# Maximal element

In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set (poset) is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some partially ordered set is defined dually as an element of S that is not greater than any other element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum. The maximum of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S, and the minimum of S is again defined dually. For totally ordered sets, the notions of maximal element and maximum on one hand and minimal element and minimum on the other hand coincide.

While a partially ordered set can have at most one each maximum and minimum it may have multiple maximal and minimal elements. Zorn's lemma states that every partially ordered set for which every totally ordered subset has an upper bound contains at least one maximal element. This lemma is equivalent to the well-ordering theorem and the axiom of choice and implies major results in other mathematical areas like the Hahn–Banach theorem and Tychonoff's theorem, the existence of a Hamel basis for every vector space, and the existence of an algebraic closure for every field.

As an example, in the collection

S = {{d, o}, {d, o, g}, {g, o, a, d}, {o, a, f}}

ordered by containment, the element {d, o} is minimal, the element {g, o, a, d} is maximal, the element {d, o, g} is neither, and the element {o, a, f} is both minimal and maximal. By contrast, neither a maximum nor a minimum exists for S.

## Definition

The definition for minimal elements is obtained by using ≥ instead of ≤.

## Existence and uniqueness

Maximal elements need not exist.

Example 1: Let S = [1,∞) ⊂ , for all mS we have s=m+1∈S but m<s (that is, ms but not m=s).
Example 2: Let S = {s: 1≤s2≤2} ⊂ ℚ and recall that ${\sqrt {2}}$ ∉ℚ.

In general ≤ is only a partial order on S. If m is a maximal element and sS, it remains the possibility that neither sm nor ms. This leaves open the possibility that there are many maximal elements.

Example 3: In the fence a1 < b1 > a2 < b2 > a3 < b3 > ..., all the ai are minimal, and all the bi are maximal, see picture.
Example 4: Let A be a set with at least two elements and let S={{a}: aA} be the subset of the power set P(A) consisting of singletons, partially ordered by ⊂. This is the discrete poset—no two elements are comparable—and thus every element {a}∈S is maximal (and minimal) and for any a‘’,a‘‘ neither {a‘} ⊂ {a‘‘} nor {a‘‘} ⊂ {a‘}.

## Maximal elements and the greatest element

It looks like $m$ should be a greatest element or maximum but in fact it is not necessarily the case: the definition of maximal element is somewhat weaker. Suppose we find $s\in S$ with $\max S\leq s$ , then, by the definition of greatest element, $s\leq \max S$ so that $s=\max S$ . In other words, a maximum, if it exists, is the (unique) maximal element.

The converse is not true: there can be maximal elements despite there being no maximum. Example 3 is an instance of existence of many maximal elements and no maximum. The reason is, again, that in general $\leq$ is only a partial order on $S$ . If $m$ is a maximal element and $s\in S$ , it remains the possibility that neither $s\leq m$ nor $m\leq s$ .

If there are many maximal elements, they are in some contexts called a frontier, as in the Pareto frontier.

Finally, let us remark that $S$ being totally ordered is sufficient to ensure that a maximal element is a greatest element, but it is not necessary. For example, every power set P(S) of a set S has only one maximal element, viz. S itself, which is also the unique greatest element; but almost no power set is totally ordered, cf. picture.

## Directed sets

In a totally ordered set, the terms maximal element and greatest element coincide, which is why both terms are used interchangeably in fields like analysis where only total orders are considered. This observation applies not only to totally ordered subsets of any poset, but also to their order theoretic generalization via directed sets. In a directed set, every pair of elements (particularly pairs of incomparable elements) has a common upper bound within the set. Any maximal element of such a subset will be unique (unlike in a poset). Furthermore, this unique maximal element will also be the greatest element.

Similar conclusions are true for minimal elements.

Further introductory information is found in the article on order theory.

## Examples

### Consumer theory

In economics, one may relax the axiom of antisymmetry, using preorders (generally total preorders) instead of partial orders; the notion analogous to maximal element is very similar, but different terminology is used, as detailed below.

$y\in B$ implies $y\preceq x$ It should be remarked that the formal definition looks very much like that of a greatest element for an ordered set. However, when $\preceq$ is only a preorder, an element $x$ with the property above behaves very much like a maximal element in an ordering. For instance, a maximal element $x\in B$ is not unique for $y\preceq x$ does not preclude the possibility that $x\preceq y$ (while $y\preceq x$ and $x\preceq y$ do not imply $x=y$ but simply indifference $x\sim y$ ). The notion of greatest element for a preference preorder would be that of most preferred choice. That is, some $x\in B$ with

$y\in B$ implies $y\prec x.$ An obvious application is to the definition of demand correspondence. Let $P$ be the class of functionals on $X$ . An element $p\in P$ is called a price functional or price system and maps every consumption bundle $x\in X$ into its market value $p(x)\in {\mathbb {R}}_{+}$ . The budget correspondence is a correspondence $\Gamma \colon P\times {\mathbb {R} }_{+}\rightarrow X$ mapping any price system and any level of income into a subset

$\Gamma (p,m)=\{x\in X\mid p(x)\leq m\}.$ $D(p,m)={\big \{}x\in X\mid x$ is a maximal element of $\Gamma (p,m){\big \}}$ .