In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
, an order isomorphism is a special kind of
monotone function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
that constitutes a suitable notion of
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
for
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
s (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 embedding In order theory, a branch of mathematics, an order embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order embeddings constitute a notion which is s ...
s and
Galois connections.
Definition
Formally, given two
posets and
, an order isomorphism from
to
is a
bijective function
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
from
to
with the property that, for every
and
in
,
if and only if
. That is, it is a bijective
order-embedding In order theory, a branch of mathematics, an order embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order embeddings constitute a notion which is stric ...
.
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
Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony.
Monotone or monotonicity may also refer to:
In economics
*Monotone preferences, a property of a consumer's preference ordering.
*Monotonic ...
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 group
In abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order "≤" that is ''translation-invariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' ≤ ''b'' ...
s (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 In order theory, a branch of mathematics, an order embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order embeddings constitute a notion which is s ...
.
Examples
* The
identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
on any partially ordered set is always an order automorphism.
*
Negation is an order isomorphism from
to
(where
is the set of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s 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
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...