论文标题
Debreu的半月式空白差距引理
Debreu's open gap lemma for semiorders
论文作者
论文摘要
自1956年在\ emph {EconiceTrica}概念中引入的R.D. Luce以来，已经研究了为半手找到（连续的）效用函数的问题。表示代表的连续性几乎没有结果。与Debreu的引理相似，但对于半月式而言，从未实现过。最近，A。Estevan提出了一些连续代表以及一些猜想的必要条件。在本文中，我们证明了这些猜想，从而实现了DebReu的开放缝隙引理的所需版本。该结果允许删除子集$ s \ subseteq \ mathbb {r} $的开放式闭合和封闭的间隙，但现在保持恒定阈值，因此$ x+1 <y $ if and＆horn if $ g（x）+g（x）+1 <g（y）因此，表征了有限的半符号的连续表示（在斯科特 - 供应意义上）。这些结果得益于$ε$ - 续签的关键概念，从而概括了半度性的连续性概念。
The problem of finding a (continuous) utility function for a semiorder has been studied since in 1956 R.D. Luce introduced in \emph{Econometrica} the notion. There was almost no results on the continuity of the representation. A similar result to Debreu's Lemma, but for semiorders, was never achieved. Recently, some necessary conditions for the existence of a continuous representation as well as some conjectures were presented by A. Estevan. In the present paper we prove these conjectures, achieving the desired version of Debreu's Open Gap Lemma for bounded semiorders. This result allows to remove the open-closed and closed-open gaps of a subset $S\subseteq \mathbb{R}$, but now keeping the constant threshold, so that $x+1<y$ if and only if $g(x)+1<g(y) \, (x,y\in S)$. Therefore, the continuous representation (in the sense of Scott-Suppes) of bounded semiorders is characterized. These results are achieved thanks to the key notion of $ε$-continuity, which generalizes the idea of continuity for semiorders.
