Difference between revisions of "Minimum"
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− | Given | + | 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>n \in S</math>, <math>M \leq n</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. |
Revision as of 13:07, 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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