# Order isomorphism

In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that one of the orders can be obtained from the other just by renaming of elements. Two strictly weaker notions that relate to order isomorphisms are order embeddings and Galois connections.[1]

## Definition

It is also possible to define an order isomorphism to be a surjective order-embedding. The two assumptions that ${\displaystyle f}$ cover all the elements of ${\displaystyle T}$ and that it preserve orderings, are enough to ensure that ${\displaystyle f}$ is also one-to-one, for if ${\displaystyle f(x)=f(y)}$ then (by the assumption that ${\displaystyle f}$ preserves the order) it would follow that ${\displaystyle x\leq y}$ and ${\displaystyle y\leq x}$, implying by the definition of a partial order that ${\displaystyle x=y}$.

Yet another characterization of order isomorphisms is that they are exactly the monotone bijections that have a monotone inverse.[3]

An order isomorphism from a partially ordered set to itself is called an order automorphism.[4]

## Order types

Two partially ordered sets are said to be order isomorphic when there exists an order isomorphism from one to the other.[8] Identity functions, function inverses, and compositions of functions correspond, respectively, to the three defining characteristics of an equivalence relation: reflexivity, symmetry, and transitivity. Therefore, order isomorphism is an equivalence relation. The class of partially ordered sets can be partitioned by it into equivalence classes, families of partially ordered sets that are all isomorphic to each other. These equivalence classes are called order types.

• Permutation pattern, a permutation that is order-isomorphic to a subsequence of another permutation

## Notes

1. This is the definition used by Template:Harvtxt. For Template:Harvtxt and Template:Harvtxt it is a consequence of a different definition.
2. This is the definition used by Template:Harvtxt and Template:Harvtxt.
3. Template:Harvtxt, p. 13.
4. See example 4 of Template:Harvtxt, p. 39., for a similar example with integers in place of real numbers.
5. Template:Harvtxt, example 1, p. 39.

## References

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