Total order (linear order)

A total order (or linear order) on a set $P$ is a partial order $\le$ such that

Total orders allow “linear” comparison of elements and are the starting point for the notion of a well-ordered set .

Examples:

• The usual $\le$ on $\mathbb{Z}$ or $\mathbb{R}$ is a total order.
• Lexicographic order on $\mathbb{R}^2$ is a total order.
• The divisibility order on $\mathbb{N}$ is not total: neither $2$ divides $3$ nor $3$ divides $2$.