[seqfan] A class of well-founded orderings

Frank Adams-Watters franktaw at netscape.net
Mon Oct 20 21:52:29 CEST 2014


I've been interested for some time in a class of orderings. I was
wondering if anybody else has studied these.

There are a number of equivalent formulations for such an ordering S:

1. S is well-founded ordering and every anti-chain of S is finite.

2. For any infinite sequence a(1), a(2), ... of members of S, there
exists i, j with i < j and a(i) <= a(j).

3. Any infinite sequence of members of S has a monotonically increasing 
subsequence.

4. Every subset of S is supported by a finite set of minimal elements.

The principle result for these is that if S is such an ordering, and we
define an ordering S' on the anti-chains of S, such that if A and B are
anti-chains, then A <= B iff for every a in A there exists b in B with
a <= b. Then S' is also an ordering of this type.

Franklin T. Adams-Watters




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