Greatest element

In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S.

Formally, given a partially ordered set (P, ≤), an element g of a subset S of P is the greatest element of S if

Hence, the greatest element of S is an upper bound of S that is contained within this subset. It is necessarily unique. By using ≥ instead of ≤ in the above definition, one defines the least element of S.

Like upper bounds, greatest elements may fail to exist. Even if a set has some upper bounds, it need not have a greatest element, as shown by the example of the negative real numbers. This example also demonstrates that the existence of a least upper bound (the number 0 in this case) does not imply the existence of a greatest element either. Similar conclusions hold for least elements. A finite chain always has a greatest and a least element.