A bit confused. Abbey is Αββαείο in Greek. Mονή is monastery actually. I think μονή and μοναστήρι are the same, monastery both. But I am not a RE specialist to be sure.
The word αββάς is used as a tittle in some monasteries, i think, the Greek Orthodox Church use the tittle ηγούμενος, that is the same. Not in all Orthodox Churches, i think. A theologist could make it clear.