In Mathematics, maximal is used in a related but distinct sense from maximum. Given some partial order "≤", an element m is said to be maximal iff for any element y, it is not the case that m≤y.
Why the strange definition? Since "≤" is only a partial order, it is not necessary that we can compare a maximal m with every element x. In particular, it is not necessarily the case that x≤m, only that either x≤m or x and m cannot be compared.
If M is an element such that y≤M for every element y, then M is called a maximum. In particular, when "≤" is a total order, every maximal element is also a maximum -- hence there's only one maximal element, known as "THE maximum". This can also happen for partial orders.
For instance, if X is some set with a partial order "⊆X" on its subsets, then X is the maximum element of "⊆X". However, if we restrict "⊆X" to the set
Y = { X \ E | ∅ ≠ E ⊆ X },
then ⊆X has |X| maximal elements
{ X \ {a} | a ∈ X }.
As another example, say we work on the natural numbers with the partial order given by "|" (a|b iff a divides b). For any n∈N, the set X=Xn={a : a|n & a<n} is partially ordered by "|". The set of its maximal elements is precisely {n/p : p|n & p is prime}.
"Maximal ____" is often a useful concept: we have maximal ideals, maximal filters, and even Zorn's lemma and Haussdorff's maximality principal to ensure the existence of various maximal objects. Note well the last: it is not enough to say "maximal" to get one, you still have to prove its existence.
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