Names for both men and women follow your patriarchal lineage. A woman from an Islamic family always keeps her full name even after marriage. Most names are religious; either one of the 99 names of God, or prophets and their wives/children.
In contrast, in Western countries where women normally take their husband’s last name. This becomes problematic for women who have gone through several marriages; when you marry or re-marry, you must legally change your name on all your official documentation.
So, in Islamic naming conventions, your name would be: your given name, your father's first name, your grandfather's first name, your great grandfather's first name, your great-great grandfather's first name (and so on); and then your tribe or family name as your last name. It really forces you to know your lineage.
This naming convention also explains why in Western countries, we tend to misunderstand names of people from Islamic countries, if not provided with enough information. For example, if someone tells you that their name is "Mohammed Ali," you can usually ask him what his family name is or their full name is; "Mohammed Ali Talal Al-Jassem" for example (there are exceptions, of course). So, some people can get away with passport changes if they only include their father's name, then they can go back and add their grandfather's name also and Western people see it as a completely different person.
Also, the "Al-" (predominantly in the Arabian Gulf Region) is the same as "Mc" or "Mac" or "O'" in Irish, Scottish names. It means "of" (whatever family name).
So..... If someone from the Gulf tells you that their name is just “Al” then they are just a jerk and you don’t need to talk to them anyways.