The Menger property for infinite ordered sets

Abstract

It is shown that, if an ordered set P contains at most k pairwise disjoint maximal chains, where k is finite, then every finite family of maximal chains in P has a cutset of size at most k. As a corollary of this, we obtain the following Menger-type result that, if in addition, P contains k pairwise disjoint complete maximal chains, then the whole family, M (P), of maximal chains in P has a cutset of size k. We also give a direct proof of this result. We give an example of an ordered set P in which every maximal chain is complete, P does not contain infinitely many pairwise disjoint maximal chains (but arbitrarily large finite families of pairwise disjoint maximal chains), and yet M (P) does not have a cutset of size <x, where x is any given (infinite) cardinal. This shows that the finiteness of k in the above corollary is essential and disproves a conjecture of Zaguia.