In order theory, a discipline within mathematics, a critical pair is a pair of elements in a partially ordered set that are incomparable but that could be made comparable without requiring any other changes to the partial order. Formally, let P = (S, ≤) be a partially ordered set. Then a critical pair is an ordered pair (x, y) of elements of S with the following three properties: * x and y are incomparable in P, * for every z in S, if z < x then z < y, and * for every z in S, if y < z then x < z. If (x, y) is a critical pair, then the binary relation obtained from P by adding the single relationship x ≤ y is also a partial order. The properties required of critical pairs ensure that, when the relationship x ≤ y is added, the addition does not cause any violations of the transitive property. A set R of linear extensions of P is said to reverse a critical pair (x, y) in P if there exists a linear extension in R for which y occurs earlier than x. This property may be used to characterize realizers of finite partial orders: A nonempty set R of linear extensions is a realizer if and only if it reverses every critical pair. (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
Set Theory 1.4 : Well Orders, Order Isomorphisms, and Ordinals
In this video, I introduce well ordered sets and order isomorphisms, as well as segments. I use these new ideas to prove that all well ordered sets are order isomorphic to some ordinal. Email : fematikaqna@gmail.com Discord: https://discord.gg/ePatnjV Subreddit : https://www.reddit.com/r/
From playlist Set Theory
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
Intermediate Algebra-Inverse Functions
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From playlist Intermediate Algebra
Orders on Sets: Part 1 - Partial Orders
This was recorded as supplemental material for Math 115AH at UCLA in the spring quarter of 2020. In this video, I discuss the concept and definition of a partial order.
From playlist Orders on Sets
Set Theory (Part 3): Ordered Pairs and Cartesian Products
Please feel free to leave comments/questions on the video and practice problems below! In this video, I cover the Kuratowski definition of ordered pairs in terms of sets. This will allow us to speak of relations and functions in terms of sets as the basic mathematical objects and will ser
From playlist Set Theory by Mathoma
Put all three properties of binary relations together and you have an equivalence relation.
From playlist Abstract algebra
Set Theory (Part 5): Functions and the Axiom of Choice
Please feel free to leave comments/questions on the video and practice problems below! In this video, I introduce functions as a special sort of relation, go over some function-related terminology, and also prove two theorems involving left- and right-inverses, with the latter theorem nic
From playlist Set Theory by Mathoma
Discrete Morse Theory -- math major seminar.
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From playlist MathMajor Seminar
Introduction to Solid State Physics, Lecture 19: Superconductivity Theory
Upper-level undergraduate course taught at the University of Pittsburgh in the Fall 2015 semester by Sergey Frolov. The course is based on Steven Simon's "Oxford Solid State Basics" textbook. Lectures recorded using Panopto, to see them in Panopto viewer follow this link: https://pitt.host
From playlist Introduction to Solid State Physics
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
Wolfram Physics Project: Axiomatization of the Computational Universe Tuesday, Feb. 16, 2021
This is a Wolfram Physics Project working session about the axiomatization of the Computational Universe. Begins at 1:36 Originally livestreamed at: https://twitch.tv/stephen_wolfram Stay up-to-date on this project by visiting our website: http://wolfr.am/physics Check out the announceme
From playlist Wolfram Physics Project Livestream Archive
Ulrich Bauer (3/4/22): Gromov hyperbolicity, geodesic defect, and apparent pairs in Rips filtrations
Motivated by computational aspects of persistent homology for Vietoris-Rips filtrations, we generalize a result of Eliyahu Rips on the contractibility of Vietoris-Rips complexes of geodesic spaces for a suitable parameter depending on the hyperbolicity of the space. We consider the notion
From playlist Vietoris-Rips Seminar
Discrete Morse Theory 2.0 [Ximena Fernández]
Is there any way to understand the homotopy type of CW-complexes from its combinatorial structure? In this tutorial we explain the main ideas behind discrete Morse theory, and present a refinement of the theory that allows to answer positively the previous question. We apply these results
From playlist Tutorial-a-thon 2021 Fall
Automated Theorem Proving and Axiomatic Mathematics
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From playlist Wolfram Technology Conference 2019
MagLab Theory Winter School 2019: Andrey Chubukov "vs Non-Fermi liquid"
Topic: Superconductivity vs non Fermi liquid near a quantum-critical point The National MagLab held it's seventh Theory Winter School in Tallahassee, FL from January 7th - 11th, 2019.
From playlist 2019 Theory Winter School
Robyn Brooks and Celia Hacker (6/24/20): Morse-based fibering of the rank invariant
Title: Morse-based fibering of the rank invariant Abstract: Given the success of single-parameter persistence in data analysis and the fact that some systems warrant analysis across multiple parameters, it is highly desirable to develop data analysis pipelines based on multi-parameter per
From playlist AATRN 2020
Wolfram Physics III: Completion Procedures and Basic Quantum Mechanics"
Find more information about the summer school here: https://education.wolfram.com/summer/school Stay up-to-date on this project by visiting our website: http://wolfr.am/physics Check out the announcement post: http://wolfr.am/physics-announcement Find the tools to build a universe: https:
From playlist Wolfram Summer Programs
MagLab Theory Winter School 2019: Nandini Trivedi "Scientific Revolutions 2"
Topic: Scientific Revolutions: How can an insulator become a superconductor? The National MagLab held it's seventh Theory Winter School in Tallahassee, FL from January 7th - 11th, 2019.
From playlist 2019 Theory Winter School
Definition of the Order of an Element in a Group and Multiple Examples
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Definition of the Order of an Element in a Group and Multiple Examples
From playlist Abstract Algebra