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## The Cancellation Problem

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. \

Bernd.S.W.Schroder