Jase, the subject pronoun is required in Hebrew as far as I know. The linking one is optional but the topic pronoun or full NP (the keys) is required. I was wondering if that's also true for the language Zaba was describing.
Zaba, are you asking about a full sentence including a subject or just the PP itself? I don't see how only the PP is a clause by itself, but it can be the predicate in a verbless sentence.
As for a term, I'm not sure of the best one. Null copula, verbless sentence, etc. in traditional Arabic grammar, there is a distinction between verbal and nominal sentences based on type of predicate. Effectively, a PP is a subtype of the nominal kind, because there is no agreement.