A 2-element antichain that is not contained in any finite retract (Q664315)

The question posed in the book [Ordered sets. An introduction. Boston, MA: Birkhäuser (2003; Zbl 1010.06001)] by the second author asks whether every finite subset of an (infinite) ordered set is actually contained in a finite retract. This question is motivated by the product problem for the fixed-point property. It is known that in a chain-complete ordered set, every two-element antichain consisting of minimal or maximal elements is contained in a finite retract. This can be generalized to an arbitrary ordered set, every two-element antichain in which one of the two elements is maximal or minimal is contained in a finite retract. Hence, it is surprising that there is an ordered set of height 2 with a two-element antichain that is not contained in any finite retract. The paper contains the construction of such an ordered set.