This paper proposes a new proof of the existence of constant threshold representations of semiorders on countably infinite sets. The construction treats each indifference-connected component of the semiorder separately. It uses a partition of such an indifference-connected component into indifference classes. Each element in the indifference-connected component is mirrored, using a “ghost” element, into a reference indifference class that is weakly ordered. A numerical representation of this weak order is used as the basis for the construction of the unit representation after an appropriate lifting operation. We apply the procedure to each indifference-connected component and assemble them adequately to obtain an overall unit representation.

Our proof technique has several original features. It uses elementary tools and can be seen as the extension of a technique designed for the finite case, using a denumerable set of inductions. Moreover, it gives us much control on the representation that is built, so that it is, for example, easy to investigate its uniqueness. Finally, we show in a companion paper that our technique can be extended to the general (uncountable) case, almost without changes, through the addition of adequate order-denseness conditions.

本文提出了一个新的证明，证明在可数无穷集合上半阶的常数阈值表示存在。该构造分别处理半阶的每个无差异连通分量。它使用将这种无差异连接组件划分为无差异类。无差异连通分量中的每个元素都使用“幽灵”元素镜像到弱排序的参考无差异类中。在适当的提升操作之后，这个弱阶的数值表示被用作构建单元表示的基础。我们将该过程应用于每个无差异连通分量，并充分组装它们以获得整体单元表示。

我们的证明技术有几个原始特征。它使用基本工具，可以看作是为有限情况设计的技术的扩展，使用了一组可数的归纳。此外，它让我们可以对构建的表示进行很多控制，因此，例如，很容易研究其独特性。最后，我们在一篇配套论文中展示了我们的技术可以扩展到一般（不可数）情况，通过添加足够的有序密度条件，几乎没有变化。