Sums and products of intervals in ordered semigroups

Pub Date : 2021-06-01 DOI:10.2478/auom-2021-0025

T. Glavosits, Zsolt Karácsony

引用次数: 2

Abstract

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.

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有序半群中区间的和与积

摘要:我们给出了一个简单的例子，对于有序半群 = (+， )，它(∈∈a, b[+] c, d[=]a + c, b + d[对于所有的a, b, c, d∈，使得a < b和c < d，但区间不是平移不变量，即方程c +]a, b[=]c + a, c + b[对于所有的元素a, b, c∈，使得a < b并不总是满足。也给出了上述例子的乘法版本。本文还研究了所有整数(记为0)的有序环上开区间的乘积。令x:= {1,2，…，x，对于所有x}∈0 +，并定义函数g: 0 +→0 +:g(x):=max {y∈0 +| y≠Ix⋅Ix} g \left (x \right):= \max\left ｛{y\in{\mathbb{Z} ＿ +}|{ I＿y }\subseteq I＿x{}\cdot I＿x{}}\right｝对于所有x∈0 +。我们用著名的切比舍夫定理隐式地给出了函数g。最后，针对上述主题提出了一些问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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