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.[1]
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.[2]
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
monotonebijections that have a monotone inverse.[3]
An order isomorphism from a partially ordered set to itself is called an order
automorphism.[4]
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.[5]
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.[6]
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.[7]
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 .[10]
Two partially ordered sets are said to be order isomorphic when there exists an order isomorphism from one to the other.[11] 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