Some cards, like Magic Lamp, Illusion, and Pollymorph Potion, let you get rid of ONE monster. If you get rid of a monster, does its mate also leave?

A Mate is like a Wandering Monster. There are now two monsters, and you have to get rid of them individually. (We have found that it is easiest to get rid of the Mate card so the actual monster is still there for reference.) If you play the get-rid-of-it card before somebody tries to play Mate, though, there's no monster left for the Mate to join, so Mate can't be played.