# Greatest and least elements

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

*s*≤*g*, for all elements*s*of*S*.

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.

The 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.[1] 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.

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

## Examples

- The subset of integers has no upper bound in the set ℝ of real numbers.
- Let the relation "≤" on {
*a*,*b*,*c*,*d*} be given by*a*≤*c*,*a*≤*d*,*b*≤*c*,*b*≤*d*. The set {*a*,*b*} has upper bounds*c*and*d*, but no least upper bound, and no greatest element. - In the rational numbers, the set of numbers with their square less than 2 has upper bounds but no greatest element and no least upper bound.
- In ℝ, the set of numbers less than 1 has a least upper bound, viz. 1, but no greatest element.
- In ℝ, the set of numbers less than or equal to 1 has a greatest element, viz. 1, which is also its least upper bound.
- In ℝ² with the product order, the set of pairs (
*x*,*y*) with 0 <*x*< 1 has no upper bound. - In ℝ² with the lexicographical order, this set has upper bounds, e.g. (1, 0). It has no least upper bound.

## References

- The notion of locality requires the function's domain to be at least a topological space.

- Davey, B. A.; Priestley, H. A. (2002).
*Introduction to Lattices and Order*(2nd ed.). Cambridge University Press. ISBN 978-0-521-78451-1.