The family of $a$-floor quotient partial orders

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 $d=\lfloor n\mathrm{/}k\rfloor$. 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.