Minimal element

A minimal element of a partially ordered set has no distinct element strictly below it. A partially ordered set can have several minimal elements, and a minimal element need not be least.

Partially ordered set: A set equipped with a reflexive, antisymmetric, transitive relation. Minimality is defined relative to the partial order on this set.
Least element: An element below every element in a partially ordered set. Unlike a minimal element, a least element must be comparable with and below the entire set.
Divisibility: The relation in which one integer divides another without remainder. Under divisibility on positive integers above one, the primes are minimal elements.
Pareto efficiency: A state is Pareto efficient when no alternative improves one objective without worsening another. Efficient outcomes are minimal under the order that compares outcomes by all objectives.
Partial order: A reflexive, antisymmetric, transitive relation on a set. Its comparison relation determines whether any element lies strictly below a candidate.
Minimum element: An element that is no greater than every element of a set. This common synonym for least element is stronger than minimality in a partial order.
Prime number: A natural number greater than one with exactly two positive divisors. Each prime is minimal among integers greater than one ordered by divisibility.
Minimal vertex: A vertex with no distinct predecessor in a directed acyclic graph. A directed acyclic graph induces an order in which minimal elements have no incoming predecessor.
Strict partial order: An irreflexive, transitive relation that represents strict precedence. The phrase “strictly smaller” can be expressed using this relation.
Maximal element: An element with no distinct element strictly above it in a partially ordered set. It reverses the direction of the defining condition for minimality.

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