Math 2.0

Order, lattices, ordered algebraic structures Proposition

Uniqueness of the supremum

Builds on Supremum

Statement

Let (P,≤)(P, \le) be a partially ordered set and let SS be a subset of PP. If bb and b′b' are both suprema of SS, then b=b′b = b'.

Remarks

Proof: bb is an upper bound and b′b' is a least upper bound, thus b′≤bb' \le b. In the same way b≤b′b \le b', and antisymmetry gives b=b′b = b'. The same argument shows that a set has at most one infimum.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Set bounded above, definition, no evidence levelSet bounded abovedefinitionSupremum, definition, no evidence levelSupremumdefinitionUniqueness of the supremum, statement, no evidence levelUniqueness of the sup…statement
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  1. Set bounded above definition
  2. Supremum definition
  3. Uniqueness of the supremum statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Infimum_and_supremum, Existence and uniqueness
About this page
Tag04ETX30HT2
URIm2://04ETX30HT2
Kindstatement, proposition
WritingW0 stub
MSC06A06
StatusThis copy of the site has no build data. Run m2 status 04ETX30HT2.
UsesSupremum

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