<abstract xmlns="http://www.w3.org/1999/xhtml">

We show a simple example for ordered semigroup 𝕊 = 𝕊 (+,⩽) that 𝕊 ⊆ℝ (ℝ denotes the real line) and ]<italic>a, b</italic>[ + ]<italic>c, d</italic>[ = ]<italic>a</italic> + <italic>c, b</italic> + <italic>d</italic>[ for all <italic>a</italic>, <italic>b</italic>, <italic>c</italic>, <italic>d</italic> ∈ 𝕊 such that <italic>a &lt; b</italic> and <italic>c &lt; d</italic>, but the intervals are no translation invariant, that is, the equation <italic>c</italic> +]<italic>a, b</italic>[ = ]<italic>c</italic> + <italic>a, c</italic> + <italic>b</italic>[ is not always fulfilled for all elements <italic>a</italic>, <italic>b</italic>, <italic>c</italic> ∈ 𝕊 such that <italic>a &lt; b</italic>.
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 <italic>Ix</italic> := {1, 2, . . ., <italic>x</italic>} for all <italic>x</italic> ∈ ℤ+ and defined the function <italic>g</italic> : ℤ+ → ℤ+ by
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2021-0025_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mo>max</mo><mrow><mo>{</mo> <mrow><mi>y</mi><mo>∈</mo><msub><mrow><mi>ℤ</mi></mrow><mo>+</mo></msub><mo>|</mo><msub><mrow><mi>I</mi></mrow><mi>y</mi></msub><mo>⊆</mo><msub><mrow><mi>I</mi></mrow><mi>x</mi></msub><mo>⋅</mo><msub><mrow><mi>I</mi></mrow><mi>x</mi></msub></mrow> <mo>}</mo></mrow></mrow>
</math>
<tex-math>g\left( x \right): = \max \left\{ {y \in {\mathbb{Z}_ + }|{I_y} \subseteq {I_x} \cdot {I_x}} \right\}</tex-math>
</alternatives>
</disp-formula>
for all <italic>x</italic> ∈ ℤ+. We give the function <italic>g</italic> implicitly using the famous Theorem of Chebishev.
Finally, we formulate some questions concerning the above topics.
</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.

Sums and products of intervals in ordered semigroups

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Sums and products of intervals in ordered semigroups"}

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...

Sums and products of intervals in ordered semigroups

Published Online: Jul 08, 2021

Page range: 187 - 198

Received: Jul 11, 2020

Accepted: Aug 31, 2020

DOI: https://doi.org/10.2478/auom-2021-0025

Keywords
interval, ordered dense Abelian group, ordered field

© 2021 T. Glavosits et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Sums and products of intervals in ordered semigroups

Published Online: Jul 08, 2021

Page range: 187 - 198

Received: Jul 11, 2020

Accepted: Aug 31, 2020

DOI: https://doi.org/10.2478/auom-2021-0025

Keywordsinterval, ordered dense Abelian group, ordered field

© 2021 T. Glavosits et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
interval, ordered dense Abelian group, ordered field