Difference between revisions of "Totally ordered set"
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Note that it is possible to impose a total ordering on any set. For example, the [[lexicographic ordering]] on the [[complex number]]s, where we say <math>a + bi > c + di</math> if <math>a > c</math> or if <math>a = c</math> and <math>b > d</math>, is a total ordering, but it is not a "natural" ordering of this set. In particular, it behaves very poorly with respect to arithmetic operations on <math>\mathbb C</math>. | Note that it is possible to impose a total ordering on any set. For example, the [[lexicographic ordering]] on the [[complex number]]s, where we say <math>a + bi > c + di</math> if <math>a > c</math> or if <math>a = c</math> and <math>b > d</math>, is a total ordering, but it is not a "natural" ordering of this set. In particular, it behaves very poorly with respect to arithmetic operations on <math>\mathbb C</math>. | ||
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| + | == See also == | ||
| + | |||
| + | * [[Order relation]] | ||
| + | * [[Binary relation]] | ||
{{stub}} | {{stub}} | ||
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| + | [[Category:Set theory]] | ||
Latest revision as of 12:52, 25 November 2007
A totally ordered set is a partially ordered set in which every two elements are comparable. Thus, the standard ordering on the real numbers 
or the integers 
is a total ordering, but if we order the subsets of the set 
by inclusion (the boolean lattice on a set of size 3), we don't get a total order because 
and 
are incomparable (there are no inclusion relations between them).
Note that it is possible to impose a total ordering on any set. For example, the lexicographic ordering on the complex numbers, where we say 
if 
or if 
and 
, is a total ordering, but it is not a "natural" ordering of this set. In particular, it behaves very poorly with respect to arithmetic operations on 
.
See also
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