2
coordinate of x} = "A with t omitted".
Here is a modification of Theorem 1.1 which we call the
‘complementation theorem for R[A\(8k)!]’.
THEOREM 1.2. For all k,n
≥
1 and strictly dominating R
[1,n]
2k
, there exists A [1,n]
k
such that R[A\(8k)!] = A’.
Furthermore, A [1,n]
k
is unique.
We now incorporate the antichain concept.
Let R [1,n]
2k
. We say that A is an antichain for R if and
only if A [1,n]
k
and A,RA are disjoint.
Of course, we have the following "maximal antichain"
theorem.
THEOREM 1.3. For all k,n
≥
1, every R [1,n]
2k
has a
maximal antichain.
Note that Theorem 1.3 is virtually contentless, since we
are in a finite context where every nonempty class has a
maximal element.
Note that Theorem 1.1 provides a much stronger kind of
antichain, which we call a ‘complete’ antichain. We refer
to the following as a ‘complete antichain’ theorem.
THEOREM 1.4. For all k,n
≥
1, every strictly dominating R
[1,n]
2k
has an antichain A such that RA contains A’.
Furthermore, A is unique.
However, the analog of Theorem 1.4 for R[A\(8k)!] is false.
THEOREM 1.5. The following is false. For all k,n
≥
1, every
strictly dominating R [1,n]
2k
has an antichain A such that
R[A\(8k)!] contains A’.
We will weaken the conclusion in a simple way.
Our development depends heavily on a very strong regularity
condition on R.
We say that R [1,n]
k
is order invariant if and only if
for all x,y in [1,n]
k
of the same order type, R(x) iff R(y).
The number of such R is bounded by an exponential
expression in k that does not depend on n.