In the
mathematics 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
and Galois connections.
[; .]
Definition
Formally, given two posets
$(S,\backslash le\_S)$ and
$(T,\backslash le\_T)$, an
order isomorphism from
$(S,\backslash le\_S)$ to
$(T,\backslash le\_T)$ is a
bijection $f$ from
$S$ to
$T$ with the property that, for every
$x$ and
$y$ in
$S$,
$x\; \backslash le\_S\; y$ if and only if
$f(x)\backslash le\_T\; f(y)$. That is, it is a bijective
orderembedding.
[This is the definition used by . For and it is a consequence of a different definition.]
It is also possible to define an order isomorphism to be a surjective orderembedding. The two assumptions that $f$ cover all the elements of $T$ and that it preserve orderings, are enough to ensure that $f$ is also onetoone, for if $f(x)=f(y)$ then (by the assumption that $f$ preserves the order) it would follow that $x\backslash le\; y$ and $y\backslash le\; x$, implying by the definition of a partial order that $x=y$.
Yet another characterization of order isomorphisms is that they are exactly the monotone that have a monotone inverse.[This is the definition used by and .]
An order isomorphism from a partially ordered set to itself is called an order automorphism.[, p. 13.]
When an additional algebraic structure is imposed on the posets $(S,\backslash le\_S)$ and $(T,\backslash le\_T)$, a function from $(S,\backslash le\_S)$ to $(T,\backslash le\_T)$ must satisfy additional properties to be regarded as an isomorphism. For example, given two partially ordered groups (pogroups) $(G,\; \backslash le\_G)$ and $(H,\; \backslash le\_H)$, an isomorphism of pogroups from $(G,\backslash leq\_G)$ to $(H,\backslash le\_H)$ is an order isomorphism that is also a group isomorphism, not merely a bijection that is an order embedding.[This definition is equivalent to the definition set forth in .]
Examples

The identity function on any partially ordered set is always an order automorphism.

Additive inverse is an order isomorphism from $(\backslash mathbb\{R\},\backslash leq)$ to $(\backslash mathbb\{R\},\backslash geq)$ (where $\backslash mathbb\{R\}$ is the set of and $\backslash le$ denotes the usual numerical comparison), since − x ≥ − y if and only if x ≤ y.
[See example 4 of , p. 39., for a similar example with integers in place of real numbers.]

The open interval $(0,1)$ (again, ordered numerically) does not have an order isomorphism to or from the closed interval $0,1$: the closed interval has a least element, but the open interval does not, and order isomorphisms must preserve the existence of least elements.
[, example 1, p. 39.]
Order types
If
$f$ is an order isomorphism, then so is its
inverse function.
Also, if
$f$ is an order isomorphism from
$(S,\backslash le\_S)$ to
$(T,\backslash le\_T)$ and
$g$ is an order isomorphism from
$(T,\backslash le\_T)$ to
$(U,\backslash le\_U)$, then the function composition of
$f$ and
$g$ is itself an order isomorphism, from
$(S,\backslash le\_S)$ to
$(U,\backslash le\_U)$.
[; .]
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 .
See also

Permutation pattern, a permutation that is orderisomorphic to a subsequence of another permutation
Notes