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On Hyperbolic Integers Hichem Gargoubi, Sayed Kossentini

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On Hyperbolic Integers Hichem Gargoubi, Sayed Kossentini On Hyperbolic Integers Hichem Gargoubi, Sayed Kossentini To cite this version: Hichem Gargoubi, Sayed Kossentini. On Hyperbolic Integers. 2021. hal-03198641 HAL Id: hal-03198641 https://hal.archives-ouvertes.fr/hal-03198641 Preprint submitted on 15 Apr 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On Hyperbolic Integers Hichem Gargoubi* and Sayed Kossentini Abstract The algebra D of hyperbolic numbers is endowed with its standard f-algebra D structure [7]. We introduce the ring of hyperbolic integers Zh as a sub f-ring of Z the ring of integers of D. We prove that Zh is the unique, up to ring isomorphism, Archimedean f-ring of quadratic integers. Our study focuses on arithmetic prop- erties of Zh related to its lattice-ordered structure. We show that many of basic properties of the ring of integers Z such as primes, unique factorization theorem and the notions of floor and ceiling functions can be extended to Zh. In particular, Some properties of the hyperbolic Gaussian integers are obtained. As an application, we discuss the Dirichlet divisor problem using hyperbolic intervals and introduce the notion of toroidal groups. Key words: Hyperbolic numbers, Hyperbolic Gaussian integers, f-ring, `-group, Lattice, Regular toroid. 1 Introduction Motivated by solving problems concerning integers, number theorists of the XIX century have been interested in the analogous of Z when the concept of arithmetic can be devel- oped. This leads Dedekind [3] to introduce and study the ring of integers OK of a number field K. Usual notions of the arithmetic of Z, such as unique factorization into prime elements and Euclidean division, can be generalized, with various degrees, to such a ring. In fact, when one studies the ring of integers Z it becomes clear to take into account the relation between its divisibility and its order properties as the existence of a unique positive gcd and the existence of unique factorization into product of positive primes. In this paper, the usual order structure of Z is seen as an Archimedean f-ring. Indeed, the general notion of f-algebra is simultaneously a Riesz space (or vector lattice) and an associative real algebra that fulfills certain "positivity" conditions. A typical example of f-algebras is the linear space of real valued continuous functions on a topological space. Obviously, the fundamental example of Archimedean f-algebras is the field of real numbers. *Universit´ede Tunis, I.P.E.I.T., Department of Mathematics, 2 Rue Jawaher Lel Nehru, Monfleury, Tunis, 1008 Tunisia. E-mail address: [email protected] Universit´ede Tunis El Manar, Facult´edes Sciences de Tunis, Department of Mathematics, 2092, Tunis, Tunisia. E-mail address: [email protected] 1 One of the points that motivates our approach is that there is no analogous of positiv- ity in the ring OK . Actually, we prove a more general result concerning ring extensions of Q (see Theorem 3.1). This leads us to seek, in the direction hyperbolic numbers, a class of lattice-ordered rings (`-rings) of algebraic integers where we are able to generalize many of the basic divisibility and order properties of Z. The hyperbolic numbers D (also called duplex numbers) are an extension of real num- bers defined in the same way that complex numbers C but with an imaginary unit j satisfying j2 = 1 (instead of i2 = −1) and so, D is not a division algebra. However, it enjoys an important order structure which makes it into the unique (up to an isomor- phism) two dimensional Archimedean f-algebra. Therefore, basic notions of real analysis as sign, absolute value, Archimedean and Dedekind completeness are extended to the hyperbolic numbers [7]. Note that complex numbers and hyperbolic numbers are the only real commutative Clifford algebras: ∼ ∼ D = ClR(1; 0) and C = ClR(0; 1) The notion of partial order on D stimulates many authors and leads to interesting applications in different areas of mathematics. Alpay et al. [1] investigated the D-normed bicomplex modulus. In probability theory it is shown in [2] that Kolmogrov's axioms and Bays theorems holds in the context of D-valued probabilities. Kumar et al. [15] intro- duced the notion of D-valued measure on a sigma algebra. As an Application to fractal geometry, a concept of Cantor sets in hyperbolic numbers was developed by Balankin et al. [4] and T´ellez-S´anchez et al. [25]. Recently, the authors of the present paper used in [8] lattice-theoretical results to go further in the development of the theory of bicomplex zeta function. Further applications can be found in [13, 14, 17, 18, 20]. The purpose of this paper is to investigate a `-ring structure of algebraic integers from an arithmetic point of view. Our main result (Theorem 3.2) is the existence of a unique, up to ring isomorphism, Archimedean f-ring of quadratic integers called the ring of hy- perbolic integers denoted Zh. The present paper is organized in the following way: In section 2, we recall some notions and terminology concerning `-group, f-ring and f-algebra and we present basic notions and properties of hyperbolic numbers that will be used throughout this article. Section 3 introduces the lattice-ordered ring Zh of hyperbolic algebraic integers, and various of its properties are established. We introduce the notions of hyperbolic floor and ceiling functions which generalize that of real numbers. Section 4 and 5 are devoted to the divisibility in Zh. Many of the basic concepts of the arithmetic of Z are extended to Zh: the Euclidean division, the existence of a unique positive gcd, the existence of a unique factorization into a product of positive primes. In the last section, we establish some properties of the hyperbolic Gaussian integers as a subring of Zh. As an application, we discuss the Dirichlet divisor problem using hyperbolic intervals and introduce the notion of toroidal groups with hyperbolic indices. 2 2 Preliminaries In this section, we recall basic properties of hyperbolic numbers which we will use through- out this paper (see [7]). First, we need to recall some notions and terminology of `-groups, f-rings [5, 23] and f-algebras [26]. A partially ordered group (G; +) is a partially ordered set under a partial order ≤ such that for any a; b 2 G, a ≤ b implies a+c ≤ b+c and c+a ≤ c+b for all c 2 G. A partially ordered group is a lattice-ordered group (an `-group) if the partial order is a lattice order (i.e., the supremum a ^ b and the infimum a _ b exist in G for all a; b 2 G). Every element a in an `-group G can be written as a = a+ − a− where a+ = a _ 0 and a− = −a _ 0. The absolute value of a is defined as jaj = a _ (−a) = a+ + a−. An `-group G is said to be Archimedean if for each nonzero a in G the set fna : n 2 Zg has no upper bound in G; equivalently, a; b in G+ and na ≤ b for all n 2 N implies a = 0. An Archimedean `-group is abelian. A real vectorial space V is said to be a vector lattice or Riesz space if V as a group is an `-group satisfying the property: a; b 2 V , a ≤ b and α 2 R+ implies αa ≤ αb. A ring R is an `-ring if R is an `-group and ab 2 R+ for every a; b 2 R+. An `-ring is Archimedean if its underlying group is Archimedean. An f-ring is an `-ring such that for every a; b 2 R+; a^b = 0 implies ac^b = ca^b = 0 for all c 2 R+. An associative real algebra is an f-algebra if it is an f-ring and its underlying group is a vector lattice. Hyperbolic numbers are defined as the set n 2 o D := z = x + jy : x; y 2 R; j 2= R; j = 1 : It is a commutative ring with the group of units defined as n o D∗ = z 2 D : kzkh 6= 0 ; where kzkh := zz¯ denotes the hyperbolic square-norm of z = x + jy,(x; y 2 R) andz ¯ is the conjugate of z given by exchanging y ! −y. The hyperbolic plane has an important basis fe1; e2g where 1 + j 1 − j e = ; e = : 1 2 2 2 e1 and e2 are mutually complementary idempotent zero divisors, i.e., 2 2 e1 = e1; e2 = e2; e1 + e2 = 1; e1e2 = 0: (2.1) In this basis, each hyperbolic number z can be written as z = π1(z)e1 + π2(z)e2; (2.2) where the maps π1; π2 : D ! R are a pair of surjective ring homomorphisms defined by π1(x + jy) = x + y and π2(x + jy) = x − y: 3 From the representation (2.2), called the spectral decomposition [22], algebraic operations correspond to coordinate-wise operations, the square norm of z is the product π1(z)π2(z) and its conjugation is given by exchanging π1(z) $ π2(z). Moreover, we can define a partial order ≤ on D that makes it into Archimedean f-algebra, where z; w 2 D; z ≤ w if and only if πk(z) ≤ πk(w); k = 1; 2: (2.3) From this ordering the lattice operation are z _ w = maxfπ1(z); π2(z)ge1 + maxfπ1(z); π2(z)ge2; z ^ w = minfπ1(z); π2(z)ge1 + minfπ1(z); π2(z)ge2: Moreover, z _w and z ^w can be expressed as a I(D)-combination of z and w, where I(D) means the set of all idempotent elements of D.
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