(α

(β

(x

)

, β

(z

)), β

(z

)

, β

), Θ(α

(β

(y

)

, β

(w

)), β

(w

)

), β

= ψ Θ β

(x

, y

)

, (α

, α

)

◦ ψ Θ β

(z

, w

)

, (β

, β

)

. It is easy to see that ψ is onto and one to one.

Theorem 4.5. There exists an exact covariant functor between the category of (Γ, n)-semihpergroup and the category of semigroups.

Proof. Suppose that G

, G

and G

are (Γ

, n), (Γ

, n) and (Γ

, n)-semihyper- groups, respectively and (ϕ

, f

) : (G

, Γ

) −→ (G

, Γ

), (ϕ

, f

) : (G

, Γ

) −→

(G

, Γ

) are epimorphisms. We define

T (G

, Γ

) = ∆[G

], T (G

, Γ

) = ∆[G

], T (G

, Γ

) = ∆[G

].

T ((ϕ

, f

)) = \ (ϕ

, f

), T ((ϕ

, f

)) = \ (ϕ

, f

).

T ((ϕ

, f

) ◦ (ϕ

, f

)) Θ β

(x

)

, α
= T ((ϕ

◦ ϕ

, f

◦ f

))Θ β

(x

)

, α

= Θ(ϕ

◦ ϕ

(β

(x

)

), f

◦ f

(α))

= T ((ϕ

, f

)) ◦ T ((ϕ

, f

))Θ(β

(x

)

, α).

for every Θ β

(x

)

, α

∈ ∆[G

]. On the other hand, if Id is an identity homomorphism, then T (Id) is an identity homomorphism. Therefore, T is a covariant functor. By Theorem 4.3, T is an exact functor. This complete the proof.

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Received 22 January 2015

Revised 25 February 2015