$a$-向下取整商偏序族
The family of $a$-floor quotient partial orders
AI总结：
该研究提出了正整数集上介于向下取整商偏序与除数偏序之间的$a$-向下取整商偏序族，探究了这类近似除数序的内部结构，丰富了数论中偏序关系的研究内容。
摘要：
近似除数序是正整数集$\n\mathbb{N}^+$上的一种偏序，它是除数序的加细，同时是加法全序的特化。此前一篇论文研究了$\n\mathbb{N}^+$上一种由向下取整函数构造的此类偏序：若存在正整数$k$使得$d = \lfloor{n / k}\rfloor$，则称正整数$d$是$n$的向下取整商，记为$d \\,\preccurlyeq_{1}\\, n$，该向下取整商关系在正整数集上定义了一种偏序。本文研究了一族偏序，即适用于$a \in \mathbb{N}^+$的$a$-向下取整商关系$\\,\preccurlyeq_{a}\\,$，它是$\n\mathbb{N}^+$上向下取整商序与除数序之间的插值结构，论文探究了这些序的内部结构。
英文摘要：
An approximate divisor order is a partial order on the positive integers $\mathbb{N}^+$ that refines the divisor order and is refined by the additive total order. A previous paper studied such a partial order on $\mathbb{N}^+$, produced using the floor function. A positive integer $d$ is a floor quotient of $n$, denoted $d \,\preccurlyeq_{1}\, n$, if there is a positive integer $k$ such that $d = \lfloor{n / k}\rfloor$. The floor quotient relation defines a partial order on the positive integers. This paper studies a family of partial orders, the $a$-floor quotient relations $\,\preccurlyeq_{a}\,$, for $a \in \mathbb{N}^+$, which interpolate between the floor quotient order and the divisor order on $\mathbb{N}^+$. The paper studies the internal structure of these orders.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
