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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.

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marked as duplicate by Asaf Karagila, t.b., Srivatsan, Matt N., robjohn Jan 3 '12 at 11:42

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You mean maximal as in en.wikipedia.org/wiki/Maximal_element? I guess one way to write about them in a way so that the reader understands the difference is to follow the definition with some examples of maximal elements that are also maximum elements, and with some examples of maximal elements that are not maximum elements. –  Jonas Meyer Jan 3 '12 at 8:33
    
Yes. But it is talking in the terms of 'smaller' and 'larger' which are closely related with real-numbers. One can use the adjective maximum and minimum here since they are more closely linked with them. But maximal and minimal - as far as I can see - are ore generic adjectives which can also be used for sets and subsets. How can one make a newcomer see such points in the first reading itself? –  Dilawar Jan 3 '12 at 8:39
    
Is "by immediately giving examples of maximal but not maximum elements" an answer? –  Jonas Meyer Jan 3 '12 at 8:40
    
@JonasMeyer I did not read your first comment fully (perhaps you were editing it). There are great many formal definitions available but can I get a definition which has least amount of confusion. I was never been able to keep a nice picture in my head of these two terms (I have a poor short term memory). I would like to see a nicely written definition of these two terms which is easy to keep in head (few jargon). –  Dilawar Jan 3 '12 at 8:43
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The maximum element in an ordering is the unique element, if any, that is larger than every other element with respect that ordering. A maximal element, and there may be many, is one that is not smaller than any other element in the ordering; it may not be larger than every other element, but no other element is larger than it. Thus, a maximum element must also be maximal, but a maximal element need not be maximum. (And similarly for minimum and minimal.) –  Brian M. Scott Jan 3 '12 at 8:58

3 Answers 3

up vote 5 down vote accepted

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 $b$.

enter image description here

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.

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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.

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"You can't find any other element which is greater" means it's the greatest element, no? –  Gigili Jan 3 '12 at 10:44
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@Gigli: No, you be unable to compare them. –  Henry Jan 3 '12 at 11:13

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.

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The best explanation by far. –  Gigili Jan 3 '12 at 11:48
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I disagree with the first sentence: a partially ordered set may well have a maximum. It is only necessary for that one element to be comparable to everything else. –  t.b. Jan 3 '12 at 12:37
    
@t.b. Yes, providing that one element is greater than all the others. But a maximum element of a partially ordered set, if it exists, is also the only maximal element. –  Henry Jan 3 '12 at 17:30

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