Реттелген жиынтардағы максималды және минималды элементтер

Element that is not ≤ (or ≥) any other element

In mathematics, especially in order theory, a maximal element of a subset of some preordered set is an element of that is not smaller than any other element in A minimal element of a subset of some preordered set is defined dually as an element of that is not greater than any other element in
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 of a preordered set is an element of which is greater than or equal to any other element of and the minimum of 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. As an example, in the collection

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} is both minimal and maximal. By contrast, neither a maximum nor a minimum exists for

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.

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.

Each finite nonempty subset has both maximal and minimal elements. An infinite subset need not have any of them, for example, the integers with the usual order. The set of maximal elements of a subset is always an antichain, that is, no two different maximal elements of are comparable. The same applies to minimal elements.

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.