RATIO MATHEMATICA 25(2013), 47-58 ISSN:1592-7415 The Divisors’ Hyperoperations Achilles Dramalidis Democritus University of Thrace, School of Education, 681 00 Alexandroupolis, Greece adramali@psed.duth.gr Abstract In the set N of the Natural Numbers we define two hyperoperations based on the divisors of the addition and multiplication of two num- bers. Then, the properties of these two hyperoperations are studied together with the resulting hyperstructures. Furthermore, from the coexistence of these two hyperoperations in N∗, an Hv-ring is result- ing which is dual. Key words: Hyperstructures, Hv-structures MSC2010: 20N20, 16Y99. 1 Introduction In 1934, F. Marty introduced the definitions of the hyperoperation and of the hypergroup as a generalization of the operation and the group respectively. Let H be a set and ◦ : H ×H → P′(H) be a hyperoperation, [2], [3], [5], [6], [8]: The hyperoperation (◦) in H is called associative, if (x◦y) ◦z = x◦ (y ◦z),∀x,y,z ∈ H. The hyperoperation (◦) in H is called commutative, if x◦y = y ◦x,∀x,y ∈ H. An algebraic hyperstructure (H,◦), i.e. a set H equipped with the hyper- operation (◦), is called hypergroupoid . If this hyperoperation is associative, then the hyperstructure is called semihypergroup. The semihypergroup (H,◦), is called hypergroup if it satisfies the reproduction axiom: Achilles Dramalidis x◦H = H ◦x,∀x ∈ H. One of the topics of great interest, in the last years, is the HV -structures, which was introduced by T. Vougiouklis in 1990 [7]. The class of HV - structures is the largest class of algebraic hyperstructures. These structures satisfy weak axioms, where the non-empty intersection replaces the equality, as bellow [8]: i) The (◦) in H is called weak associative, we write WASS , if (x◦y) ◦z ∩x◦ (y ◦z) 6= ∅,∀x,y,z ∈ H. ii) The (◦) is called weak commutative, we write COW , if (x◦y) ∩ (y ◦x) 6= ∅,∀x,y ∈ H. iii) If H is equipped with two hyperoperations (◦) and (∗), then (∗) is called weak distributive with respect to (◦), if [x∗ (y ◦z)] ∩ [(x∗y) ◦ (x∗z)] 6= ∅,∀x,y,z ∈ H. The hyperstructure (H,◦) is called H v-semigroup if it is WASS and it is called H v-group if it is a reproductive (i.e. x◦H = H ◦x = H,∀x ∈ H) Hv-semigroup. It is called commutative H v-group if (◦) is commutative and it is called H v- commutative group if (◦) is weak commutative. The hyperstructure (H,◦,∗) is called H v-ring if both hyperstructures (◦) and (∗) are WASS, the reproduction axiom is valid for (◦), and (∗) is weak distributive with respect to (◦). It is denoted [4] by E∗ the set of the unit elements with respect to (∗) and by I∗(x,e) the set of the inverse elements of x associated with the unit e, with respect to (∗). An Hv-ring (R, +, ·) is called Dual H v-ring, if (R, ·, +) is an Hv-ring, too [4]. Let (H, ·) be a hypergroupoid. An element e ∈ H is called right unit element if a ∈ a ·e,∀a ∈ H and is called left unit element if a ∈ e ·a,∀a ∈ H. The element e ∈ H is called unit element if a ∈ a ·e∩e ·a,∀a ∈ H. Let (H, ·) be a hypergroupoid endowed with at least one unit element. An element a′ ∈ H is called an inverse element of the element a ∈H, if there exists a unit element e ∈ H such that e ∈ a′ ·a∩a ·a′. Moreover, let us define here: If x ∈ x · y(resp.x ∈ y ·x)∀y ∈ H, then, x is called left absorbing-like element (resp. right absorbing-like element). An element a ∈ H is called idempotent element if a2 = a. The nth power of an element h, denoted hs, is defined to be the union of all expressions of n times 48 The Divisors’ Hyperoperations of h, in which the parentheses are put in all possible ways. An Hv-group (H,·) is called cyclic with finite period with respect to h ∈ H, if there exists a positive integer s, such that H = h1 ∪ h2 . . . ∪ hs. The minimum such s is called period of the generator h. If all generators have the same period, then H is cyclic with period. If there exists h ∈ H and s positive integer, the minimum one, such that H = hs, then H is called single-power cyclic and h is a generator with single-power period s. The cyclicity in the infinite case is defined similarly. Thus, for example, the Hv-group (H, ·) is called single-power cyclic with infinite period with generator h if every element of H belongs to a power of h and there exists s0 ≥ 1, such that ∀s ≥ s0 we have: h1 ∪h2 ∪·· ·∪hs−1 ⊂ hs. 2 The divisors’ hyperoperation due to addi- tion in N Let N be the set of the Natural Numbers. Let us define the hyperoperation (�) in N as follows: Definition 2.1. For every x,y ∈ N �: N × N →P(N) −{∅} : (x,y) 7→ x�y ⊂ N such that x�y = {z ∈ N : x + y = z ·λ,λ ∈ N} where (+) and (·) are the usual operations of the addition and multiplication in N, respectively. We call the above hyperoperation, divisors’ hyperoperation due to addition. Some properties of the divisors’ hyperoperation due to addition 1. x�y = y �x,∀x,y ∈ N 2. 0 � 0 = N 3. 0 � 1 = 1 � 0 = 1 4. {1,x + y}⊂ x�y,∀x,y ∈ N 5. If x + y = κ ·ν ⇒{1,κ,ν,κ ·ν}⊂ x�y,x,y,κ,ν ∈ N. 49 Achilles Dramalidis Remark 2.2. If x + y = p, where p ∈ N is a prime number, then x � y = {1,p},x,y ∈ N. Proposition 2.3. The number 0 is a unit element of the divisors’ hyperop- eration due to addition. Proof. Indeed, for x ∈ N x� 0 = {z ∈ N : x + 0 = z ·λ,λ ∈ N} = {z ∈ N : x = z ·λ,λ ∈ N}3 x. Then, x ∈ (x� 0) ∩ (0 �x),∀x ∈ N. Remark 2.4. Since, there is no x′ ∈ N such that 0 ∈ (x � x′) ∩ (x′ � x) when x 6= 0, the number 0 is the only one in N having an inverse element (and that is 0) associated with the unique unit element 0 of the divisors’ hyperoperation due to addition, i.e. 0 ∈ 0 � 0. Proposition 2.5. The number 1 is an absorbing-like element of the divisors’ hyperoperation due to addition. Proof. Indeed, 1 ∈ x�y,∀x,y ∈ N ⇒ 1 ∈ 1�y,∀y ∈ N ⇒ 1 ∈ (1�y)∩(y�1),∀y ∈ N. Proposition 2.6. If y = n ·x, x,n ∈ N then {1,x, 1 + n,x(1 + n)}⊂ x�y. Proof. Let y = n ·x, x,n ∈ N then x�y = {z ∈ N : x + y = z ·λ,λ ∈ N} = {z ∈ N : x + nx = z ·λ,λ ∈ N} = {z ∈ N : x(1 + n) = z ·λ,λ ∈ N}⊃{1,x, 1 + n,x(1 + n)}. Proposition 2.7. If x ∈ N is a prime number then x2 = {1, 2,x, 2x}. Proof. Let x ∈ N, be a prime number then x2 = x�x = {z ∈ N : x + x = z ·λ,λ ∈ N} = {z ∈ N : 2x = z ·λ,λ ∈ N}. According to property 5, {1, 2,x, 2x} ⊂ x2, but since x is prime, x2 = {1, 2,x, 2x}. Proposition 2.8. x� N∗ = N∗ �x = N∗,∀x ∈ N∗. Proof. Let x ∈ N∗, then x� N∗ ⊃ x� (nx) 3 n + 1, n ∈ N∗, according to Proposition 2.6. 50 The Divisors’ Hyperoperations So, we proved that n + 1 ∈ x�N∗,∀x,n ∈ N∗ and since 1 ∈ x�N∗,∀x ∈ N∗, we get x� N∗ = N∗ �x = N∗,∀x ∈ N∗. Remark 2.9. Notice that, for x ∈ N∗ x� N = ⋃ n∈IN (x�n) = ⋃ n∈IN {z ∈ N : x + n = z ·λ,λ ∈ N∗}⊇ ⊇ ⋃ n∈IN {z ∈ N : x + nx = z ·λ,λ ∈ N∗} = ⋃ n∈IN (x�nx). But from Proposition 2.6,⋃ n∈IN (x�nx) ⊃ ⋃ n∈IN {1,x,n + 1,x(n + 1)}⊃ ⋃ n∈IN {n + 1} = N∗. So, x�N = N �x = N∗,∀x ∈ N∗. Proposition 2.10. The divisors’ hyperoperation due to addition is a weak associative one in N∗. Proof. For x,y,z ∈ N∗ (x�y) �z = {w ∈ N∗ : x + y = w ·λ,λ ∈ N∗}�z = ⋃ w∈IN∗ (w �z) = = ⋃ w∈IN∗ {w′ ∈ N∗ : w + z = w′ ·λ′,λ′ ∈ N∗} ⊃{w′ ∈ N∗ : x + y + z = w′ ·λ′,λ′ ∈ N∗} (I) On the other hand x� (y �z) = x�{v ∈ N∗ : y + z = v ·ρ,ρ ∈ N∗} = ⋃ v∈IN∗ (x�v) = = ⋃ v∈IN∗ {v′ ∈ N∗ : x + v = v′ ·ρ′,ρ′ ∈ N∗} ⊃{v′ ∈ N∗ : x + y + z = v′ ·ρ′,ρ′ ∈ N∗} (II) From (I) and (II) we get: (x�y)�z∩x�(y�z) = {n ∈ N∗ : x+y+z = n·µ,µ ∈ N∗} 6= ∅,∀x,y,z ∈ N∗. Since the divisors’ hyperoperation due to addition is commutative, ac- cording to Propositions 2.8 and 2.10, we get the following: 51 Achilles Dramalidis Proposition 2.11. The hyperstructure (N∗,�) is a commutative Hv-group. Proposition 2.12. For (x,y,z) ∈ N∗×N∗×N∗, if x = z, then the divisors’ hyperoperation due to addition is strong associative. Proof. Let (x,y,z) ∈ N∗ × N∗ ×N∗ such that x = z, then due to commuta- tivity we get: (x�y) �z = (x�y) �x = x� (x�y) = x� (y �x) = x� (y �z). Proposition 2.13. The (N∗,�), is a single-power cyclic Hv-group with in- finite period where every x ∈ N∗ is a generator. Proof. For x ∈ N∗, notice that x1 = {x} x2 = x�x = {z ∈ N∗ : 2x = z ·λ,λ ∈ N∗}⊃{1, 2} x3 = x2 �x = {z ∈ N∗ : 2x = z ·λ,λ ∈ N∗}�x = ⋃ z∈IN∗ (z �x) = ⋃ z∈IN∗ {w ∈ N∗ : z + x = w ·ρ,ρ ∈ N∗}⊃ ⊃{w′ ∈ N∗ : 2x + x = w′ ·ρ′,ρ′ ∈ N∗}∪ ∪{w′′ ∈ N∗ : x + x = w′′ ·ρ′′,ρ′′ ∈ N∗}⊃ ⊃{1, 3}∪{1, 2} = {1, 2, 3}. We shall prove that xn ⊃ {1, 2, 3, . . . ,n}, ∀x ∈ N∗, n ∈ N∗,n ≥ 2, by induction. Suppose that for n = k,k ∈ N∗,k ≥ 2 : xk ⊃{1, 2, 3, . . .,k} We shall prove that the above is valid for n = k + 1, i.e. xk+1 ⊃{1, 2, 3, . . .,k,k + 1}. Indeed, xk+1 = (xk �x) ∪ (xk−1 �x2) ∪ . . .∪ (x�xk). Then xk+1 ⊃ (xk−1 �x2) ⊃{1, 2, 3, . . . ,k − 1}�{1, 2}⊃ ⊂{1, 2, 3, . . . ,k}∪{k + 1} = {1, 2, 3, . . . ,k,k + 1}. Therefore every element of N∗ belongs to a special power of x, thus, is a generator of the single-power cyclic Hv-group. 52 The Divisors’ Hyperoperations 3 The divisors’ hyperoperation due to multi- plication in N Now, let us define the hyperoperation (⊗) in N as follows: Definition 3.1. For every x,y ∈ IN ⊗ : N × N → P(N) −{∅} : (x,y) 7→ x⊗y ⊂ N such that x�y = {z ∈ N : x ·y = z ·λ,λ ∈ N} where (·) is the usual operation of the multiplication in N. We call the above hyperoperation, divisors’ hyperoperation due to multi- plication. Some properties of the divisors’ hyperoperation due to multi- plication 1. x⊗y = y ⊗x,∀x,y ∈ N 2. 0 ⊗x = x⊗ 0 = N,∀x ∈ N 3. 1 ⊗ 1 = 1, i.e 1 is an idempotent element 4. {1,x,y,xy}⊂ x⊗y,∀x,y ∈ N Remark 3.2. If x is a prime number, then 1 ⊗x = x⊗ 1 = {1,x}. Proposition 3.3. E⊗ = N. Proof. For x,e ∈ N,x⊗e = {z ∈ N : x ·e = z ·λ,λ ∈ N}3 x. So, according to property 1, we get x ∈ (x⊗e) ∩ (e⊗x),∀x,e ∈ N That means that the set of the unit elements with respect to (⊗) is the set N, i.e. E⊗ = N. Proposition 3.4. i) I⊗(x, 0) = {0},y ∈ N ii) I⊗(x, 1) = N. Proof. i) Straightforward from property 2. ii) Take the unit element 1, then from property 4, we get 1 ∈ (x⊗y) ∩ (y ⊗x),∀x,y ∈ N which means that I⊗(x, 1) = N. 53 Achilles Dramalidis Proposition 3.5. If a unit element p is a prime number, then I⊗(x,p) = { N, x = np,n ∈ N pN, x 6= np,n ∈ N. Proof. Let p ∈ N be a unit element and p = prime number. Then p has no other divisors than 1 and itself. So, let x = np, n ∈ N, then for x′ ∈ N x⊗x′ = {z ∈ N : x ·x′ = z ·λ,λ ∈ N} = = {z ∈ N : (np) ·x′ = z ·λ,λ ∈ N}3 p,∀x′ ∈ N. That means that I⊗(x,p) = N. Let x 6= np,n ∈ N, then p ∈ {z ∈ N : x ·x′ = z ·λ,λ ∈ N}⇔ x′ = pn,n ∈ N ⇔ I⊗(x,p) = pN. Seems to be particularly interesting, one to study cases where the unit element is not a prime number. The following two examples study the cases where the unit element is 6 and 9. Example 3.6. Let 6 be the unit element. Assume that x = 6n, n ∈ N, then x⊗x′ = {z ∈ N : (6n) ·x′ = z ·λ,λ ∈ N}3 6,∀x′ ∈ N. Then, I⊗(x, 6) = N. Assume that x = 3m 6= 6n, n,m ∈ N, then 6 ∈ x⊗x′ = {z ∈ N : (3m) ·x′ = z ·λ,λ ∈ N}⇔ x′ = 2n,n ∈ N ⇔ I⊗(x, 6) = 2N. Assume that x = 2m 6= 6n, n,m ∈ N, then 6 ∈ x⊗x′ = {z ∈ N : (2m) ·x′ = z ·λ,λ ∈ N}⇔ x′ = 3n,n ∈ N ⇔ I⊗(x, 6) = 3N. Assume that x = 2m + 1 6= 3n, n,m ∈ N, then 6 ∈ x⊗x′ = {z ∈ N : (2m + 1) ·x′ = z ·λ,λ ∈ N} ⇔ x′ = 6n,n ∈ N ⇔ I⊗(x, 6) = 6N. Example 3.7. Let 9 be the unit element. Assume that x = 9n, n ∈ N, then x⊗x′ = {z ∈ N : (9n) ·x′ = z ·λ,λ ∈ N}3 9,∀x′ ∈ N. Then, I⊗(x, 9) = N. Assume that x = 3m 6= 9n, n,m ∈ N, then 9 ∈ x⊗x′ = {z ∈ N : (3m) ·x′ = z ·λ,λ ∈ N} ⇔ x′ = 3n,n ∈ N ⇔ I⊗(x, 9) = 3N. Assume that x 6= 3m, m ∈ N, then 9 ∈ x⊗x′ = {z ∈ N : x·x′ = z·λ,λ ∈ N}⇔ x′ = 9n,n ∈ N ⇔ I⊗(x, 9) = 9N. 54 The Divisors’ Hyperoperations Proposition 3.8. Every element x ∈ N is an absorbing-like element of the divisors’ hyperoperation due to multiplication. Proof. According to property 4, x ∈ x ⊗ y, ∀x,y ∈ N, which means, that for every x∈ N, x ∈ x ⊗ y, ∀y ∈ N and due to property 1, ∀x ∈ N,x ∈ (x ⊗ y) ∩ (y ⊗ x),∀y ∈ N. Then, every natural number is an absorbing-like element of the divisors’ hyperoperation due to multiplication. Proposition 3.9. The divisors’ hyperoperation due to multiplication is a strong associative one in N. Proof. For x,y,z ∈ N (x⊗y) ⊗z = {w ∈ N : x ·y = w ·λ,λ ∈ N}⊗z = ⋃ w∈IN (w ⊗z) = = ⋃ w∈IN {w′ ∈ N : w ·z = w′ ·λ′,λ′ ∈ N} = = ⋃ λ∈IN {w′ ∈ N : [1 λ (xy) ] z = w′ ·λ′,λ′ ∈ N} = = ⋃ λ∈IN {w′ ∈ N : x [1 λ (yz) ] = w′ ·λ′,λ′ ∈ N} = = ⋃ v∈IN {w′ ∈ N : x ·v = w′ ·λ′,λ′ ∈ N} = = ⋃ v∈IN (x⊗v) = x⊗{v ∈ N : y ·z = v ·λ,λ ∈ N} = x⊗ (y ⊗z). So, (x⊗y) ⊗z = x⊗ (y ⊗z),∀x,y,z ∈ N. Proposition 3.10. The hyperstructure (N,⊗) is a commutative hypergroup. Proof. Indeed, for x ∈ N, x⊗ N = (x⊗ 0) ∪ [ ⋃ n∈IN∗ (x⊗n)] = N ∪ [ ⋃ n∈IN∗ (x⊗n) ] = N. So, x⊗ N = N ⊗x = N, ∀x ∈ N. Also, according to property 1 and Proposition 3.9 we get that (N,⊗) is a commutative hypergroup. Remark 3.11. For x ∈ N∗, x⊗N∗ = ⋃ n∈IN∗ (x⊗n) = ⋃ n∈IN∗ {z ∈ N : x·n = z·λ,λ ∈ N∗}⊃ ⋃ n∈IN∗ {n} = N∗. 55 Achilles Dramalidis So, x⊗ N∗ = N∗ ⊗x = N∗,∀x ∈ N∗. Proposition 3.12. For every x ∈ N,xn−1 ⊆ xn,n ∈ N,n ≥ 2. Proof. For x ∈ N and n ∈ N,n ≥ 2 xn = (xn−1 ⊗x) ∪ (xn−2 ⊗x2) ∪ . . .∪ (xn−p ⊗xp) where p = [ n 2 ] the integer part of n 2 , [1]. Then, xn ⊇ xn−1 ⊗x ⊇ xn−1 ⊗ 1 ⊇ xn−1. 4 On a dual Hv-ring in N∗ Proposition 4.1. (x⊗y) � (x⊗z) ⊃ x⊗ (y �z),∀x,y,z ∈ N∗. Proof. For x,y,z ∈ N∗, we get x⊗ (y �z) = x⊗{w ∈ N∗ : y + z = w ·λ,λ ∈ N∗} = ⋃ w∈IN∗ (x⊗w) = = ⋃ w∈IN∗ {w′ ∈ N∗ : x ·w = w′ ·λ′,λ′ ∈ N∗} = = ⋃ λ∈IN∗ {w′ ∈ N∗ : x · y + z λ = w′ ·λ′,λ′ ∈ N∗}. On the other hand, (x⊗y) � (x⊗z) = = {v ∈ N∗ : x ·y = v ·ρ,ρ ∈ N∗}�{v′ ∈ N∗ : x ·z = v′ ·ρ′,ρ′ ∈ N∗} = ⋃ v,v′∈IN∗ (v �v′) = ⋃ v,v′∈IN∗ {k ∈ N∗ : v + v′ = k ·µ,µ ∈ N∗} = = ⋃ ρ,ρ′∈IN∗ {κ ∈ N∗ : xy ρ + xz ρ′ = κ ·µ,µ ∈ N∗}⊃ ⊃ ⋃ µ′∈IN∗ {κ′ ∈ N∗ : x · y + z µ′ = κ′ · τ ′,τ ′ ∈ N∗} = x⊗ (y �z). So, (x⊗y) � (x⊗z) ⊃ x⊗ (y �z) and then, x⊗ (y �z) ∩ (x⊗y) � (x⊗z) 6= ∅,∀x,y,z ∈ N∗. Proposition 4.2. The divisors’ hyperoperation due to addition is weak dis- tributive with respect to the divisors’ hyperoperation due to multiplication in N∗. 56 The Divisors’ Hyperoperations Proof. For x,y,z ∈ N∗, we get x� (y ⊗z) = x�{w ∈ N∗ : y ·z = w ·λ,λ ∈ N∗} = ⋃ w∈IN∗ (x�w) = = ⋃ w∈IN∗ {w′ ∈ N∗ : x + w = w′ ·λ′,λ′ ∈ N∗}⊃ ⊃{w′′ ∈ N∗ : x + y = w′′ ·λ′′,λ′′ ∈ N∗}. On the other hand, (x�y) ⊗ (x�z) = = {v ∈ N∗ : x + y = v ·ρ,ρ ∈ N∗}⊗{v′ ∈ N∗ : x + z = v′ ·ρ′,ρ′ ∈ N∗} = = ⋃ v,v′∈IN∗ (v ⊗v′) = ⋃ v,v′∈IN∗ {v′′ ∈ N∗ : v ·v′ = v′′ ·ρ′′,ρ′′ ∈ N∗}⊃ ⊃{κ ∈ N∗ : (x + y) · (x + z) = κ ·µ,µ ∈ N∗}⊃ ⊃{κ′ ∈ N∗ : x + y = κ′ ·µ′,µ′ ∈ N∗}. So, x�(y⊗z)∩(x�y)⊗(x�z) ⊃{τ ∈ N∗ : x + y = τ ·σ,σ ∈ N∗} and then x� (y ⊗z) ∩ (x�y) ⊗ (x�z) 6= ∅,∀x,y,z ∈ N∗. Proposition 4.3. The hyperstructure (N∗,�,⊗) is a commutative dual Hv- ring. Proof. Indeed, according to Propositions 2.11 and 3.10 the hyperstructures (N∗,�) and (N∗,⊗) are commutative Hv-group and commutative hypergroup respectively. From Propositions 4.1 and 4.2 we get that (⊗) is weak distribu- tive with respect to (�) and (�) is weak distributive with respect to (⊗), respectively. References [1] N. Antampoufis, Widening Complex number addition, Proc. “Struc- ture Elements of Hyper-structures”, Alexandroupolis, Greece, Spanidis Press, (2005), 5–16. [2] P. Corsini, V. 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