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=X_{n}={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.