Order isomorphism

Definition

Let $(P, \le_P)$ and $(Q, \le_Q)$ be partially ordered sets. An order isomorphism from $P$ to $Q$ is a bijective function $f \colon P \to Q$ such that, for all $a, b \in P$, $$a \le_P b \iff f(a) \le_Q f(b).$$

Remarks

The inverse of an order isomorphism is an order isomorphism. The map $x \mapsto 2x$ is an order isomorphism from $\mathbb{Z}$ to the even integers. The ordered sets $\mathbb{N}$ and $\mathbb{Z}$ are not order isomorphic, because $\mathbb{N}$ has a least element and $\mathbb{Z}$ has none. A monotone bijection need not be an order isomorphism, when the orders are not total.

Explanations

L1 human

Order isomorphic sets have the same shape as ordered sets. Only the names of the elements differ.