The purpose of this paper is to investigate some properties of the crossing number chi(P) of a poset P. We first study the crossing numbers of the product and the lexicographical sum of posets. The results are similar to the dimensions of these posets. Then we consider the problem of what happens to the crossing number when a point is taken away from a poset. We show that if P is a poset such that x is-an-element-of P and chi(P - x) greater-than-or-equal-to 1, then 1/2 chi(P) less-than-or-equal-to chi(P - x) less-than-or-equal-to chi(P). We don't know yet how to improve the lower bound. We also determine the crossing numbers of some subposets of the Boolean lattice B(n) which consist of some specified ranks. Finally we show that PSI(n) is crossing critical where PSI(n) is the subposet of B(n) which is restricted to rank 1, rank n - 1 and middle rank(s). Some open problems are raised at the end of this paper.
關聯:
ORDER-A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS
