We introduce SO-HORNr which is a revised version of SO-HORN and show that SO-HORNr captures equation image on ordered finite structures. We also introduce second-order extended Horn logic SO-EHORN and a superclass SO-EHORNr of it. We show that both of them capture equation image on ordered finite structures by proving that SO-EHORN and SO-EHORNr have the same expressive power when only consider ordered structures.