与区间集有关的一类代数结构
A type of algebraic structure related to sets of intervals
摘要：
F. Wehrung问道：给定一个集合$Ω$的子集族$\mathcal{C}$，在什么条件下$Ω$上存在全序，使得$\mathcal{C}$的每个成员都是凸的？<p>请注意，如果$A$和$B$是一个全序集的不相交凸子集，两者都不包含另一个，则$A\cup B$、$A\cap B$和$A\setminus B$也是凸的。因此，令$\mathcal{C}$是集合$Ω$的子集的任意集合，并在形成下形成其闭包$\mathcal{P}$，只要$A$和$B$不相交且不包含另一个，则形成集合$A\cup B$，$A\cap B$和$A\setminus B$。我们确定的形式$\mathcal{P}$可以采取时，$\mathcal{C}$，因此$\mathcal{P}$，是有限的，并为这种情况下得到的充要条件，存在一个订购的$Ω$的所需的排序。由此我们得到一个不需要有限性假设的条件。<p>我们建立了由$n$ -元素集$\mathcal{C}$如上生成的子集$\mathcal{P}$的基数上的界限。<p>我们注意到与<i>区间图</i>和<i>超图</i>理论的联系，这导致了回答Wehrung问题的其他方法。
英文摘要：
F. Wehrung has asked: Given a family $\mathcal{C}$ of subsets of a set $Ω$, under what conditions will there exist a total ordering on $Ω$ under which every member of $\mathcal{C}$ is convex? <p> Note that if $A$ and $B$ are nondisjoint convex subsets of a totally ordered set, neither of which contains the other, then $A\cup B$, $A\cap B$, and $A\setminus B$ are also convex. So let $\mathcal{C}$ be an arbitrary set of subsets of a set $Ω$, and form its closure $\mathcal{P}$ under forming, whenever $A$ and $B$ are nondisjoint and neither contains the other, the sets $A\cup B$, $A\cap B$, and $A\setminus B$. We determine the form $\mathcal{P}$ can take when $\mathcal{C}$, and hence $\mathcal{P}$, is finite, and for this case get necessary and sufficient conditions for there to exist an ordering of $Ω$ of the desired sort. From this we obtain a condition which works without the finiteness hypothesis. <p> We establish bounds on the cardinality of the subset $\mathcal{P}$ generated as above by an $n$-element set $\mathcal{C}$. <p> We note connections with the theory of <i>interval graphs</i> and <i>hypergraphs</i>, which lead to other ways of answering Wehrung's question.
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