Order theory | Duality theories
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.e. if one poset is order isomorphic to the dual of the other. The importance of this simple definition stems from the fact that every definition and theorem of order theory can readily be transferred to the dual order. Formally, this is captured by the Duality Principle for ordered sets: If a given statement is valid for all partially ordered sets, then its dual statement, obtained by inverting the direction of all order relations and by dualizing all order theoretic definitions involved, is also valid for all partially ordered sets. If a statement or definition is equivalent to its dual then it is said to be self-dual. Note that the consideration of dual orders is so fundamental that it often occurs implicitly when writing ≥ for the dual order of ≤ without giving any prior definition of this "new" symbol. (Wikipedia).
The elements of a set can be ordered by a relation. Some relation cause proper ordering and some, partial ordering. Have a look at some examples.
From playlist Abstract algebra
15 Properties of partially ordered sets
When a relation induces a partial ordering of a set, that set has certain properties with respect to the reflexive, (anti)-symmetric, and transitive properties.
From playlist Abstract algebra
Duality in Higher Categories-I by Pranav Pandit
PROGRAM DUALITIES IN TOPOLOGY AND ALGEBRA (ONLINE) ORGANIZERS: Samik Basu (ISI Kolkata, India), Anita Naolekar (ISI Bangalore, India) and Rekha Santhanam (IIT Mumbai, India) DATE & TIME: 01 February 2021 to 13 February 2021 VENUE: Online Duality phenomena are ubiquitous in mathematics
From playlist Dualities in Topology and Algebra (Online)
Order of Elements in a Group | Abstract Algebra
We introduce the order of group elements in this Abstract Algebra lessons. We'll see the definition of the order of an element in a group, several examples of finding the order of an element in a group, and we will introduce two basic but important results concerning distinct powers of ele
From playlist Abstract Algebra
Orders and Ordered Sets | Axiomatic Set Theory, Section 2.3
We discuss order relations on sets, and isomorphisms of ordered sets. My Twitter: https://twitter.com/KristapsBalodi3
From playlist Axiomatic Set Theory
Duality in Algebraic Geometry by Suresh Nayak
PROGRAM DUALITIES IN TOPOLOGY AND ALGEBRA (ONLINE) ORGANIZERS: Samik Basu (ISI Kolkata, India), Anita Naolekar (ISI Bangalore, India) and Rekha Santhanam (IIT Mumbai, India) DATE & TIME: 01 February 2021 to 13 February 2021 VENUE: Online Duality phenomena are ubiquitous in mathematics
From playlist Dualities in Topology and Algebra (Online)
Duality In Higher Categories II by Pranav Pandit
PROGRAM DUALITIES IN TOPOLOGY AND ALGEBRA (ONLINE) ORGANIZERS: Samik Basu (ISI Kolkata, India), Anita Naolekar (ISI Bangalore, India) and Rekha Santhanam (IIT Mumbai, India) DATE & TIME: 01 February 2021 to 13 February 2021 VENUE: Online Duality phenomena are ubiquitous in mathematics
From playlist Dualities in Topology and Algebra (Online)
Math 101 090817 Introduction to Analysis 04 Ordered fields
Ordered sets. Examples. Ordered fields. Properties of ordered fields.
From playlist Course 6: Introduction to Analysis (Fall 2017)
Deconfined quantum critical points: symmetries and dualities by Senthil Todadri
Date : Friday, July 7, 2017 Time : 3:00 PM Venue : Madhava Lecture Hall, ICTS Campus, Bangalore Abstract : It has been recognized for many years that at certain quantum phase transitions the standard Landau-Ginzburg-Wilson paradigm will break down. The most striking
From playlist Seminar Series
Duality In Higher Categories III by Pranav Pandit
PROGRAM DUALITIES IN TOPOLOGY AND ALGEBRA (ONLINE) ORGANIZERS: Samik Basu (ISI Kolkata, India), Anita Naolekar (ISI Bangalore, India) and Rekha Santhanam (IIT Mumbai, India) DATE & TIME: 01 February 2021 to 13 February 2021 VENUE: Online Duality phenomena are ubiquitous in mathematics
From playlist Dualities in Topology and Algebra (Online)
Symmetries, Duality, and the Unity of Physics (Lecture – 01) by Nathan Seiberg
DATE: 08 January 2018, 16:00 to 17:30 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Lecture 1: 8 January 2018, 16:00 to 17:30 Title: Symmetries, Duality, and the Unity of Physics Abstract: Global symmetries and gauge symmetries have played a crucial role in physics. The idea of duality d
From playlist Infosys-ICTS Chandrasekhar Lectures
Supersymmetric Gauge Dynamics, Part 3 - Nathan Seiberg
Supersymmetric Gauge Dynamics, Part 3 Nathan Seiberg Institute for Advanced Study July 30, 2010
From playlist PiTP 2010
Matrix Models, Gauge-Gravity Duality, and Simulations on the Lattice (Lecture 3) by Georg Bergner
PROGRAM NONPERTURBATIVE AND NUMERICAL APPROACHES TO QUANTUM GRAVITY, STRING THEORY AND HOLOGRAPHY (HYBRID) ORGANIZERS: David Berenstein (University of California, Santa Barbara, USA), Simon Catterall (Syracuse University, USA), Masanori Hanada (University of Surrey, UK), Anosh Joseph (II
From playlist NUMSTRING 2022
Shay Shadovsky - Involutions and costs: A zoo of dualities - IPAM at UCLA
Recorded 08 February 2022. Shay Sadovsky of Tel Aviv University presents "Involutions and costs: A zoo of dualities" at IPAM's Calculus of Variations in Probability and Geometry Workshop. Abstract: Motivated by Boroczky and Schneider's characterization of set polarity, and Artstein-Avidan
From playlist Workshop: Calculus of Variations in Probability and Geometry
2020 Theory Winter School: Srinivas Raghu
Topic: Boson-ferimon duality in strongly coupled field theories For more information on the 2020 Theory Winter School: https://nationalmaglab.org/news-events/events/for-scientists/winter-theory-school
From playlist 2020 Theory Winter School
Scattering Amplitudes in Maximally Supersymmetric Gauge Theory and a New Duality
Topic: Scattering Amplitudes in Maximally Supersymmetric Gauge Theory and a New Duality Speaker: Lance Dixon Affiliation: Stanford University Date: May 2, 2022 Lance Dixon 2022-05-02
From playlist IAS High Energy Theory Seminar
Michael Green -- String Scattering Amplitudes, Feynman diagrams, and M-theory
This workshop seeks to explore connections between geometric flows and other areas of mathematics and physics. Geometric flows refer to ways in which geometry can be deformed smoothly in time, rather analogous to the way in which the geometry of the surface of a balloon becomes smooth and
From playlist Research Lectures
Rings 12 Duality and injective modules
This lecture is part of an online course on rings and modules. We descibe some notions of duality for modules generalizing the dual of a vector space. We first discuss duality for free and projective modules, which is very siilar to the vector space case. Then we discuss duality for finit
From playlist Rings and modules
Quantum Inequivalence, Evanescent Operators and Gravity Divergences: Zvi Bern
https://strings2015.icts.res.in/talkTitles.php
From playlist Strings 2015 conference
Put all three properties of binary relations together and you have an equivalence relation.
From playlist Abstract algebra