supremum
The supremum^{} of a set $X$ having a partial order^{} is the least upper bound of $X$ (if it exists) and is denoted $supX$.
Let $A$ be a set with a partial order $\u2a7d$, and let $X\subseteq A$. Then $s=supX$ if and only if:

1.
For all $x\in X$, we have $x\u2a7ds$ (i.e. $s$ is an upper bound).

2.
If ${s}^{\prime}$ meets condition 1, then $s\u2a7d{s}^{\prime}$ ($s$ is the least upper bound).
There is another useful definition which works if $A=\mathbb{R}$ with $\u2a7d$ the usual order on $\mathbb{R}$, supposing that s is an upper bound:
$$ 
Note that it is not necessarily the case that $supX\in X$. Suppose $X=]0,1[$, then $supX=1$, but $1\notin X$.
Note also that a set may not have an upper bound at all.
Title  supremum 
Canonical name  Supremum 
Date of creation  20130322 11:48:12 
Last modified on  20130322 11:48:12 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  11 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06A06 
Related topic  Infimum 
Related topic  MinimalAndMaximalNumber 
Related topic  InfimumAndSupremumForRealNumbers 
Related topic  ExistenceOfSquareRootsOfNonNegativeRealNumbers 
Related topic  LinearContinuum 
Related topic  NondecreasingSequenceWithUpperBound 
Related topic  EssentialSupremum 