论文标题
良好封面的方向
Orientation of good covers
论文作者
论文摘要
我们研究满足以下所谓内部状态的三元组的方向系统：$ \ CrocklearRowLowleft（ABD）=〜\ crocclearRowLowleft（bcd）=〜\ crocclearrowleft（cad）= 1 $ insciles $ \ crocklearrowleft（abc）= 1 $ for noth $ a，b，b，b，b，c，c，c，c，c，c，c，c，c，c，c，c，c，c，c，c，c，c，c，c，c，c，c，d $。我们将这种取向称为P3O（部分3阶），这是Poset的自然概括，具有几个有趣的特殊情况。例如，平面点集的顺序类型（可以具有共线三元）为p3O；我们将p3O表示为p-p3O。如果我们不允许$ \ circlearRowleft（ABC）= 0 $，我们将获得T3O（总计3阶）。与线性顺序相反，T3O可以具有丰富的结构。 T3O可按点（P-T3O）实现的t3O是一般位置设置的点的顺序类型。在我们的论文“凸集的方向”中，我们在成对相交的凸组上定义了3阶；这样的P3O称为C-P3O。在本文中，我们将这三阶扩展到成对相交的良好盖；这样的P3O称为GC-P3O。如果我们不允许$ \ circlearrowleft（ABC）= 0 $，则分别获得C-T3O和GC-T3O。本文的主要结果是，有一个不是GC-T3O的P-T3O，这也意味着它不是C-T3O，这是我们较早的论文中的后一个问题。我们的证明涉及几种可能具有独立感兴趣的组合和几何观测。一路上，我们定义了GC-T3O的几个特殊家庭。
We study systems of orientations on triples that satisfy the following so-called interiority condition: $\circlearrowleft(ABD)=~\circlearrowleft(BCD)=~\circlearrowleft(CAD)=1$ implies $\circlearrowleft(ABC)=1$ for any $A,B,C,D$. We call such an orientation a P3O (partial 3-order), a natural generalization of a poset, that has several interesting special cases. For example, the order type of a planar point set (that can have collinear triples) is a P3O; we denote a P3O realizable by points as p-P3O. If we do not allow $\circlearrowleft(ABC)=0$, we obtain a T3O (total 3-order). Contrary to linear orders, a T3O can have a rich structure. A T3O realizable by points, a p-T3O, is the order type of a point set in general position. In our paper "Orientation of convex sets" we defined a 3-order on pairwise intersecting convex sets; such a P3O is called a C-P3O. In this paper we extend this 3-order to pairwise intersecting good covers; such a P3O is called a GC-P3O. If we do not allow $\circlearrowleft(ABC)=0$, we obtain a C-T3O and a GC-T3O, respectively. The main result of this paper is that there is a p-T3O that is not a GC-T3O, implying also that it is not a C-T3O -- this latter problem was left open in our earlier paper. Our proof involves several combinatorial and geometric observations that can be of independent interest. Along the way, we define several further special families of GC-T3O's.
