On the Boolean dimension of spherical orders
On the Boolean dimension of spherical orders
复制标题
关于球阶的布尔维数
作者：
G. Brightwell;P. G. Franciosa
众所周知，如果平面阶P是有界的，即只有一个最小值和一个最大值，那么P(LD(P))的维数至多为2，如果我们去掉P只有一个最大值的限制，那么LD(P)≤3。然而，在球体上绘制的有界阶的维数可以是任意大。偏序集 P 的布尔维数 BD(P) 是线性阶的最小数量，使得 P 的阶关系可以写成线性阶的某种布尔组合。我们证明有界球阶的布尔维数永远不会大于 4，并且在偏序集具有多个最大元素但只有一个最小值的情况下不大于 5。这些结果是通过根据圆弧之间的包容性对球阶进行表征而获得的。
It is well known that if a planar order P is bounded, i.e. has only one minimum and one maximum, then the dimension of P (LD(P)) is at most 2, and if we remove the restriction that P has only one maximum then LD(P)≤3. However, the dimension of a bounded order drawn on the sphere can be arbitrarily large.The Boolean dimension BD(P) of a poset P is the minimum number of linear orders such that the order relation of P can be written as some Boolean combination of the linear orders. We show that the Boolean dimension of bounded spherical orders is never greater than 4, and is not greater than 5 in the case the poset has more than one maximal element, but only one minimum. These results are obtained by a characterization of spherical orders in terms of containment between circular arcs.