1. The family of $a$-floor quotient partial orders
- Author
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Lagarias, Jeffrey C. and Richman, David Harry
- Subjects
Mathematics - Number Theory ,06A06, 11A05 (Primary) 05A16, 15B36, 26D07 (Secondary) - Abstract
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., Comment: 30 pages, 3 figures, comments welcome! arXiv admin note: text overlap with arXiv:2212.11689
- Published
- 2024