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The family of $a$-floor quotient partial orders

Jeffrey C. Lagarias, David Harry Richman

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.

The family of $a$-floor quotient partial orders

Abstract

An approximate divisor order is a partial order on the positive integers that refines the divisor order and is refined by the additive total order. A previous paper studied such a partial order on , produced using the floor function. A positive integer is a floor quotient of , denoted , if there is a positive integer such that . The floor quotient relation defines a partial order on the positive integers. This paper studies a family of partial orders, the -floor quotient relations , for , which interpolate between the floor quotient order and the divisor order on . The paper studies the internal structure of these orders.
Paper Structure (33 sections, 24 theorems, 89 equations, 3 figures)

This paper contains 33 sections, 24 theorems, 89 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The a-floor quotient order
  3. Hierarchy of a-floor quotient orders
  4. Structure of a-floor quotient initial intervals and almost complementation
  5. Prior work
  6. Contents of paper
  7. Preliminary results on 1-floor quotient order
  8. The 1-floor quotient order compared with divisibility order
  9. Scaling-invariance properties: divisor and 1-floor quotient order
  10. Self-duality and almost complementation properties: divisor order
  11. Self-duality and almost complementation properties: 1-floor quotient order
  12. The family of a-floor quotient orders
  13. Characterizations of a-floor quotients
  14. Hierarchy of a-floor quotient orders: Proof of Theorem \ref{['thm:hierarchy-00']}
  15. Hierarchy of a-floor quotient orders: Stabilization bounds
  16. ...and 18 more sections

Key Result

Theorem 1.3

The $a$-floor quotient relation ${\,\preccurlyeq_{a}\,}$ defines a partial order ${\mathcal{Q}}_a := ( {\mathbb{N}^+}, {\,\preccurlyeq_{a}\,})$. For each $a \ge 1$, the partial order ${\mathcal{Q}}_a$ is an approximate divisor order.

Figures (3)

  • Figure 1: Plot of set- and poset-stabilization thresholds for ${\mathcal{Q}}_a[1,n]$ for $1 \leq n \leq 100$. The horizontal axis is the $n$-variable and the vertical axis is the $a$-variable.
  • Figure 2: Plot of the number of small $2$-floor quotients $\left\vert{ {\mathcal{Q}}_2^{-}(n) }\right\vert$ (blue) versus number of large $2$-floor quotients $\left\vert{ {\mathcal{Q}}_2^{+}(n) }\right\vert$ (orange) for $n$ up to $300$. The upper envelope line is $y = \sqrt{n}$.
  • Figure 3: Plot of the total number of 2-floor quotients $\left\vert{ {\mathcal{Q}}_2[1,n] }\right\vert$ for $n$ up to $300$. The upper smooth curve is $y = 2 \sqrt{n}$ and the lower curve is $y = \sqrt{n/2}$.

Theorems & Definitions (57)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4: $a$-floor quotient order hierarchy
  • Theorem 1.5: Structure of scaling sets
  • Theorem 1.6: $a$-floor quotient order stabilization
  • Theorem 1.7: Average value of $a$-floor initial interval, variable $n$
  • Theorem 1.8: Numerical semigroup of $a$-floor multiples
  • Proposition 2.1: Almost complementation for initial $1$-floor quotient intervals
  • Remark 2.2
  • ...and 47 more