greatest and least elements

Distinguishing elements of a set equipped with a preorder:

A greatest element is an upper bound contained in the set, i.e. it is greater than or equal to any other element of the set.
A least element is a lower bound contained in the set, i.e. it is less than or equal to any other element of the set.

Formally,
for a preorder $\le$ over set $S$,

• an element $g\in S$ is a greatest element of $S$ if $s\le g$ for all $s\in S$
• an element $l\in S$ is a least element of $S$ if $l\le s$ for all $s\in S$

N.B. by definition, a greatest or least element must be comparable to every other element.

If $S$ is equipped with a partial order, then $S$ can have at most one greatest element and at most one least element, equal to the supremum and infimum, and they are the maximum and minimum of $S$, respectively.

If $S$ is equipped with a total order, the maximum and minimum necessarily exist.