Sciendo Logo
  1. bookTom 29 (2021): Zeszyt 2 (June 2021)
Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică's Cover Image
Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică
Informacje o czasopiśmie
License
Format
Czasopismo
eISSN
1844-0835
Pierwsze wydanie
17 May 2013
Częstotliwość wydawania
1 raz w roku
Języki
Angielski
access type Otwarty dostęp

Sums and products of intervals in ordered semigroups

T. Glavosits oraz
T. Glavosits oraz
Zs. Karácsony
Zs. Karácsony
Data publikacji: 08 Jul 2021
Tom & Zeszyt: Tom 29 (2021) - Zeszyt 2 (June 2021)
Zakres stron: 187 - 198
Otrzymano: 11 Jul 2020
Przyjęty: 31 Aug 2020
Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică's Cover Image
Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică
Informacje o czasopiśmie
License
Format
Czasopismo
eISSN
1844-0835
Pierwsze wydanie
17 May 2013
Częstotliwość wydawania
1 raz w roku
Języki
Angielski
Abstract

We show a simple example for ordered semigroup 𝕊 = 𝕊 (+,⩽) that 𝕊 ⊆ℝ (ℝ denotes the real line) and ]a, b[ + ]c, d[ = ]a + c, b + d[ for all a, b, c, d ∈ 𝕊 such that a < b and c < d, but the intervals are no translation invariant, that is, the equation c +]a, b[ = ]c + a, c + b[ is not always fulfilled for all elements a, b, c ∈ 𝕊 such that a < b.

The multiplicative version of the above example is shown too.

The product of open intervals in the ordered ring of all integers (denoted by ℤ) is also investigated. Let Ix := {1, 2, . . ., x} for all x ∈ ℤ+ and defined the function g : ℤ+ → ℤ+ by g(x):=max{ y∈ℤ+|Iy⊆Ix⋅Ix } g\left( x \right): = \max \left\{ {y \in {\mathbb{Z}_ + }|{I_y} \subseteq {I_x} \cdot {I_x}} \right\} for all x ∈ ℤ+. We give the function g implicitly using the famous Theorem of Chebishev.

Finally, we formulate some questions concerning the above topics.

Keywords

  • interval
  • ordered dense Abelian group
  • ordered field

MSC 2010

  • Primary 39B22

[1] M. El Bachraoui, Primes in the interval [2n; 3n], Int. J. Contemp. Math. Sciences, 1(13) (2006), 617-621.10.12988/ijcms.2006.06065 Search in Google Scholar

[2] A.M. Bruckner, J.B. Bruckner, B.S. Thomson, Elementary Real Analysis, Prentice-Hall (2001) Search in Google Scholar

[3] P. Erd¨os, Beweis eines satzes von tschebyschef. Acta Litt. Univ. Sci., Szeged, Sect. Math., 5 (1932), 194–198. Search in Google Scholar

[4] L. Fuchs, Partially Ordered Algebraic Systems, Dover Publications, Inc. Minesota, New York. (1963) Search in Google Scholar

[5] P. Erdei, T. Glavosits On the functional equation f (x + y) = g(xy), (In preparation) Search in Google Scholar

[6] T. Glavosits, Short remark to the Rim´an’s Theorem, (In preparation) Search in Google Scholar

[7] T. Glavosits, Zs. Kar´acsony, On the restricted Pexider additive functional equations on rectangulars of ℤ2, (In preparation) Search in Google Scholar

[8] T. Glavosits, Zs. Kar´acsony, Sums and products of intervals in ordered semigroups and fields, (accepted in Acta Univ. Sap., Math. (2021)10.2478/ausm-2021-0010 Search in Google Scholar

[9] T. Glavosits, À. Sz´az, On the existence of nonnegativity domains of subsets of groups. Demonstratio Math. 37 (2004), 505–516. Search in Google Scholar

[10] F. W. Levi, Arithmetische Gesetze im Gebiete diskreter Gruppen, Rend. Circ. Mat. Palermo 35 (1913), 225–236.10.1007/BF03015602 Search in Google Scholar

[11] P. Lorenzen, Abstrakte Begründung der multiplikativen Idelatheorie, Math. Z. 45 (1939), 533–553.10.1007/BF01580299 Search in Google Scholar

[12] R. E. Moore, Automatic error analysis in digital computation. Technical Report LMSD-48421 Lockheed Missiles and Space Co, Palo Alto, CA., (1959) Search in Google Scholar

[13] H. Simbireva, On the theory of partially ordered groups, Mat. Sb.20 (1947), 145–178. (in Russian) Search in Google Scholar

Polecane artykuły z Trend MD

Zaplanuj zdalną konferencję ze Sciendo