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 either 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.

Definition
Formally, given two posets  and , an order isomorphism  from  to  is a bijective function  from  to  with the property that, for every  and  in ,  if and only if . That is, it is a bijective order-embedding.

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

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

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

When an additional algebraic structure is imposed on the posets  and , a function from  to  must satisfy additional properties to be regarded as an isomorphism. For example, given two partially ordered groups (po-groups)  and , an isomorphism of po-groups from  to  is an order isomorphism that is also a group isomorphism, not merely a bijection that is an order embedding.

Examples
 The identity function on any partially ordered set is always an order automorphism.
 Negation is an order isomorphism from  to  (where  is the set of real numbers and  denotes the usual numerical comparison), since −x ≥ −y if and only if x ≤ y.
 The open interval  (again, ordered numerically) does not have an order isomorphism to or from the closed interval : the closed interval has a least element, but the open interval does not, and order isomorphisms must preserve the existence of least elements.
By Cantor's isomorphism theorem, every unbounded countable dense linear order is isomorphic to the ordering of the rational numbers. Explicit order isomorphisms between the quadratic algebraic numbers, the rational numbers, and the dyadic rational numbers are provided by Minkowski's question-mark function.

Order types
If  is an order isomorphism, then so is its inverse function.
Also, if  is an order isomorphism from  to  and  is an order isomorphism from  to , then the function composition of  and  is itself an order isomorphism, from  to .

Two partially ordered sets are said to be order isomorphic when there exists an order isomorphism from one to the other. 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.

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

Notes

References
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Morphisms
Order theory