Which posets have a scattered MacNeille completion?

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”: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta \to [\eta ]^{2}_{3} ,$$\end{document} a famous result of F. Galvin.