Bottom 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.

A greatest element of a partially ordered subset must not be confused with maximal elements of the set, which are elements that are not smaller than any other elements. A set can have several maximal elements without having a greatest element. However, if it has a greatest element, it can't have any other maximal element.

In a totally ordered set both terms coincide; it is also called maximum; in the case of function values it is also called the absolute maximum, to avoid confusion with a local maximum. The dual terms are minimum and absolute minimum. Together they are called the absolute extrema.

The least and greatest element of the whole partially ordered set plays a special role and is also called bottom and top, or zero (0) and unit (1), or ⊥ and ⊤, respectively. If both exists, the poset is called a bounded poset. The notation of 0 and 1 is used preferably when the poset is even a complemented lattice, and when no confusion is likely, i.e. when one is not talking about partial orders of numbers that already contain elements 0 and 1 different from bottom and top. The existence of least and greatest elements is a special completeness property of a partial order.

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