upper and lower bounds

Distinguishing elements of a set equipped with a preorder, relative to some subset:

An upper bound is greater than or equal to every element of the subset.
A lower bound is less than or equal to every element of the subset.

Formally,
for a preorder $\le$ over a set $K$ and subset $S\subseteq K$,

• an element $u\in K$ is an upper bound for $S$ if $s\le u$ for all $s\in S$
• an element $l\in K$ is a lower bound for $S$ if $l\le s$ for all $s\in S$

If the upper and lower bounds are contained in the subset, then they are the greatest and least elements of the subset, respectively.