Don't lean against automatic door
Don't hold the door open
I usually see this kind of prohibition on the train. What makes me confused is why the first sentence doesn't use the and the second sentence does? I mean both of the sentences refer to the same door in the train.
Well in trains, there is one automatic door, so when they say don't lean on automatic door, they mean don't lean on any of the doors, even though it's grammatically incorrect.
For example,
Cardi B is a queen, but Nicki Minaj is the queen.
The "the" puts more emphasis.
The first case is a reference to the special type of door, "automatic". It's a more "standardized" warning that one should not lean against an automatic door, any automatic door, for that matter. Thus, there's no article. It's also sometimes written: "Don't lean against: Automatic Door!". The idea is to draw attention to the distinctness of the door.
In the second, the reference is to the particular door in front of the passenger. So it needs the definite article.
HTH.
