There is a sign at my work that says "Join the fight for Alzheimer's first survivor" and I am wondering about the use of "Alzheimer's" here. They are not reffering to a survivor of Alzheimer, they are referring to a survivor of Alzheimer's disease and using Alzheimer's as a short form of the disease. What would the possessive form of Alzheimer's be? I don't think it should be Alzheimer's's but i also don't think that it is correct as is. When searching for "double possessive" rules I can only find articles saying to avoid phrases like "St Paul's Cathedral's arches" is the only correct form to rearrange the syntax to be "Join the fight for the first survivor of Alzheimer's?"
Alzheimer's is correct. There is no double possessive in the English language. You can chain possessives (as in your example St Paul's Cathedral's arches). You can also shorten noun phrases (Alzheimer's disease routinely gets shortened to Alzheimer's).