maximum versus maximal

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.

One thought on “maximum versus maximal

  1. In a partially ordered set S, an element x is a maximal element of S if there is no y in S such that x is less than y. There may still be an element y in S not related to x (i.e. x is not less than y, x is not equal to y, and x is not greater than y). As an example. consider ideals of a ring.

    An element x is a maximum of S if x is greater or equal than y for every y in S. If it exists, it is unique.

    A maximum (if there is one) is maximal, but there may be other maximal elements. If your set is totally ordered (i.e. there are not non-related elements), then maximal is the same as maximum.

    The concept of local maximum is a bit different, because of the local adjective. Let f be a function from A to R, for some topological space A (assume A is a real interval if you want). We say that f has a (global) maximum at a point x in A when f(x) is a maximum of the set f(A). However, we say that f has a local maximum at a point x when there is a neighborhood U of x in A such that f(x) is a maximum of the set f(U). That is why we may have multiple local maxima (or minima). In addition, different points x and y in A may satisfy f(x)=f(y), so you can even have a function f with more than one (global) maximum.

    Does this help?

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