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.