In the set N of the Natural Numbers we define two hyperoperations based on the divisors of the addition and multiplication of two numbers. 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 H v -ring is resulting which is dual

Dramalidis, Achilles

English

Achilles Dramalidis

Directory of Open Access Journals

RATIO MATHEMATICA 25(2013), 47-58 ISSN:1592-7415The Divisors’ HyperoperationsAchilles DramalidisDemocritus University of Thrace, School of Education,681 00 Alexandroupolis, Greeceadramali@psed.duth.grAbstractIn the set N of the Natural Numbers we define two hyperoperationsbased on the divisors of the addition and multiplication of two num-bers. Then, the properties of these two hyperoperations are studiedtogether with the resulting hyperstructures. Furthermore, from thecoexistence of these two hyperoperations in N∗, an Hv-ring is result-ing which is dual.Key words: Hyperstructures, Hv-structuresMSC2010: 20N20, 16Y99.1 IntroductionIn 1934, F. Marty introduced the definitions of the hyperoperation and ofthe 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 , ifx ◦ 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 Dramalidisx ◦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 structuressatisfy 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 (∗) iscalled 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 itis 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 commutativeand it is called H v- commutative group if (◦) is weak commutative. Thehyperstructure (H, ◦, ∗) is called H v-ring if both hyperstructures (◦) and (∗)are WASS, the reproduction axiom is valid for (◦), and (∗) is weak distributivewith 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 unitelement 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 thereexists 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, xis called left absorbing-like element (resp. right absorbing-like element). Anelement a ∈ H is called idempotent element if a2 = a. The nth power of anelement h, denoted hs, is defined to be the union of all expressions of n times48The Divisors’ Hyperoperationsof 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 existsa positive integer s, such that H = h1 ∪ h2 . . . ∪ hs. The minimum suchs 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 cyclicand h is a generator with single-power period s. The cyclicity in the infinitecase is defined similarly. Thus, for example, the Hv-group (H, ·) is calledsingle-power cyclic with infinite period with generator h if every element ofH belongs to a power of h and there exists s0 ≥ 1, such that ∀s ≥ s0 wehave:h1 ∪ h2 ∪ · · · ∪ hs−1 ⊂ hs.2 The divisors’ hyperoperation due to addi-tion in NLet 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 ⊂ Nsuch thatx y = {z ∈ N : x+ y = z · λ, λ ∈ N}where (+) and (·) are the usual operations of the addition and multiplicationin N, respectively.We call the above hyperoperation, divisors’ hyperoperation due to addition.Some properties of the divisors’ hyperoperation due to addition1. x y = y  x,∀x, y ∈ N2. 0 0 = N3. 0 1 = 1 0 = 14. {1, x+ y} ⊂ x y,∀x, y ∈ N5. If x+ y = κ · ν ⇒ {1, κ, ν, κ · ν} ⊂ x y, x, y, κ, ν ∈ N.49Achilles DramalidisRemark 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 ∈ Nx 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 thenx 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 thenx2 = 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∗, thenx N∗ ⊃ x (nx) 3 n+ 1, n ∈ N∗, according to Proposition 2.6.50The Divisors’ HyperoperationsSo, we proved that n+ 1 ∈ xN∗,∀x, n ∈ N∗ and since 1 ∈ xN∗,∀x ∈ N∗,we getx 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 weakassociative 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 handx (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:51Achilles DramalidisProposition 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 thatx1 = {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, byinduction.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).Thenxk+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 agenerator of the single-power cyclic Hv-group.52The Divisors’ Hyperoperations3 The divisors’ hyperoperation due to multi-plication in NNow, 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 ⊂ Nsuch thatx 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-plication1. x⊗ y = y ⊗ x,∀x, y ∈ N2. 0⊗ x = x⊗ 0 = N, ∀x ∈ N3. 1⊗ 1 = 1, i.e 1 is an idempotent element4. {1, x, y, xy} ⊂ x⊗ y,∀x, y ∈ NRemark 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, accordingto property 1, we getx ∈ (x⊗ e) ∩ (e⊗ x),∀x, e ∈ NThat means that the set of the unit elements with respect to (⊗) is theset 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 get1 ∈ (x⊗ y) ∩ (y ⊗ x),∀x, y ∈ Nwhich means that I⊗(x, 1) = N.53Achilles DramalidisProposition 3.5. If a unit element p is a prime number, thenI⊗(x, p) ={N, x = np, n ∈ NpN, x 6= np, n ∈ N.Proof. Let p ∈ N be a unit element and p = prime number. Then p has noother divisors than 1 and itself. So, let x = np, n ∈ N, then for x′ ∈ Nx⊗ 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 unitelement is not a prime number. The following two examples study the caseswhere the unit element is 6 and 9.Example 3.6. Let 6 be the unit element. Assume that x = 6n, n ∈ N, thenx⊗ 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, then6 ∈ 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, then6 ∈ 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, then6 ∈ 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, thenx⊗ 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, then9 ∈ 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, then9 ∈ x⊗x′ = {z ∈ N : x·x′ = z ·λ, λ ∈ N} ⇔ x′ = 9n, n ∈ N⇔ I⊗(x, 9) = 9N.54The Divisors’ HyperoperationsProposition 3.8. Every element x ∈ N is an absorbing-like element of thedivisors’ hyperoperation due to multiplication.Proof. According to property 4, x ∈ x ⊗ y, ∀x, y ∈ N, which means, thatfor 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-likeelement of the divisors’ hyperoperation due to multiplication.Proposition 3.9. The divisors’ hyperoperation due to multiplication is astrong 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 :[1λ(xy)]z = w′ · λ′, λ′ ∈ N} ==⋃λ∈IN{w′ ∈ N : x[1λ(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 acommutative 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∗.55Achilles DramalidisSo, 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 ≥ 2xn = (xn−1 ⊗ x) ∪ (xn−2 ⊗ x2) ∪ . . . ∪ (xn−p ⊗ xp)where p =[n2]the integer part of n2, [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 getx⊗ (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 + zλ= 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∗ : xyρ+xzρ′= κ · µ, µ ∈ N∗} ⊃⊃⋃µ′∈IN∗{κ′ ∈ N∗ : x · y + zµ′= 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 multiplicationin N∗.56The Divisors’ HyperoperationsProof. For x, y, z ∈ N∗, we getx (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 thenx (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 hypergrouprespectively. 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, SpanidisPress, (2005), 5–16.[2] P. Corsini, V. Leoreanu, Applications of Hypergroup Theory, KluwerAcademic Publishers, 2003.[3] B. Davvaz, V. Leoreanu-Fotea, Hyperring Theory and Applications, In-ternational Academic Press, 2007.57Achilles Dramalidis[4] A. Dramalidis, Dual H v-rings (MR1413019), Rivista di MatematicaPura ed Applicata, Italy, v. 17, (1996), 55–62.[5] A. Dramalidis, On geometrical hyperstructures of finite order, RatioMathematica, Italy, v. 21(2011), 43–58.[6] A. Dramalidis, T. Vougiouklis,H v-semigroups an noise pollution modelsin urban areas, Ratio Mathematica, Italy, v. 23(2012), 39–50.[7] T. Vougiouklis, The fundamental relation in hyperrings. The generalhyperfield, 4thAHA Congress, World Scientific (1991), 203–211.[8] T. Vougiouklis, Hyperstructures and their Representations, Monographsin Mathematics, Hadronic Press, 1994.58

The Divisors’ Hyperoperations

http://eiris.it/ojs/index.php/ratiomathematica/article/download/49/75

The Divisors’ Hyperoperations

Abstract

Similar works

Full text

Available Versions

Directory of Open Access Journals