Difference between revisions of "Minimum"
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| − | Given a [[partially ordered set]] <math>S</math>, the '''minimum''' [[element]] of <math>S</math>, if it exists, is some <math> | + | Given a [[partially ordered set]] <math>S</math>, the '''minimum''' [[element]] of <math>S</math>, if it exists, is some <math>m \in S</math> such that for all <math>s \in S</math>, <math>m \leq s</math>. |
For example, the minimum element of the [[set]] <math>S_1 = \{0, e, \pi, 4\}</math> of [[real number]]s is <math>0</math>, since it is smaller than every other element of the set. | For example, the minimum element of the [[set]] <math>S_1 = \{0, e, \pi, 4\}</math> of [[real number]]s is <math>0</math>, since it is smaller than every other element of the set. | ||
Latest revision as of 12:12, 9 February 2007
Given a partially ordered set 
, the minimum element of , if it exists, is some 
such that for all 
, 
.
For example, the minimum element of the set 
of real numbers is 
, since it is smaller than every other element of the set.
Every finite subset of the reals (or any other totally ordered set) has a minimum. However, many infinite subsets do not. The integers, 
have no minimum, since for any 
we can find 
such that 
. (Taking 
works nicely.)
A more subtle example of this phenomenon is the set 
. While this set has a greatest lower bound 0, it has no minimum.
The previous example suggests the following formulation: if is a set contained in some larger ordered set 
with the greatest lower bound property, then has a minimum if and only if the greatest lower bound of is a member of .
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