Difference between revisions of "Totally ordered set"
(No difference)
|
Revision as of 13:00, 9 February 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
.
This article is a stub. Help us out by expanding it.