Order isomorphism

Order isomorphism

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Encyclopedia
In the mathematical
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

field of order theory
Order theory
Order theory is a branch of mathematics which investigates our 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 introduces the field and gives some basic definitions...

an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

for partially ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set 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 relation that indicates that, for certain pairs of elements in the...

s (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 connection
Galois connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . The same notion can also be defined on preordered sets or classes; this article presents the common case of posets. Galois connections generalize the correspondence...

s.

Formally, given two posets S=(S, ≤S) and T=(T, ≤T) an order isomorphism from (S, ≤S) to (T, ≤T) is a surjective function h : ST such that for all u and v in S,
h(u) ≤T h(v) if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

uS v.

In this case, the posets S and T are said to be order isomorphic. Note that the above definition characterizes order isomorphisms as surjective order embeddings. It should also be remarked that order isomorphisms are necessarily injective. Hence, yet another characterization of order isomorphisms is possible: they are exactly those monotone bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

s that have a monotone inverse.

An order isomorphism from (S, ≤) to itself is called an order automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...

.

Examples

• Negation
Negation (disambiguation)
Negation may refer to:* Negation , the operation in mathematics that takes x to −x, mapping positive numbers to negative numbers and the other way around, equivalent to reversing the sign of a number...

is an order isomorphism from (R,≤) to (R,≥), since -x ≥ -y if and only if xy.
• The function f(x) = x-1 is an order automorphism on (R,≤), since x-1 ≤ y-1 if and only if xy.
• A utility
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....

function is an order isomorphism from some consumption set into the real line.