Least element
A least element of a partially ordered set is an element less than or equal to every element in the set. If it exists, it is unique.
Partial order: A reflexive, antisymmetric, transitive relation on a set. Leastness is defined relative to the relation ordering the set.
Natural numbers: The counting numbers, with conventions about whether zero is included. Under the usual order, the set has a least element: zero or one, depending on convention.
Uniqueness of least elements: The theorem that a partially ordered set has at most one least element. Antisymmetry forces any two candidates below everything to be equal.
Initial object: An object from which there is exactly one morphism to every object in a category. In a poset viewed as a category, a least element is an initial object.
Partially ordered set: A set equipped with a partial order. The least element is defined within this structure.
Divisibility: A relation in which one integer divides another without remainder. On positive integers, divisibility orders the set with least element one.
Order ideal: A subset of a partially ordered set that contains every element below each of its members. A least element, when present, belongs below every element in the set.
Bottom element: A least element in an ordered structure, often denoted by ⊥. This name is common in logic, semantics, and lattice theory.
Minimal element: An element with no strictly smaller element in a partially ordered set. Minimality does not require being below incomparable elements.
Power set: The set of all subsets of a given set. Ordered by inclusion, its empty subset is the least element.