Definition of “maximal” and “minimal”

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difference between maximal element and greatest element

When I first encountered the terms maximal and minimal, I confused them with maximum and minimum. Many of my classmates also got confused about these terms (although they did not realise it). One usually do not find good definition of these two terms in literature (especially in Engineering books).

How would you define these two terms such that any person reading about them would understand the difference between um and it's corresponding al (minimal/minimum and maximal/maximum) easily?

P.S.: I may have got confused because English is not my native-tongue.

Answer 1

I would think that a good way for a newcomer to be clear on the difference is to build a stock of examples. For example, when maximal ideals are studied, it would be good to know of examples of rings that have a maximum proper ideal and contrast with examples of rings that have several distinct maximal ideals.

Perhaps a visual like the following could help to distinguish maximum and maximal, where in each of the two examples the vertices are partially ordered by the relation $a<b$ if there is an upward path from $a$ to $b$.

If this is more about the words themselves, perhaps it will help to link the words with corresponding definite articles, keeping in mind that "maximum" is (often) a noun while "maximal" is an adjective. "The maximum" is unique if it exists while "a maximal element" may be one among many.

Answer 2

Here's how I remember it:

Maximal element. You can't find any other element which is greater.

Maximum (= greatest element). Can be compared to all other elements, and is greater than all of them.

Answer 3

The distinction is between partially ordered sets and totally ordered sets.

You can have several maximal elements of a partially ordered set because they are each greater than the elements they can be compared to individually, but they cannot be compared between them.

You cannot have more than one maximum element of a totally ordered set since all elements are comparable.