**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Concept# Order isomorphism

Summary

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.
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.
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.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related courses (36)

Related lectures (220)

Related publications (508)

Related people (41)

Related concepts (16)

Related MOOCs (3)

MATH-351: Advanced numerical analysis

The student will learn state-of-the-art algorithms for solving differential equations. The analysis and implementation of these algorithms will be discussed in some detail.

AR-202(a): Studio BA4 (Bakker & Blanc A.)

MANSLAB se concentre sur la question de l'assemblage programmatique et spatial entre des territoires différents afin de provoquer la manufacture d'une densité physique et métaphysique.

COM-401: Cryptography and security

This course introduces the basics of cryptography. We review several types of cryptographic primitives, when it is safe to use them and how to select the appropriate security parameters. We detail how

Elementary Algebra: Numeric Sets

Explores elementary algebra concepts related to numeric sets and prime numbers, including unique factorization and properties.

Monster Group: Representation

Explores the Monster group, a sporadic simple group with a unique representation theory.

Mild Dissipative Surface Dynamics

Explores mild dissipative surface dynamics, including conservative behavior and ergodic functions.

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 strictly weaker than the concept of an order isomorphism. Both of these weakenings may be understood in terms of . Formally, given two partially ordered sets (posets) and , a function is an order embedding if is both order-preserving and order-reflecting, i.e.

Dense order

In mathematics, a partial order or total order < on a set is said to be dense if, for all and in for which , there is a in such that . That is, for any two elements, one less than the other, there is another element between them. For total orders this can be simplified to "for any two distinct elements, there is another element between them", since all elements of a total order are comparable. The rational numbers as a linearly ordered set are a densely ordered set in this sense, as are the algebraic numbers, the real numbers, the dyadic rationals and the decimal fractions.

Duality (order theory)

In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the same set, but with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P. It is easy to see that this construction, which can be depicted by flipping the Hasse diagram for P upside down, will indeed yield a partially ordered set. In a broader sense, two partially ordered sets are also said to be duals if they are dually isomorphic, i.

Introduction to optimization on smooth manifolds: first order methods

Learn to optimize on smooth, nonlinear spaces: Join us to build your foundations (starting at "what is a manifold?") and confidently implement your first algorithm (Riemannian gradient descent).

Geographical Information Systems 2

This course is the second part of a course dedicated to the theoretical and practical bases of Geographic Information Systems (GIS).
It offers an introduction to GIS that does not require prior compu

Geographical Information Systems 2

This course is the second part of a course dedicated to the theoretical and practical bases of Geographic Information Systems (GIS).
It offers an introduction to GIS that does not require prior compu

Fabio Zoccolan, Gianluigi Rozza

In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the Peclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Str ...

Simon Nessim Henein, Loïc Benoît Tissot-Daguette

The present invention concerns a pivot comprising two assemblies, namely a central assembly (401) and a peripheral assembly (400). These two assemblies are mobile in rotation relative to each other around an axis of rotation (A). The pivot is characterized ...

2024Alessandro Mapelli, Radoslav Marchevski, Alina Kleimenova

A sample of 3984 candidates of the K+ -> pi(+)gamma gamma decay, with an estimated background of 291 +/- 14 events, was collected by the NA62 experiment at CERN during 2017-2018. In order to describe the observed di-photon mass spectrum, the next-to-leadin ...