Maximal and minimal elements

In mathematics, especially in order theory, a maximal element of a subset ${\displaystyle S}$ of some preordered set is an element of ${\displaystyle S}$ that is not smaller than any other element in ${\displaystyle S}$. A minimal element of a subset ${\displaystyle S}$ of some preordered set is defined dually as an element of ${\displaystyle S}$ that is not greater than any other element in ${\displaystyle S}$.

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 ${\displaystyle S}$ of a preordered set is an element of ${\displaystyle S}$ which is greater than or equal to any other element of ${\displaystyle S,}$ and the minimum of ${\displaystyle S}$ 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.^[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.

As an example, in the collection $${\displaystyle S:=\left\{\{d,o\},\{d,o,g\},\{g,o,a,d\},\{o,a,f\}\right\}}$$ 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 ${\displaystyle S.}$

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.