Necessary and sufficient conditions under which semiorders on uncountable sets can be represented by a real-valued function and a constant threshold are known. We show that the proof strategy that we used for constructing representations in the case of denumerable semiorders can be adapted to the uncountable case. We use it to give an alternative proof of the existence of strict unit representations. In contrast to the countable case, semiorders on uncountable sets that admit a strict unit representation do not necessarily admit a nonstrict unit representation, and conversely. By adapting the proof strategy used for strict unit representations, we establish a characterization of the semiorders that admit a nonstrict representation. Conditions for the existence of other special unit representations are also obtained.

已知不可数集上的半阶可以由实值函数和恒定阈值表示的充分必要条件。我们表明，我们用于在可数半阶情况下构造表示的证明策略可以适用于不可数情况。我们用它来给出严格单位表示存在的另一种证明。与可数情况相反，不可数集上的半序允许严格单位表示并不一定允许非严格单位表示，反之亦然。通过调整用于严格单位表示的证明策略，我们建立了允许非严格表示的半阶的表征。还获得了其他特殊单位表示存在的条件。