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 $\mathbb{R}$ or the integers $\mathbb Z$ is a total ordering, but if we order the subsets of the set $\{1, 2, 3\}$ by inclusion (the boolean lattice on a set of size 3), we don't get a total order because $\{1, 2\}$ and $\{3\}$ 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 $a + bi > c + di$ if $a > c$ or if $a = c$ and $b > d$ , 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 $\mathbb C$ .

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Totally ordered set