supremum
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:
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1.
For all , we have (i.e. is an upper bound).
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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:
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.
Title | supremum |
Canonical name | Supremum |
Date of creation | 2013-03-22 11:48:12 |
Last modified on | 2013-03-22 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 |