Extensions of ordered sets having the finite cutset property

✍ Scribed by John Ginsburg

• Publisher: Elsevier Science
• Year: 1986
• Tongue: English
• Weight: 892 KB
• Volume: 58
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(86)90157-3

✦ Synopsis

Let P be an ordered set. P is said to have the finite cutset property if for every x in P there is a finite set F of elements which are noncomparable to x such that every maximal chain in P meets {x} t.J F. It is well known that this property is equivalent to the space of maximal chains of P being compact. We consider the following question: Which ordered sets P can be embedded in an ordered set Q which has the finite cutset property?

For x e P, we let x ÷ = {t9 ~ P: x ~<p}. Our main results are the following:

Theorem. Suppose P can be embedded in an ordered set having the finite cutset property and that A is an uncountable antichain in P. Then there exist distinct elements a, b, c in A such that a + fq b + =a + f3 c +.

Corollary. There exists an ordered set P which cannot be embedded in an ordered set having the finite cutset property whereas every countable subset of P can be embedded in such an ordered set.