On the existence of hyperrings associated to arithmetic functions

Accepted Manuscript On the existence of hyperrings associated to arithmetic functions

M. Al Tahan, B. Davvaz

PII: DOI: Reference:

S0022-314X(16)30300-6 http://dx.doi.org/10.1016/j.jnt.2016.10.017 YJNTH 5621

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Journal of Number Theory

Received date: Accepted date:

15 June 2016 25 October 2016

Please cite this article in press as: M. Al Tahan, B. Davvaz, On the existence of hyperrings associated to arithmetic functions, J. Number Theory (2017), http://dx.doi.org/10.1016/j.jnt.2016.10.017

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On the existence of hyperrings associated to arithmetic functions M. Al Tahan1 and B. Davvaz2 1 Department of Mathematics, Lebanese International University, Lebanon [email protected] 2 Department of Mathematics, Yazd University, Iran [email protected] Abstract After introducing the definition of hyperring by Krasner, the study of hyperrings has been of great importance. In this paper, we define a new hyperoperation associated to the set G of all arithmetic functions and give a complete analysis of its properties. Also, we combine our hyperoperation (+) with an already defined hyperoperation (·) and define (G, +, ·) in which we study its properties and the possibility of containing hyperring . First, we prove that (G, +, ·) is a weak hyperring. Then we find the largest hyperring contained in (G, +, ·).

Keywords and phrases: Arithmetic functions, hyperstructure, hyperring. AMS Mathematics Subject Classification: 20N20, 16Y99.

1

Introduction

Hypergroup theory was born in 1934, when Marty [8] gave the definition of hypergroup and illustrated some applications and showed its utility in the study of groups, algebraic functions and relational fractions. Nowadays the hypergroups are studied from the theoretical point of view and for their applications to many subjects of pure and applied mathematics: geometry, topology, cryptography and code theory, graphs and hypergraphs, probability theory, binary relations, theory of fuzzy and rough sets, automata theory, economy, etc. (see [3, 5, 15]). A hypergroup is an algebraic structure similar to a group, but the composition of two elements is a non-empty set. Hyperrings are essentially rings, with approximately modified axioms in which addition or multiplication is a hyperoperation. This concept has been studied by a variety of authors. A well known type of a hyperring is called the Krasner hyperring [7]. This concept has been studied by a variety of authors. Some principal notions of hyperring theory can be found in [4, 6, 9, 10, 11, 16]. In [1], Asghari and Davvaz introduced a hyperoperation associated to the set of all arithmetic functions and analyzed the properties of this hyperoperation. This paper is a continuation of [1]. The paper is structured as follows. After an 1

introduction, in Section 2, we present some basic definitions concerning hyperstructures that are used throughout this paper. In Section 3, we define a new hyperoparation (∗) on the set G of arithmetic functions and study its properties. In particular, we prove that (G, ∗) is a commutative hypergroup. In Section 4, we introduce ◦, a hyperoperation on the set of arithmetic functions defined in [1]. We prove that the identities are unique for both hyperoperations ∗ and ◦. Also, we characterize all elements in G that admit inverses under ◦ and ∗. In Section 5, we set the hyperoperations ◦ and ∗ on G as addition (+) and multiplication (·). We introduce the notion of weak hyperring and we prove that (G, +, ·) is a weak hyperring. In addition, we characterize all elements in G that satisfy the distributive property. Finally, we find the largest hyperring contained in (G, +, ·) and prove that (G, +, ·) contains no hyperfield. Throughout this paper, G is defined as the set of all arithmetic functions.

2

Basic definitions

In this section, we present some definitions related to hyperstructures that are used throughout the paper. Let H be a non-empty set. Then a mapping ◦ : H ×H → P ∗ (H) is called a hyperoperation on H, where P ∗ (H) is the family of all non-empty subsets of H. The couple (H, ◦) is called a hypergroupoid. In the above definition, if A and B are two non-empty subsets of H and x ∈ H, then we define:  a ◦ b, x ◦ A = {x} ◦ A and A ◦ x = A ◦ {x}. A◦B = a∈A b∈B

An element e ∈ H is called an identity of (H, ◦) if x ∈ x ◦ e ∩ e ◦ x, for all x ∈ H and it is called an scalar identity of (H, ◦) if x ◦ e = e ◦ x = {x}, for all x ∈ H. If e is a scalar identity of (H, ◦), then e is the unique identity of (H, ◦). The hypergroupoid (H, ◦) is said to be commutative if x ◦ y = y ◦ x, for all x, y ∈ H. A hypergroupoid (H, ◦) is called a semihypergroup if for every x, y, z ∈ H, we have x ◦ (y ◦ z) = (x ◦ y) ◦ z and is called a quasihypergroup if for every x ∈ H, x ◦ H = H = H ◦ x. This condition is called the reproduction axiom. The couple (H, ◦) is called a hypergroup if it is a semihypergroup and a quasihypergroup. A subhypergroup K of a hypergroup (H, ◦) is normal if a ◦ K = K ◦ a for all a ∈ H. A canonical hypergroup [12] is a non-empty set H endowed with an additive hyperoperation +:H × H → P ∗ (H), satisfying the following properties: (1) for any x, y, z ∈ H, x + (y + z) = (x + y) + z, (2) for any x, y ∈ H, x + y = y + x, (3) there exists 0 ∈ H such that 0 + x = x + 0 = x, for any x ∈ H,(4) for every x ∈ H, there exists a unique element x ∈ H, such that 0 ∈ x + x (we shall write −x for x and we call it the opposite of x), (5) z ∈ x + y implies that y ∈ −x + z and x ∈ z − y, that is (H, +) is reversible. We can consider several definitions for a hyperring, by replacing at least one of the two operations by hyperoperations. In general case, (R, +, ·) is a hyperring if + and · are two hyperoperations such that (R, +) is a hypergroup, (R, ·) is a semihypergroup and the hyperoperation · is distributive over the hyperoperation +, which means that for all x, y, z of R we have: x · (y + z) = x · y + x · z and (x + y) · z = x · z + y · z. We call (R, +, ·) 2

a hyperfield if (R, +, ·) is a hyperring and (R, ·) is a hypergroup. There are different types of hyperrings. If only the addition + is a hyperoperation and the multiplication · is a usual operation, then we say that R is an additive hyperring. A special case of this type is the Krasner hyperring. For more detail about hyperrings we refer to [6]. A Krasner hyperring [6, 7] is an algebraic hypersructure (R, +, ·) which satisfies the following axioms: (1) (R, +) is a canonical hypergroup,(2) (R, ·) is a semigroup having zero as a bilaterally absorbing element, i.e., x · 0 = 0 · x = x, (3) The multiplication is distributive with respect to the hyperoperation +.

3

Arithmetic functions and hyperstructures

An arithmetic function is a function in which its domain of definition is the set of natural numbers and its codomain is the set of complex numbers. An arithmetic function f is said to be additive if whenever m and n are coprime, f (mn) = f (m) + f (n). An arithmetic function f is said to be multiplicative if whenever m and n are coprime, f (mn) = f (m)f (n). Remark 1. If f is an additive function and g is a multiplicative function then f (1) = 0 and g(1) = 1. Denote by AF (G) the set of all additive functions of G and by ℘∗ (G) the set of all non empty subsets of G. Now, we define a hyperoperation ∗ on G. Definition 3.1. Define a hyperoperation on G as follows ∗ : G × G → ℘∗ (G) (α, β) −→ α ∗ β such that

   n n α(d) + β( ). (α ∗ β)(n) = α(d) + β( ) : d | n = d d d|n

Remark 2. Let α and β be two elements in G. If α(n) = β(n) for all natural numbers n, then α = β. Proposition 3.2. (G, ∗) is commutative. Proof. Let n be a naturalnumber and α and β be two elements in G. Let n be a natural is equivalent to the number then α ∗ β(n) = d|n α(d) + β( nd ). Since the set {d : d | n}  set { nd : nd | n}, it follows that α ∗ β(n) can be written as α ∗ β(n) = n |n α( nd ) + β(d) = d β ∗ α(n). Proposition 3.3. (G, ∗) is associative.

3

Proof. Let n be a natural number and α, β and γ be elements in G. We consider first the expression α ∗ (β ∗ γ). We have 

n α(d) + (β ∗ γ)( ) d d|n   n  α(d) + β(m) + γ( = ) md n d|n m| d  n  α(d) + β(m) + γ( ) = md d|n m| n  d n (α(d) + β(m) + γ( )). = md

α ∗ (β ∗ γ)(n) =

md|n

We consider now the expression (α ∗ β) ∗ γ. We have  n (α ∗ β)(d) + γ( ) d d|n    d  n = α(s) + β( ) + γ( ) s d d|n s|d  d n = α(s) + β( ) + γ( ). s d

(α ∗ β) ∗ γ(n) =

d|n s|d

Since s | d then there exists a positive integer t such that d = st. Thus the above expression can be written as  n α(s) + β(t) + γ( ). α ∗ (β ∗ γ)(n) = st st|n

Therefore, α ∗ (β ∗ γ) = (α ∗ β) ∗ γ. Theorem 3.4. (G, ∗) is a commutative hypergroup. Proof. By Propositions 3.2 and 3.3, (G, ∗) is commutative and associative. So, we only need to show that (G, ∗) satisfies the reproduction axiom, i.e., α ∗ G = G ∗ α = G for all α ∈ G. Since α ∗ G ⊆ G and G is commutative, it suffices to show that G ⊆ α ∗ G. For any β ∈ G, define γ(n) = β(n) − α(1) for all natural numbers n. It is clear that γ ∈ G. We have that β(n)  = α(1) + γ(n) ∈ α ∗ γ(n). This implies that β ∈ α ∗ γ. Using the definition α ∗ H = γ∈H α ∗ γ, it is easy to see that β ∈ α ∗ H. Lemma 3.5. (AF (G), ∗) is a normal subhypergroup of (G, ∗). Proof. Since (G, ∗) is commutative, it suffices to prove that (AF (G), ∗) is a subhypergroup of (G, ∗). This is equivalent to proving the reproduction axiom, i.e., α ∗ AF (G) = AF (G) for all α in AF (G). Let β be any element in AF (G). Since α(1) = 0 and α(1) + β(n) ∈ α ∗ β(n) for all natural numbers n, it follows that β ∈ α ∗ β. Thus, AF (G) ⊆ α ∗ AF (G). 4

Definition 3.6. Let G be the set of all arithmetic functions. Define a map ‘’ on G ∗ G as follows  : (G ∗ G) × (G ∗ G) → ℘∗ (G) ((α1 ∗ β1 ), (α2 ∗ β2 )) −→ (α1 ∗ β1 )  (α2 ∗ β2 ) such that for all natural numbers m and n



((α1 ∗ β1 )  (α2 ∗ β2 ))(m, n) =

α + β.

α∈(α1 ∗β1 )(m),β∈(α2 ∗β2 )(n)

It is easy to prove the following proposition by using Definition 3.6. Proposition 3.7. Let G be the set of all arithmetic functions and m,n be natural numbers. Then 1. ((α1 ∗ β1 )  (α2 ∗ β2 ))(m, n) = ((α2 ∗ β2 )  (α1 ∗ β1 ))(n, m) for all α1 , α2 , β1 and β2 ∈ G. 2. (G ∗ G, ) is associative. Definition 3.8. Let α and β ∈ G. Then α ∗ β is a multiplicative function in G ∗ G if for all coprime natural numbers m and n the following condition holds (α ∗ β)(mn) = (α ∗ β)(m)  (α ∗ β)(n). We denote by AF (G ∗ G) the set of all additive functions in G ∗ G. Using the above definition, it is easy to prove the following lemma. Lemma 3.9. Let α, β ∈ G and m, n be two natural numbers. Then    α(d) + β(D) = α(d)  β(D). d|m,D|n

d|m

D|n

Proposition 3.10. If α and β ∈ AF (G) then α ∗ β ∈ AF (G ∗ G). Proof. Let m and n be coprime natural numbers. We have that  mn α(d) + β( ). α ∗ β(mn) = d d|mn

Since gcd(m, n) = 1 and d | mn, it follows that d can be uniquely written as d = st, n where s | m, t | n and gcd( m s , t ) = gcd(s, t) = 1. We get that α ∗ β(mn) =  m n m n s|m,t|n α(st) + β( s · t ). Since α and β are additive functions and gcd( s , t ) = m n m n gcd(s, t) = 1, it follows that α(st) = α(s) + α(t) and β( s · t ) = β( s ) + β( t ). Substituting in α ∗ β(mn), we get that  m n α ∗ β(mn) = α(s) + α(t) + β( ) + β( ) s t s|m,t|n  m n α(s) + β( ) + α(t) + β( ). = s t s|m,t|n

5

Using Lemma 3.9, we can write the above expression as  m  n α(s) + β( )  α(t) + β( ) α ∗ β(mn) = s t s|m

t|n

= (α ∗ β)(m)  (α ∗ β)(n). Therefore, we obtain α ∗ β ∈ AF (G ∗ G).

4

Identity and inverse functions under (G, ∗) and (G, ◦)

In this section, we find the identity under ∗ and prove that it is unique and we characterize all the functions that admit inverses under both hyperoperations; our new hyperoperation ∗ and the hyperoperation ◦ defined in [1]. Definition 4.1. [1] Define a hyperstructure on G as follows ◦ : G × G → ℘∗ (G)

such that (α ∗ β)(n) =



(α, β) −→ α ◦ β

n d|n α(d)β( d ).

Lemma 4.2. The identity ı defined in [1] for (G, ◦) is unique. Proof. Suppose that ı and j are both identities for (G, ◦). Having ı and j as identities implies that for any natural number n, we have that (ı◦j)(n) = {ı(n), 0} and (j◦ı)(n) = {j(n), 0} respectively. Since (G, ◦) is commutative, it follows that ı(n) = j(n) for any n. Thus, ı = j. Lemma 4.3. Let α be a multiplicative function that admits an inverse in (G, ◦). Then α = ı. Proof. Since α is a multiplicative function, it follows from Remark 1 that α(1) = 1. By using the results in [1], we get that α−1 is multiplicative and α−1 (1) = 1. We have that α ◦ α−1 = ı which implies that for any natural number n, α ◦ α−1 (n) = ı(n). For n > 1, α(n) = α(n)α−1 (1) ∈ α ◦ α−1 (n) = ı(n) = 0 which implies that α(n) = 0 for all n > 1. Therefore, α = ı. Lemma 4.4. Let α ∈ (G, ◦) such that α−1 ∈ G. Then α(1) = 0 and α−1 (1) =

1 α(1) .

Proof. Having α ◦ α−1 = ı implies that for any natural number n, we have that α ◦ α−1 (n) = ı(n). For n = 1, we have that α ◦ α−1 (1) = ı(1). This is equivalent to 1 . α(1)α−1 (1) = 1. Thus α(1) = 0 and α−1 (1) = α(1) Lemma 4.5. Let α ∈ (G, ◦) such that α(1) = a = 0 and α−1 ∈ G. Then  a, if n=1 α(n) = 0, otherwise. 6

Proof. By Lemma 4.4, α−1 (1) = a1 . For n > 1, α ◦ α−1 (n) = ı(n) = {0}. Having that α(n)α−1 (1) ∈ α ◦ α−1 (n), we get that α(n) a = 0. This implies that for all n > 1, α(n) = 0. Theorem 4.6. Let α ∈ (G, ◦) with α(1) = a = 0. Then α−1 ∈ G if and only if  a, if n=1 α(n) = 0, otherwise. Proof. The necessary condition results from Lemma 4.5. For the sufficient condition, we have that a = 0 then there exists β ∈ G such that  1 a , if n=1 β(n) = 0, otherwise. It is easy to see that α ◦ α−1 = ı. Next, we prove similar results for (G, ∗). Definition 4.7. Define O∗ : N → C as O∗ (n) = 0 for all n ∈ N. Remark 3. O∗ ∈ AF (G). Proposition 4.8. If α ∈ G then α ∈ α ∗ O∗ .   Proof. For every α ∈ G, α ∗ O∗ (n) = d|n α(d) + O∗ ( nd ) = d|n α(d). This implies that α(d) ∈ α∗O∗ (n) for every d | n. In particular when d = n, α(n) ∈ α∗O∗ (n). Therefore, α ∈ α ∗ O∗ . Definition 4.9.  An element λ is said to be identity in (G, ∗) if for all natural numbers n, α ∗ λ(n) = d|n α(d). Lemma 4.10. (G, ∗) has unique identity. In particular, O∗ is its identity.  Proof. For all natural numbers n, we have that α ∗ O∗ (n) = d|n α(d). This implies that O∗ is an  identity. Let λ be an identity in (G, ∗). Then for all natural numbers n, α  ∗ λ(n) = d|n  α(d). In particular, forα = O∗ and α = λ we get that O∗ ∗ λ(n) = O (d) = d|n ∗ d|n 0 and λ ∗ O∗ (n) = d|n λ(d). The latter expressions imply that λ(n) = 0 for all natural numbers n. Thus λ = O∗ . Definition 4.11. An element α−1 is said to be an inverse of α in (G, ∗) if α∗α−1 = O∗ . Lemma 4.12. Let α be an element in (G, ∗) that admits an inverse α−1 in (G, ∗). Then α−1 is unique. Proof. Let β and γ be two inverses of α. Then for all natural numbers  n, we have that α∗β(n) = α∗γ(n) = 0∗ (n){0}. This implies that d|n α(d)+β( nd ) = d|n α(d)+γ( nd ) = {0}. It is easy to see that α(1) + β(n) = α(1) + γ(n) = 0 for all natural numbers n. Thus β = γ. 7

Theorem 4.13. Let α ∈ AF (G) such that its inverse α−1 exists. Then α−1 ∈ AF (G). Proof. Suppose that α−1 is not an additive function. Then there exist coprime integers m and n such that α−1 (mn) = α−1 (m) + α−1 (n). Let mn be the least positive integer such that the above relation holds, i.e., for all coprime natural numbers a and b with ab < mn α−1 (ab) = α−1 (a) + α−1 (b). It is clear that mn > 1 since α(1) = α−1 (1) = 0. We have that {α−1 (m) + α−1 (n), 0} = {α−1 (m) + α−1 (n), O∗ (mn)} = {α−1 (m) + α−1 (n), α ∗ α−1 (mn)}  mn = {α−1 (m) + α−1 (n)} ∪ α−1 (d) + α( ). d d|mn

If d | mn, then there exist coprime natural numbers s and t such that d = st, s | m, t | n and gcd(s, t) = 1. So, {α−1 (m) + α−1 (n), 0} can be written as follows 

{α−1 (m) + α−1 (n)} ∪ {α−1 (mn)} ∪

α−1 (st) + α(

s|m,t|n,st
mn ). st

The latter can be written as {α−1 (mn)} ∪ {α−1 (m) + α(1) + α−1 (n) + α(1)}∪  m n α−1 (s) + α−1 (t) + α( ) + α( ) s t s|m,t|n,st
By Lemma 3.9, we have 

m  −1 n  α (t) + α( ) ) s t t|n  s|m  = {α−1 (mn)} ∪ (α−1 ∗ α)(m)  (α−1 ∗ α)(n)

{α−1 (m) + α−1 (n), 0} = {α−1 (mn)} ∪

α−1 (s) + α(

= {α−1 (mn)} ∪ (0∗ (m)  0∗ (n)) = {α−1 (mn), 0}. Then α−1 (mn) = α−1 (m) + α−1 (n) which contradicts the choice of m and n. Lemma 4.14. Let α be an additive function that admits an inverse in (G, ∗). Then α = O∗ . 8

Proof. Since α is an additive function that admits an inverse then by Theorem 4.13 that for n = 1, α−1 (n) = α−1 (1) = 0. For α−1 is also an additive function.  This implies −1 −1 n > 1, O∗ (n) = α ∗ α (n) = d|n α(d) + α ( nd ). It is easy to see that α(n) + α−1 (1) ∈ α ∗ α−1 (n) = {0}. This implies that α(n) = 0 for all n > 1. Therefore, α = O∗ . Theorem 4.15. Let α ∈ (G, ∗). Then α−1 ∈ G if and only if α is a constant function. Proof. For the sufficient condition, let α be a constant function. Then there exists a complex number a such that α(n) = a. Define β ∈ G as β(n) = −a. It is easy to see that α ∗ β(n) = 0∗ (n). Thus β is the inverse of α. For the necessary condition, suppose that α and α−1 are in G then α ∗ α−1 (n) = 0∗ (n) for all n ≥ 1. For n = 1, we get that α ∗ α−1 (1) = 0∗ (1). This implies that α−1 (1) = −α(1). For n > 1, we have that α(n) + α−1 (1) ∈ 0∗ (n) = {0}. This implies that for all n > 1, α(n) = −α−1 (1) = α(1). Therefore, α(n) = α(1) for all n ≥ 1.

5

Hyperrings of arithmetic functions

In this section, we find the largest hyperring of arithmetic functions. We set the hyperoperation ∗ defined in Definition 3.1 as + and the hyperoperation ◦ defined in [1] as · and we define now (G, +, ·) as the set of arithmetic function with the addition and multiplication hyperoperations. Remark 4. Weak hyperstructures or Hv -structures were introduced by Vougiouklis [16]. The concept of weak hyperstructures constitutes a generalization of the well known algebraic hyperstructures (hypergroup, hyperring, and so on). Actually some axioms concerning the above hyperstructures such as associative law, the distributive law, and so on are replaced by their corresponding weak axioms. Definition 5.1. A set W associated to the the hyperoperations + and · is said to be weak distributive if x · (y + z) ∩ x · y + x · z = ∅ and (x + y) · z ∩ x · z + y · z = ∅ whenever x, y and z are in W . Definition 5.2. A set W associated to the the hyperoperations + and · is said to be weak hyperring if the following conditions are satisfied: 1. (W, +) is a hypergroup; 2. (W, ·) is a semihypergroup; 3. (W, ·) is weak distributive. Lemma 5.3. (G, +, ·) is weak distributive. Proof. Let α, β and γ be elements in G. Since (G, +, ·) is commutative, it suffices to show that α · (β + γ) ∩ α · β + α · γ = ∅. It is easy to see that α(1)(β(1) + γ(n)) ∈ α.(β + γ) ∩ α.β + α.γ. Therefore, (G, +, ·) is weak distributive Theorem 5.4. (G, +, ·) is a weak hyperring. 9

Proof. The proof results from having that (G, +) and (G, ·) are hypergroup and semihypergroup respectively and that (G, +, ·) is weak distributive (by Lemma 5.3). Denote by (M, +, ·) the set of all constant arithmetic functions under the hyperoperations of G and by (N, +, ·) the largest distributive set contained in (G, +, ·). Theorem 5.5. Let M be the set of all constant arithmetic functions in G. Then (M, +, ·) is a Krasner hyperring. Proof. Let α, β and γ be elements in M . Then α(n), β(n) and γ(n) are equal to a, b and c respectively. The conditions for having (M, +, ·) a Krasner hyperring are satisfied since: • (M, +) is associative: Since (G, +) is associative and M ⊂ G, it follows that (M, +) is associative. • (M, +) has an identity: We have that O∗ (n) is a constant function. This implies that O∗ ∈ M . For every natural number n, α(n) = a which implies that (O∗ + α)(n) =

 d|n

n O∗ (d) + α( ) = a = α(n). d

• Elements of (M, +) admit unique inverses: The additive inverse of α, by Theorem 4.15, is in M . • Elements of (M, +) are reversible: If α ∈ β + γ then α(n) ∈ (β + γ)(n). This implies that a = b + c which is equivalent to c = −b + a and to b = a − c. Thus γ ∈ −β + α and β ∈ α − γ. Here −β and −γ denote the additive inverses of β and γ respectively.  • (M, ·) is a semigroup: We have that α.β(n) = d|n α(d)β( nd ) = ab is a constant function. • O∗ is bilaterally absorbing element: We have that  n O∗ · α(n) = α · O∗ (n) = α(d)O∗ ( ) = {0} = O∗ (n). d d|n

• Distributive property: It is easy to see that the distributive property is satisfied as α · (β + γ)(n) = ab + ac = α · β + α · γ(n). Remark 5. Join spaces were introduced by Prenowitz [13, 14] to provide a common algebraic framework in which classical geometries could be axiomatized. If a, b are elements of a hypergroupoid (H, ◦), then we denote a/b = {x ∈ H : a ∈ x ◦ b}. A commutatice hypergroup (H, ◦) is called a join space if the following conditions holds for all a, b, c, d in H: a/b ∩ c/d = ∅ ⇒ a ◦ d ∩ b ◦ c = ∅.

10

Theorem 5.6. (Theorem 2.4.8 of [6]) A commutative hypergroup is canonical if and only if it is a join space with scalar identity. Corollary 5.7. (M, +) is a join space with scalar identity. Lemma 5.8. Let α ∈ N with α(1) = 0. Then α(n) = α(1) ∀ n ∈ N. Proof. We do our proof by Mathematical induction. It is clear that for n = 1, α(n) = α(1). We assume now that α(k) = α(1) ∀ k < n. Set α(1) = a. Since α ∈ N , it follows that α · (α + α) = α · α + α · α. This implies that for every n, (α · (α + α))(n) = (α · α + α · α)(n). We have that 

n α(d)(α + α)( ) d d|n   n  α(d) α(s) + α( ) . = ds n

(α · (α + α))(n) =

d|n

s| d

n It is easy to see that α(s) = α( ds ) = α(d) = a when d = 1 and d = n. This implies that (α · (α + α))(n) = {2aα(n), a(a + α(n)), 2a2 }. On the other side, we have that  (α · α + α · α)(n) = d|n (α · α)(d) + (α · α)( nd ). We can simplify (α · α + α · α)(n) as

{(α · α)(n) + (α · α)(1)}

 d=1,d=n,d|n

n (α · α)(d) + (α · α)( ). d

 Since d = 1 and d = n, it follows that d=1,d=n,d|n (α.α)(d) + (α.α)( nd ) = {2a2 }. We have that (α.α)(n) + (α · α)(1) = {(α · α)(n) + a2 }. The latter can be simplified as (α · α)(n) + (α · α)(1) = {aα(n) + a2 , 2a2 }. This implies that (α · α + α · α)(n) = {aα(n) + a2 , 2a2 }. Comparing the results of (α.(α + α))(n) and (α · α + α · α)(n), we get that 2aα(n) = 2a2 or 2aα(n) = aα(n) + a2 . Since a = 0, it follows that α(n) = a. Lemma 5.9. Let α ∈ N with α(1) = 0. Then α(n) = 0 ∀ n ∈ N. Proof. By Theorem 5.5, the set of constant functions is distributive. And using the definition of N as the largest distributive set in G, we get that N = ∅. Let β ∈ N such that β(1) = b = 0. It follows from Lemma 5.8 that for all natural numbers m, β(m) = b. We prove our Lemma by means of Mathematical induction. It is clear that α(n) = 0 for n = 1. We assume now that α(k) = 0 ∀ k < n. Since α and β are both elements of N , it follows that α · (β + β) = α · β + α · β. This implies that for every n, (α · (α + β))(n) = (α · β + α · β)(n).  We have that (α · (β + β))(n) = d|n α(d)(β + β)( nd ). We can rewrite the latter as (α · (β + β))(n) = {α(1)(β + β)(n), α(n)(β + β)(1)}

 d=1,d=n,d|n

n α(d)(β + β)( ). d

Since for all natural numbers m and k < n, β(m) = b and α(k) = 0, it follows that (α · (β + β))(n) = {0, 2bα(n)}. On the other side, we have that (α · β + α · β)(n) = 11



d|n (α

· β)(d) + (α · β)( nd ). We can simplify (α · β + α · β)(n) as 

(α · β + α · β)(n) = {(α · β)(n) + (α · β)(1)}

d=1,d=n,d|n

n ((α · β)(d) + (α · β)( ). d

Using the given and our assumption, we get that (α.β +α.β)(n) = {bα(n), 0}. Comparing the results of (α · (β + β))(n) to that of (α · β + α · β)(n), we get that 2bα(n) = bα(n). Since b = 0, it follows that α(n) = 0. Theorem 5.10. If α ∈ N then α is a constant function. Proof. The proof results from Lemma 5.8 and 5.9. The next theorem, Theorem 5.11 results from Theorems 5.5 and 5.10. Theorem 5.11. The largest hyperring contained in (G, +, ·) is (M, +, ·). Corollary 5.12. There is no hyperfield contained in (G, +, ·). Proof. Suppose that there exists a hyperfield F contained in (G, +, ·) and let α be a nonzero element in F . Then α is a constant function. By means of Theorem 4.6, α has no multiplicative inverse.

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Conclusion

After the introduction of the concept of hyperstructure by F. Marty, there have been a number of generalizations of this fundamental concept especially in the fields of hypergroups and hyperrings. Moreover, the concept of hyperstructures associated to arithmetic functions was first introduced by Asghari and Davvaz. In this paper, we presented the hyperoperation defined by Asghari and Davvaz on the set G of all arithmetic functions and defined a new hyperoparation on G. We used the two hyperoperations on G, characterized their properties and studied the existence of hyperrings over the set of arithmetic functions by finding the largest hyperring it contains.

References [1] M. Asghari-Larimi and B. Davvaz, Hyperstructures associated to arithmetic functions, ARS Combinatoria, 97 (2010), 51-63. [2] D. M. Burton, Elementary number theory, International Edition 2007. [3] P. Corsini and V. Leoreanu, Applications of hyperstructures theory, Advanced in Mathematics, Kluwer Academic Publisher, 2003. [4] I. Cristea and S. Jancic-Rasovic, Composition hyperrings, An. Stiint. Univ. Ovidius Constanta Ser. Mat., 21(2) (2013), 81-94. [5] B. Davvaz, Polygroup Theory and Related Systems, World Scientific, 2013.

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[6] B. Davvaz and V. Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press, USA, 2007. [7] M. Krasner, A class of hyperrings and hyperfields, International J. Math. and Math. Sci., 6 (1983), 307-312. [8] F. Marty, Sur une generalization de la notion de group, In 8th Congress Math. Scandenaves, (1934), 45-49. [9] S. Mirvakili, B. Davvaz and V. Leoreanu Fotea, A note on “New fundamental relation of hyperrings”, European J. Combin., 41 (2014), 258-261. [10] S. Mirvakili, B. Davvaz, Applications of the α∗ -relation to Krasner hyperrings, J. Algebra, 362 (2012), 145-156. [11] S. Mirvakili and B. Davvaz, Relationship between rings and hyperrings by using the notion of fundamental relations, Comm. Algebra, 41 (2013), 70-82. [12] J. Mittas, Hypergroups canoniques, Math. Balkanica, 2 (1972), 165-179. [13] W. Prenowitz, Spherical geometries and multigroups, Canadian J. Math., 2 (1950), 100-119. [14] W. Prenowitz, A contemporary approach to classical geometry, Amer. Math. Monthly, 68 (1961), part II v+67 pp. [15] T. Vougiouklis, Hyperstructures and Their Representations, Aviani editor. Hadronic Press, Palm Harbor, USA, 1994. [16] T. Vougiouklis, The fundamental relation in hyperrings. The general hyperfield, Algebraic hyperstructures and applications (Xanthi, 1990), 203-211, World Sci. Publ., Teaneck, NJ, 1991.

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