Is this a prepositional phrase?

Question

I'm trying to remember the grammatical term used to describe this part of speech. The term "prepositional phrase" comes to mind, but I think it might be something different. It's the part of a sentence that introduces the sentence and gives context, but grammatically it can always be removed and the sentence will still be valid. Examples:

Being in jail, he is unable to attend.
The car having two doors, there wasn't a third one to open.
The store being closed on weekends, he would have to wait until Monday.

Answer 1

In jail and on weekends are prepositional phrases. The italicized clauses in your example sentences are all participial clauses using a present active (-ing) participle verb form. Note that adverbial participles like these can occur in many places of the sentence besides the beginning. – John Lawler

Supplemental to what @JohnLawler says: The clause which introduces the first example is assumed to take the subject of the matrix clause as its subject. The other two, which have their distinct subjects specified, are not so tightly 'integrated' with the matrix clause; such loosely connected clauses are called absolute. – StoneyB

Answer 2

• A prepositional phrase begins with a preposition like 'in' or 'on'. Two of your sentences include prepositional phrases, but the phrases in all aren't prepositional.
• What is common among the three is that they all include a present participle '-ing'. But since there is no helper verb, like 'is', it could be a gerund. But since it does not act like a noun, it is simply a present participle as part of a present participle phrase. The following is a conversion of the first sentence so that it has a gerund phrase: "Being in jail makes him unable to attend", where 'being in jail' now acts as a noun.
• The present participle phrase is not an appositive because it is modifying the entire sentence, and does not act like a noun. In "His current condition, being in jail, makes him unable to attend", the phrase is now appositive.