Totally ordered set
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 .
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