"The notebook of Carmen is full."
Translation:Caietul lui Carmen este plin.
Yes, it's lui Carmen. I explained this in your other topic. Here's the explanation again.
Female names not ending in a vowel and names that came from another language use the possessive pronoun "lui". For example: lui Carmen (Carmen's), lui Ingrid, lui Naomi, lui Beatrice etc.
here are some "normal" names:
Andreea - Andreei
Roxana - Roxanei
Iulia - Iuliei
Ioana - Ioanei
Maria - Mariei
Dana - Danei
Alexandra - Alexandrei etc.
Later Edit (for others who see this)
keep it simple stupid: native romanian female names end in "A". If you see a female name ending in anything else you use "lui".
Unfortunately that would create a new class of problems, since "lui Andreea" is not correct.