论文标题
与一组间隔有关的一种代数结构
A type of algebraic structure related to sets of intervals
论文作者
论文摘要
F. Wehrung问：给定一个家庭$ \ MATHCAL {C} $的子集的$ω$的子集，在哪些条件下，在$ \ Mathcal {c} $的每个成员中都有$ω$的总订单？ <p>请注意，如果$ a $ and $ b $是完全有序集的nondisjoint凸出子集，它们都不包含另一个，则$ a \ cup b $，$ a \ cap b $和$ a \ setminus b $也是convex。因此，让$ \ Mathcal {C} $为组合$ω$的一组任意集，并在形成下形成其闭合$ \ Mathcal {p} $，只要$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a \ a \ cap b $，$ a \ cap b $，$ a \ cap b $ a \ a $ a \ a \ a $ b $。我们确定$ \ MATHCAL {P} $的形式可以在$ \ Mathcal {C} $时，因此是$ \ Mathcal {p} $是有限的，在这种情况下，有必要且足够的条件，以便在那里存在$ω$的订购。由此，我们获得了一个没有有限假设的条件。 <p>我们在子集$ \ Mathcal {p} $的基数上建立了界限，如$ n $ element Set $ \ Mathcal {C} $生成的上述界限。 <p>我们注意到与<i>间隔图理论的联系</i>和<i> hypergraphs </i>，这导致了其他回答Wehrung问题的方法。
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.
