A note on coherence of dcpos

In this note, we prove that a well-filtered dcpo L   is coherent in its Scott topology if and only if for every x,y∈Lx,y∈L, ↑x∩↑y↑x∩↑y is compact in the Scott topology. We use this result 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 ↑x∩↑y↑x∩↑y is compact in the Scott topology for every x,y∈Lx,y∈L.