MacNeille completions of \(D\)-posets and effect algebras (Q1586482)

A characterization of difference posets (and hence of effect algebras, too) with MacNeille completion is given by the property of the so-called strong D-continuity. A difference poset \((P,\leq,\ominus,0,1)\) is called strongly D-continuous if for every \(A,B \subseteq P\) such that \(a \leq b\) whenever \(a \in A\) and \(b \in B\) the following condition holds: \(\bigvee \{ b \ominus a \colon\;a \in A,\;b \in B \} = 0\) iff every lower bound of \(B\) is under the upper bound of \(A\).