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
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
 on page 21 show. For an overview on such
arithmetic topics in ordered sets and the cancellation problem
in particular, cf. Jónsson's survey .
Uniqueness of cores, which was discussed in ,  and  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 .) 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. \