So I know I learned this at one point, but I can’t rederive the logical argument explaining when to use the words “maximal” and “maximum.” Certainly maximum is both an adjective and a noun, and maximal is just an adjective.
One explanation I remember was that there can be many maximal things, but only one maximum thing. I know that you call it a maximal ideal in algebra, and it need not be unique (unless it’s a local ring?), but then why say “local minima” if a minimum is unique?
I just noticed I’m a bit inconsistent in my usage in this paper I’m writing, and I can’t tell if I should call it the “maximum probability of error criterion” or the “maximal probability of error criterion.” I was leaning to the former, but now thinking about it has got me all muddled.