linguistic:voice_change
absolute maximums

A greatest element of a subset of a preordered set should not be confused with a maximal element of the set, which are elements that are not strictly smaller than any other element in the set.

Let (P, ≤) {\displaystyle (P,\leq)} be a preordered set and let S ⊆ P. {\displaystyle S\subseteq P.} An element m ∈ S {\displaystyle m\in S} is said to be a maximal element of S {\displaystyle S} if the following condition is satisfied:

If (P, ≤) {\displaystyle (P,\leq)} is a partially ordered set then m ∈ S {\displaystyle m\in S} is a maximal element of S {\displaystyle S} if and only if there does not exist any s ∈ S {\displaystyle s\in S} such that m ≤ s {\displaystyle m\leq s} and s ≠ m. {\displaystyle s\neq m.} A maximal element of (P, ≤) {\displaystyle (P,\leq)} is defined to mean a maximal element of the subset S:= P. {\displaystyle S:=P.}

A set can have several maximal elements without having a greatest element. Like upper bounds and maximal elements, greatest elements may fail to exist.

In a totally ordered set the maximal element and the greatest element coincide; and it is also called maximum; in the case of function values it is also called the absolute maximum, to avoid confusion with a local maximum. The dual terms are minimum and absolute minimum. Together they are called the absolute extrema. Similar conclusions hold for least elements.

One of the most important differences between a greatest element g {\displaystyle g} and a maximal element m {\displaystyle m} of a preordered set (P, ≤) {\displaystyle (P,\leq)} has to do with what elements they are comparable to. Two elements x, y ∈ P {\displaystyle x,y\in P} are said to be comparable if x ≤ y {\displaystyle x\leq y} or y ≤ x {\displaystyle y\leq x}; they are called incomparable if they are not comparable. Because preorders are reflexive (which means that x ≤ x {\displaystyle x\leq x} is true for all elements x {\displaystyle x}), every element x {\displaystyle x} is always comparable to itself. Consequently, the only pairs of elements that could possibly be incomparable are distinct pairs. In general, however, preordered sets (and even directed partially ordered sets) may have elements that are incomparable.

By definition, an element g ∈ P {\displaystyle g\in P} is a greatest element of (P, ≤) {\displaystyle (P,\leq)} if s ≤ g, {\displaystyle s\leq g,} for every s ∈ P {\displaystyle s\in P}; so by its very definition, a greatest element of (P, ≤) {\displaystyle (P,\leq)} must, in particular, be comparable to every element in P. {\displaystyle P.} This is not required of maximal elements. Maximal elements of (P, ≤) {\displaystyle (P,\leq)} are not required to be comparable to every element in P. {\displaystyle P.} This is because unlike the definition of "greatest element", the definition of "maximal element" includes an important if statement. The defining condition for m ∈ P {\displaystyle m\in P} to be a maximal element of (P, ≤) {\displaystyle (P,\leq)} can be reworded as:

Suppose that S {\displaystyle S} is a set containing at least two (distinct) elements and define a partial order ≤ {\displaystyle \,\leq \,} on S {\displaystyle S} by declaring that i ≤ j {\displaystyle i\leq j} if and only if i = j. {\displaystyle i=j.} If i ≠ j {\displaystyle i\neq j} belong to S {\displaystyle S} then neither i ≤ j {\displaystyle i\leq j} nor j ≤ i {\displaystyle j\leq i} holds, which shows that all pairs of distinct (i.e. non-equal) elements in S {\displaystyle S} are in comparable. Consequently, (S, ≤) {\displaystyle (S,\leq)} can not possibly have a greatest element (because a greatest element of S {\displaystyle S} would, in particular, have to be comparable to every element of S {\displaystyle S} but S {\displaystyle S} has no such element). However, every element m ∈ S {\displaystyle m\in S} is a maximal element of (S, ≤) {\displaystyle (S,\leq)} because there is exactly one element in S {\displaystyle S} that is both comparable to m {\displaystyle m} and ≥ m, {\displaystyle \geq m,} that element being m {\displaystyle m} itself (which of course, is ≤ m {\displaystyle \leq m}).

In contrast, if a preordered set (P, ≤) {\displaystyle (P,\leq)} does happen to have a greatest element g {\displaystyle g} then g {\displaystyle g} will necessarily be a maximal element of (P, ≤) {\displaystyle (P,\leq)} and moreover, as a consequence of the greatest element g {\displaystyle g} being comparable to every element of P, {\displaystyle P,} if (P, ≤) {\displaystyle (P,\leq)} is also partially ordered then it is possible to conclude that g {\displaystyle g} is the only maximal element of (P, ≤). {\displaystyle (P,\leq).} However, the uniqueness conclusion is no longer guaranteed if the preordered set (P, ≤) {\displaystyle (P,\leq)} is not also partially ordered. For example, suppose that R {\displaystyle R} is a non-empty set and define a preorder ≤ {\displaystyle \,\leq \,} on R {\displaystyle R} by declaring that i ≤ j {\displaystyle i\leq j} always holds for all i, j ∈ R. {\displaystyle i,j\in R.} The directed preordered set (R, ≤) {\displaystyle (R,\leq)} is partially ordered if and only if R {\displaystyle R} has exactly one element. All pairs of elements from R {\displaystyle R} are comparable and every element of R {\displaystyle R} is a greatest element (and thus also a maximal element) of (R, ≤). {\displaystyle (R,\leq).} So in particular, if R {\displaystyle R} has at least two elements then (R, ≤) {\displaystyle (R,\leq)} has multiple distinct greatest elements.