Total order (linear order)

A total order (or linear order) on a set $X$ is a partial order $\le$ on $X$ such that for all $x,y\in X$,

This property is called comparability (or trichotomy when strengthened appropriately).

Total orders allow one to speak of intervals, monotonicity, and order convergence, which are central in one-variable real analysis.

Examples:

• The usual order $\le$ on $\mathbb{R}$ is a total order.
• Lexicographic order on $\mathbb{R}^2$: define $(a,b)\le_{\mathrm{lex}}(c,d)$ iff either $a<c$, or $a=c$ and $b\le d$; this is a total order.
• $\subseteq$ on $\mathcal{P}(\{1,2\})$ is not a total order since $\{1\}$ and $\{2\}$ are incomparable.