The cancellation problem is a long standing open problem by
G. Birkhoff. If we denote the set of all order-preserving maps
from the ordered set Q to the ordered set P by 
(which naturally is an ordered set under the pointwise order),
the problem is
the following:
Let P,Q,R be finite ordered sets.
Does the fact that is isomorphic to
imply that P is isomorphic to R?
The finiteness assumption is needed, as the examples 4 and 5 in
[66] on page 21 show. For an overview on such
arithmetic topics in ordered sets and the cancellation problem
in particular, cf. Jónsson's survey [66].
Uniqueness of cores, which was discussed in
[31], [39] and [119]
yields a new insight, though it is not clear how far
Proposition 5.5 can be pushed (cf. open question 8).
Proof:
By Lemma 5.4 we have that
can be dismantled via comparative retractions to
a subset that is isomorphic to P.
(This also answers question (2) on p.54 in [46].)
Thus can be dismantled via comparative retractions to
a subset that is isomorphic to 
.
By the uniqueness of the core, we infer 
is isomorphic to
. Similarly 
is isomorphic to 
, and
the isomorphic sets and 
naturally must have
isomorphic cores. \
