A note on coherence of dcpos (Q2630456)

It is proven that a well-filtered directed-complete partial ordered set (dcpo) \(L\) is coherent in its Scott topology if and only if for every \(x,y\in L\), \(\uparrow{x} \,\cap\uparrow y\) is compact in the Scott topology. This result is then used to prove that a well-filtered dcpo \(L\) is Lawson-compact if and only if it is patch-compact if and only if \(L\) is finitely generated and \(\uparrow x \cap\uparrow y\) is compact in the Scott topology for all \(x,y\in L\).