Total order

Definition

Let $A$ be a set. A total order on $A$ is a partial order $\le$ on $A$ such that, for all $a, b \in A$, $a \le b$ or $b \le a$.

Remarks

Other names are linear order and simple order. The usual orders on $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$ are total orders. The order $\subseteq$ on the power set of $\{1, 2\}$ is not total.

Explanations

L1 human

In a total order, all elements stand on one line. For any two of them, one comes before the other, or they are equal.