Continuity in partially ordered sets (Q938481)

The author studies posets which need not be (directed) complete partial orders or lattices but which satisfy the condition that each element is the sup of elements way below it. He shows that some properties that are known to hold for cpos do hold for a more general kind of a partially ordered set. The following special case of a poset is central for this paper. A partially ordered set \(P\) is called a \(C\)-poset if, for any \(x,y\in P\), with \(x\not\leq y\), there exist a \(u\) such that \(u\not\leq y\) and an up-complete lower set \(J\) such that \(x\notin J\) and \(\uparrow u \cup J = P\). Probably, the author could have invented a more memorable name for his poset.