Does Admor = Rabbi?

“Rabbi” is poorly defined. Arguably is acts more as a courtesy title such as “Mister” in English, or “Monsieur” in French (both of which once indicated the you were addressing person higher up on the feudal ladder than a mere peasant). Academically, it seems to indicate you can teach older children (meaning it is similar to a Bachelor’s degree, or in modern times with watering down of credential, a Master’s).

“Admor” (literally, My Lord and Teacher and Rabbi”, and “Lord” is in the fedual English meaning showing respect to someone whom you acknowledge outranks you substantially) implies you are accepting the person as a leader.

As to whether it is inherited, one should note that in most traditional cultures, our’s included, positions of leadership will tend to stay in a family, with flexibility to get rid of the unworthy and raise up the worthy. An OTD from an hashuvah (“important”, though “noble” is a better translation) family is seen as having died a horrible death and ignored, and a BT who distinguishes himself gets to be the distinguished ancestor from whom people are proud to trace descent (so in many ways we are more like the Americans, than the English in such matters, even though we are still a very non-republican (small “r”) culture).

Note that among Americans, names such as Brown of California, Cuomo of New York, Kennedy, Bush, etc., appear to end up in positions important based on inheriting from their parents (so while the son of a leader isn’t necessarily a leader, it sure helps).

Admor stands for אדונינו מורינו ורבינו could be a Rav who paskens shalos or a Rebbi who has chasidim. A Rav is not an inheried position but a Rebbi is, if no sons exist, the son in law usually inherits the position.

YABIA OMER, why are you asking?
is for yourself? you don’t want to become a Rebbel?
or you are asking for your son?

The Rambam rules that not only the kingship, but rather every position of authority and all appointees of Israel are inherited to a son and to a son’s son, forever. The Ginas Veradim quotes an anecdote of how the rabbinate in Tzefas was passed on by inheritance to a son who was under bar mitzvah when Rav Shlomo Alkabetz ordered the community to wait for the young boy to come of age, and when the son turned thirteen he took over his father’s position! The Shulchan Aruch bases a ruling on the same halachah, this time in connection with the local cantor, who retains the right to bring in his son as an assistant and groom him for taking over his position (Orach Chaim 53:25).

In Yoreh Deah (245:22), the Rema states that somebody who serves as the rabbi of a city cannot be ousted from his standing even if somebody greater than him comes to town. Even his son and his son’s son, forever, take precedence over others. In Orach Chaim 13 the Chasam Sofer writes that he upheld the ruling of Rema that the son inherits his father’s rabbinical position and another Chasam Sofer (Choshen Mishpat 21) also implies this. The Kesav Sofer (the Chasam Sofer’s son, Yoreh Deah 123) confirms that his father implemented the Rema’s ruling in all Hungarian rabbinates that the son inherits the rabbanus. The Mishna Berura (53:83) cites both opinions.