Sure, due to the history of the region(which is old as the history of humanity and religions) it's really not easy to attribute anything to any particular civilisation. It's almost a sport to argue over the origin of the dish between Greeks, Turks and Arabs.

Indeed, lahmacun (lahmajoon) clearly comes from Arabic "lahm b ajin" (meat with bread) which is just another variation of the Mankoushe (منقوشة) - a flat bread with different toppings such as thyme, cheese and the famous meat variation.