maximal

Maximal is an adjective used in mathematics and related disciplines to describe an object that cannot be extended within a given context without losing a specified property. It conveys local maximality rather than a global maximum. Unlike maximum, which denotes the single largest element with respect to a relation and may not exist or be unique, maximality only asserts that no larger element with the same property can be found.

In order theory, a partially ordered set (P, ≤) contains a maximal element if there is an element m such that no element p satisfies m < p. A maximum (greatest) element, when it exists, is an element that is greater than or equal to every element of P. The two concepts can diverge: a poset can have many maximal elements and no maximum.

In algebra, a prominent use is maximal ideals. A proper ideal M of a ring R is maximal if there is no other proper ideal strictly between M and R. A common corollary is that every proper ideal is contained in a maximal ideal (via Zorn’s lemma), and in commutative rings with unity, maximal ideals are prime. The set of maximal ideals often motivates geometric viewpoints through the spectrum of a ring, Spec(R).

Other applications appear in group theory (maximal subgroups), graph theory (maximal matchings and maximal independent sets), and general mathematical reasoning, where maximality signals that a construction cannot be enlarged without violating a given condition.