***** NOT A TEACHER *****
One of my favorite books says that an absolute phrase "is an independent expression consisting of a substantive with modifying participle."
I think that it means something like this in simpler English: It is an expression that is NOT grammatically connected to the main sentence. It consists of a noun + an -ing / -ed word.
"The teacher being ill, we had no school on Monday."
The main sentence is "We had no school on Monday."
The absolute phrase is "The teacher being ill."
This kind of construction (sentence) is usually reserved for writing.
In speech, we would usually express the idea like this: "Because the teacher was ill, we had no school on Monday."
Source: House and Harman, Descriptive English Grammar. (I have the 1950 edition, but I believe that this information is still valid today.)
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