Definition:Order Isomorphism

This page is about isomorphisms in order theory. For other uses, see Definition:Isomorphism.

Definition

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be posets.

Let $\phi: S \to T$ be a bijection such that:

• $\phi: S \to T$ is order-preserving;
• $\phi^{-1}: T \to S$ is order-preserving.

Then $\phi$ is an order isomorphism.

That is, $\phi$ is an order isomorphism iff:

$\forall x, y \in S: x \, \preceq_1 \, y \iff \phi \left({x}\right) \, \preceq_2 \, \phi \left({y}\right)$

So an order isomorphism can be described as a bijection that preserves ordering in both directions.

Two posets are (order) isomorphic if there exists such an order isomorphism between them.

Warning

It does not necessarily follow that if:

$\phi: S \to T$ is a bijection which is order-preserving

then:

$\phi^{-1}: T \to S$ is order-preserving.

Example

Let $S = \mathcal P \left({\left\{{a, b}\right\}}\right), T = \left\{{1, 2, 3, 4}\right\}$.

From Subset Relation on Power Set is Partial Ordering, we have that $\left({S, \subseteq}\right)$ is a poset.

Clearly so is $\left({T, \le}\right)$ (although its ordering is in fact total, it is still technically a poset).

Let $\phi: S \to T$ be defined as:

• $\phi \left({\varnothing}\right) = 1$
• $\phi \left({\left\{{a}\right\}}\right) = 2$
• $\phi \left({\left\{{b}\right\}}\right) = 3$
• $\phi \left({\left\{{a, b}\right\}}\right) = 4$

It is easily verified that $\phi: \left({S, \subseteq}\right) \to \left({T, \le}\right)$ is a bijection which is order-preserving.

But note that while $2 \le 3$, it is not the case that $\left\{{a}\right\} \subseteq \left\{{b}\right\}$ and so $\phi^{-1}$ is not order-preserving.

Also see

• Relation Isomorphism, from which it can be seen that order isomorphism is a special case.

• Results about order isomorphisms can be found here.

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 14$
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 3.2$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 7$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.7$