Theorems 3.2, 3.4 and
3.5 suggest a reduction procedure when
determining the fixed point property for a set:
Find a suitable retraction on the set and then decide if the
retract has the fixed point property instead.
Iterate this procedure
until no suitable retraction can be found. In finite sets this procedure
is well understood (cf. e.g., [7], [31],
[39],
[95]).
In infinite sets one might be able to iterate the above idea infinitely
often, thus being faced with the problem how to get past the
limit ordinal.
This problem has been addressed successfully by
Li and Milner (cf. [70]-[77])
for large classes of ordered sets. We will not pursue their work
in full generality here, but rather present a
very simple-minded approach to the same problem.
Our approach bypasses the problems at the limit ordinal
in a brute force fashion, namely by demanding
that there are no problems at the limit ordinal.
While this might be very simplistic, the exposition is freed
of many technical details and we can also ``mix" several different kinds of
retractions in the dismantling process.
