Can you explain the difference among these 3 sentences?
a. 窓（まど）を開（あ）けて います。
b. 窓（まど）が開（あ）いて います。
c. 窓（まど）が開（あ）けて あります。
a and b are both in the form of [てform]+います. What is the difference between the two then?
The difference is the nature of the verb. a. is with the transitive verb （～を）開（あ）ける and b. is with the intransitive verb （～が）開（あ）く.
If a transitive verb is used in the form of [てform]+います, that usually means an action is in progress*, so a. means “I am opening the window(s).”
If an intransitive verb is used in the form of [てform]+います, that usually describes a current state, so b. means “The window is open.”
If a transitive verb is used in the form of [てform]+あります, like in c, that also describes the current state but that implies the will of the doer. So c. means “The window has been opened intentionally (by someone/me).” So if somebody says 窓が開けてあります to you, that kind of implies “so please don’t close it.”
(The place) has been decorated with flowers.
Something is written.
As the smoke from the bush fire is awful, the windows have been intentionally closed.
The car key has always been placed here.
My mum’s diary had had lock on.
When I got home, the dinner had already been cooked.
The room had been cleaned thoroughly.
When I got up this morning, money had been placed on the table.
* As it doesn’t take long to open a window or two, depending on the context, 窓（まど）を開（あ）けて います can mean “I/we have left the window(s) open [the current state].” However, 窓（まど）が開（あ）けて あります implies a lot stronger “intention.”
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