minelement

Minelement is a term used in mathematics and computer science to denote the minimum element of a set under a given order, or to refer to an operation that selects that element from a collection.

Definition: In a partially ordered set (P, ≤), a nonempty subset S has a minimum element m ∈ S if m ≤ x for every x ∈ S. If such an element exists, it is unique and is often called the minimum or least element of S. In a general poset, a subset may have no minimum element, and a set may have several minimal elements that are not comparable with each other, in which case no minimum exists.

Minimal versus minimum: A minimal element has no smaller element within the set, but there can be several minimal elements. A minimum element is a single element that is less than or equal to all elements of the set.

Computing: In programming, a minelement operation returns the minimum element according to a comparator. It is typically implemented as a linear scan with time complexity O(n). Data structures such as a min-heap or a priority queue maintain the minimum element more efficiently for repeated access or updates.

Examples: minElement([3, 7, 2, 9]) = 2. In a lexicographic ordering, minElement(["apple","banana","apricot"]) = "apple".

Limitations: Some sets, such as the positive real numbers under ≤, have no minimum element even though they have an infimum. The existence of a minelement depends on the chosen order and the set’s structure.

See also: minimum element, least element, minimal element, max element.