Abstract
A poset is order-scattered if it does not embed the chain η of the rational numbers. We prove that there are eleven posets such that N(P), the MacNeille completion of P, is order-scattered if and only if P embeds none of these posets. Moreover these posets are pairwise non-embeddable in each other. This result completes a previous characterisation due to Duffus, Pouzet, Rival [4]. The proof is based on the “bracket relation”: $$\eta \to [\eta ]^{2}_{3} ,$$ a famous result of F. Galvin.
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