In mathematics, especially in order theory, a maximal element of a subset of some preordered set. A minimal element of some preordered set is defined dually.

The notions of maximal and minimal elements are weaker than those of greatest element and least element is again defined dually. In the particular case of a partially ordered set, while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements. Specializing further to totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide.

ordered by containment, the element {d, o} is minimal as it contains no sets in the collection, the element {g, o, a, d} is maximal as there are no sets in the collection which contain it, the element {d, o, g} is neither, and the element {o, a, f

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, the Kirszbraun theorem, Tychonoff's theorem, the existence of a Hamel basis for every vector space, and the existence of an algebraic closure for every field.

Definition

be a preordered set such that

Similarly, such that

).

Explicitly, a (respectively, )

also happens to be a partially ordered set

Existence and uniqueness

Maximal elements need not exist.

• Example 1: denotes the real numbers).

• Example 2: denotes the rational numbers and where \sqrt{2} is irrational.

This leaves open the possibility that there exist more than one maximal elements.

• Example 3: In the fence are maximal, as shown in the image.

• Example 4: Let A be the subset of the power set consisting of singleton subsets

Greatest and least elements

the irreflexive kernel exactly one of the following cases applies:

1. ;

2. ;

3. ;

4. are incomparable.

5. as defined above;

6. is called a

Thus the definition of a greatest element is stronger than that of a maximal element.

A subset may have at most one greatest element.

and the only one. By contraposition satisfies the ascending chain condition has a greatest element if, and only if, it has one maximal element.

is a total order in the topmost picture is an example), then the notions of maximal element and greatest element coincide.

Dual to greatest is the notion of least element that relates to minimal in the same way as greatest to maximal.

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 partially ordered set, 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. If a directed set has a maximal element, it is also its greatest element, and hence its only maximal element. For a directed set without maximal or greatest elements, see examples 1 and 2 above.

Similar conclusions are true for minimal elements.

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

Properties

• has both maximal and minimal elements. An infinite subset need not have any of them, for example, the integers with the usual order.

• is always an antichain are comparable. The same applies to minimal elements.

Examples

• In Pareto efficiency, a Pareto optimum is a maximal element with respect to the partial order of Pareto improvement, and the set of maximal elements is called the Pareto frontier.

• In decision theory, an admissible decision rule is a maximal element with respect to the partial order of dominating decision rule.

• In modern portfolio theory, the set of maximal elements with respect to the product order on risk and return is called the efficient frontier.

• In set theory, a set is finite if and only if every non-empty family of subsets has a minimal element when ordered by the inclusion relation.

• In abstract algebra, the concept of a maximal common divisor is needed to generalize greatest common divisors to number systems in which the common divisors of a set of elements may have more than one maximal element.

• In computational geometry, the maxima of a point set are maximal with respect to the partial order of coordinatewise domination.

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.

In consumer theory, usually the positive orthant represents a quantity of consumption specified for each existing commodity in the economy. Preferences of a consumer are usually represented by a total preorder is said to be a maximal element

). The notion of greatest element for a preference preorder would be that of most preferred

is called a price functional or price system. The budget correspondence

The demand correspondence

given, the rational choice

Related notions

is said to be cofinal Every cofinal subset of a partially ordered set with maximal elements must contain all maximal elements.

is said to be a lower set

See also

Notes

1. Proofs

References