A \emph{chain} is a partially ordered set $\mathbf{C}=\langle C,\le\rangle$ such that

$\le$ is a total order: $x\le y \mbox{ or } y\le x$

Remark:

Let $\mathbf{C}$ and $\mathbf{D}$ be chains. A morphism from $\mathbf{C}$ to $\mathbf{D}$ is a function $h:C\rightarrow D$ that is a orderpreserving:

$x\le y\Longrightarrow h(x)\le h(y)$

Example 1:

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &1 f(4)= &1 f(5)= &1 f(6)= &1 \end{array}$

Well-ordered chains

Dense linear orders

Trees