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The supremum of a set having a partial order is the least upper bound of (if it exists) and is denoted .
Let be a set with a partial order , and let
. Then
if and only if:
- 1.
- For all
, we have
(i.e. is an upper bound).
- 2.
- If
meets condition 1, then
( is the least upper bound).
There is another useful definition which works if
with the usual order on
, supposing that s is an upper bound:
 if and only if 
Note that it is not necessarily the case that
. Suppose
, then
, but
.
Note also that a set may not have an upper bound at all.
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