# Greatest element and least element

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.

A greatest element of a subset of a preordered set should not be confused with a maximal element of the set, which are elements that are not strictly smaller than any other element in the set.

A set can have several maximal elements without having a greatest element. Like upper bounds and maximal elements, greatest elements may fail to exist.

In a totally ordered set the maximal element and the greatest element coincide; and 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.^{[1]} The dual terms are **minimum** and **absolute minimum**. Together they are called the **absolute extrema**. Similar conclusions hold for least elements.

The least and greatest element of the whole partially ordered set play a special role and are also called **bottom** (⊥) and **top** (⊤), or **zero** (0) and **unit** (1), respectively.
If both exist, the poset is called a **bounded poset**.
The notation of 0 and 1 is used preferably when the poset is 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.

Further introductory information is found in the article on order theory.