Maximal and minimal elements
In mathematics, especially in order theory, a maximal element of a subset
of some preordered set is defined dually as an element of
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.
[1][2] Specializing further to totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide.
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[3] 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.
also happens to be a partially ordered set (or more generally, if the restriction
contains no element strictly greater than
This leaves open the possibility that there exist more than one maximal elements.
satisfies the ascending chain condition, a subset
has a greatest element, the notions coincide, too, as stated above.
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.
In a directed set, every pair of elements (particularly pairs of incomparable elements) has a common upper bound within the set.
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.
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 the consumption space is some set
, usually the positive orthant of some vector space so that each
represents a quantity of consumption specified for each existing commodity in the economy.
Preferences of a consumer are usually represented by a total preorder
It should be remarked that the formal definition looks very much like that of a greatest element for an ordered set.
with the property above behaves very much like a maximal element in an ordering.
An obvious application is to the definition of demand correspondence.
mapping any price system and any level of income into a subset
The demand correspondence maps any price
It is called demand correspondence because the theory predicts that for
Every cofinal subset of a partially ordered set with maximal elements must contain all maximal elements.
is equal to the smallest lower set containing all maximal elements of
