Infinite ordered sets spanned by finitely many chains

✍ Scribed by J-M. Brochet

• Publisher: Springer Netherlands
• Year: 1991
• Tongue: English
• Weight: 691 KB
• Volume: 8
• Category: Article
• ISSN: 0167-8094
• DOI: 10.1007/bf00385815

✦ Synopsis

We say that an ordered set P is spanned by a family %? of chains if P = (P, <) is the transitive closure of lJ{(c, < 1 C). C E U}. It is shown that there is a function h: w +w such that if P is spanned by k < o chains, then P has a finite cutset-number <h(k) (i.e. for any x E P, there is a finite set F of size IFI < h(k) -1, such that the elements of F are incomparable with x and {x) u F meets every maximal chain of P). The function h is exponentially bounded but eventually dominates any polynomial function, even if it is only required that there are at most h(k) pairwise disjoint maximal chains in P. whenever P is spanned by k i w chains.