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There will be no extensions given for this project.

In the above sentence I need some explanations about grammatical role of "given".

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The sentence is written in the passive voice and is equivalent to any of the following:

No extensions will be given for this project.

The teacher will not give any extensions for this project.

The verb "to give" serves the same role in each sentence.

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More than one explanation exists. 

 

If it is a coherent phrase, the entire phrase "no extensions given for this project" has a grammatical role.  If we regard "there" as a pronoun and a subject, then "no extensions given for this project" is a subject complement and the clause is in canonical order.  If we regard "there" as an intransitive preposition, then it is the subject complement, the long phrase is the subject and the clause is in inverted order. This inversion is triggered by the existential semantics of "there". 

If it is not a coherent phrase, then "no extensions" acts as either subject or complement on its own, "given" is free to be part of the verb of the clause, and "for this project" is an adverbial prepositional phrase.  Godel9 has already posted an answer under this assumption, showing one active voice equivalent of the passive voice interpretation. 

Under this interpretation, the grammatical role of "given" is to be a part of the passive future construction "will be given".

 

I prefer a different interpretation: 

"No extensions given for this project" is a coherent noun phrase, and the keyword noun is "extensions".  This noun has two direct modifiers: the adjective "no" and the participial phrase "given for this project". 

The participial phrase has constituents.  It is formed by the so-called past participle "given", which is modified by the prepositional phrase "for this project".  Delving deeper, the preposition "for" takes "this project" as its object, and the demonstrative adjective "this" modifies the common noun "project". 

"Given" is a non-finite form of the verb to give.  As a non-finite form, it has no tense, forms no predicate, and takes no subject.  Instead, in this case it takes a prepositional phrase as an adjunct or adverbial modifier, forming a participial phrase. 

Just as the prepositional phrase modifies the participle that precedes it, the participial phrase modifies the noun that precedes it. 

Under this interpretation, the grammatical role of "given" is to form an adjectival participial phrase.

 

There you go.  You now have two explanations about the grammatical role of "given". 

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