We show that under the proper forcing axiom the class of all Aronszajn lines behave like $\sigma$-scattered orders under the embeddability relation. In particular, we are able to show that the class of better-quasi-order labeled fragmented Aronszajn lines is itself a better-quasi-order. Moreover, we show that every better-quasi-order labeled Aronszajn line can be expressed as a finite sum of labeled types which are algebraically indecomposable. By encoding lines with finite labeled trees, we are also able to deduce a decomposition result, that for every Aronszajn line $L$ there is an integer $n$ such that for any finite coloring of $L$ there is subset $L^{\prime }$ of $L$ isomorphic to $L$ which uses no more than $n$ colors.

我们表明，在适当的强制公理下，所有 Aronszajn 线的类表现得像$\sigma$- 可嵌入关系下的分散订单。特别是，我们能够证明带有更好准序标记的碎片化 Aronszajn 线本身就是更好准序。此外，我们表明，每个更好准有序标记的 Aronszajn 线都可以表示为代数不可分解的标记类型的有限和。通过用有限标记树对线进行编码，我们还能够推导出分解结果，对于每个 Aronszajn 线$\mathrm{大号}$有一个整数$n$这样对于任何有限的着色$\mathrm{大号}$有子集${\mathrm{大号}}^{‘}$的$\mathrm{大号}$同构于$\mathrm{大号}$使用不超过$n$颜色。