The problem of finding a utility function for a semiorder has been studied since 1956, when the notion of semiorder was introduced by Luce. But few results on continuity and no result like Debreu’s Open Gap Lemma, but for semiorders, was found. In the present paper, we characterize semiorders that accept a continuous representation (in the sense of Scott–Suppes). Two weaker theorems are also proved, which provide a programmable approach to Open Gap Lemma, yield a Debreu’s Lemma for semiorders, and enable us to remove the open-closed and closed-open gaps of a set of reals while keeping the threshold.

中文翻译：

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半序和连续 Scott–Suppes 表示。带阈值的 Debreu 开隙引理

自 1956 年 Luce 引入半序的概念以来，就一直在研究寻找半序的效用函数的问题。但是关于连续性的结果很少，也没有像 Debreu 的 Open Gap Lemma 这样的结果，但对于半序，被发现。在本文中，我们描述了接受连续表示的半序（在 Scott–Suppes 的意义上）。还证明了两个较弱的定理，它们为开隙引理提供了一种可编程的方法，产生了半阶的德布鲁引理，并使我们能够在保持阈值的同时移除一组实数的开-闭和闭-开间隙。