Dave4Math / Courses / Well-Founded Confluence

# Well-Founded Confluence

Master well-founded confluence. Learn well-founded relations, well-founded induction, Newman's Lemma, and the unique normal forms property. With this course you get an introduction to reduction structures such as polynomial rings and modules over reduction rings.

### Created by Dave

This course begins Summer 2021.
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\$39.99
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Have you ever had a professor that you really enjoyed and wanted to take their classes over and over? Introducing Dave’s online math course Well-Founded Confluence. Dave knows his subject very well and is fun to watch and learn from.

## Who This Course is For

• Students in college who want to learn more mathematics.
• Anyone interested in well-founded confluence.
• Students who are enrolled in Introduction to Proofs and want to improve their grade.
• Anyone interested in majoring in mathematics, physics, or engineering.
• Anyone wanting to learn about reduction relations, well-founded induction, recursion, confluent relations, and reduction rings and other structures.

## Requirements

There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A calculator is not necessary.

## What You’ll Learn in Well-Founded Confluence

• Well-Founded Induction
• Descending Chains
• Recursion
• Antisymmetric and Irreflexive
• Reduction Relations
• Newman’s Lemma
• Buchberger-Winkler’s Property
• Reduction in the Integers
• Reduction in Vector Spaces
• Reduction in Polynomial Rings
• What are Reduction Rings?
• The Critical-Pair Completion Algorithm
• Quotients of Reduction Rings
• Sums of Reduction Rings
• Modules over Reduction Rings
• Polynomial Rings over Reduction Rings

## Course Description

We begin this course by studying well-founded relations. In particular, we study well-founded recursion and well-founded induction. We also discuss anti-symmetric relations and irreflexive relations.

Next, we examine reduction relations by bringing together well-founded relations and partial ordering relations. We then examine Newman’s Lemma and the Buchberger-Winkler generalization, which discusses various types of properties for reduction relations.

Closures are an essential part of this theory. We detail the reflexive, transitive closure of a reduction relation and the reflexive, transitive closure of the symmetric closure of a reduction relation. Using well-founded confluence to study the underlying equivalence relation is emphasized.

And the next part of the course discusses various reduction structures, including reduction rings, modules over reduction rings, polynomial rings over reduction rings, and further quotients, products, and sums of reduction rings. In each of the structures, we emphasize that establishing a critical-pair completion procedure can be carried over to other algebraic structures.

## Recommended Prerequisites for Well-Founded Confluence

I recommend the prerequisite course Ordered Sets. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free articles on introduction to proofs.

Have you ever had a professor that you really enjoyed and wanted to take their classes over and over? Introducing Dave’s online math course Well-Founded Confluence. Dave knows his subject very well and is fun to watch and learn from.

## Who This Course is For

• Students in college who want to learn more mathematics.
• Anyone interested in well-founded confluence.
• Students who are enrolled in Introduction to Proofs and want to improve their grade.
• Anyone interested in majoring in mathematics, physics, or engineering.
• Anyone wanting to learn about reduction relations, well-founded induction, recursion, confluent relations, and reduction rings and other structures.

## Requirements

There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A calculator is not necessary.

## What You’ll Learn in Well-Founded Confluence

• Well-Founded Induction
• Descending Chains
• Recursion
• Antisymmetric and Irreflexive
• Reduction Relations
• Newman’s Lemma
• Buchberger-Winkler’s Property
• Reduction in the Integers
• Reduction in Vector Spaces
• Reduction in Polynomial Rings
• What are Reduction Rings?
• The Critical-Pair Completion Algorithm
• Quotients of Reduction Rings
• Sums of Reduction Rings
• Modules over Reduction Rings
• Polynomial Rings over Reduction Rings

## Course Description

We begin this course by studying well-founded relations. In particular, we study well-founded recursion and well-founded induction. We also discuss anti-symmetric relations and irreflexive relations.

Next, we examine reduction relations by bringing together well-founded relations and partial ordering relations. We then examine Newman’s Lemma and the Buchberger-Winkler generalization, which discusses various types of properties for reduction relations.

Closures are an essential part of this theory. We detail the reflexive, transitive closure of a reduction relation and the reflexive, transitive closure of the symmetric closure of a reduction relation. Using well-founded confluence to study the underlying equivalence relation is emphasized.

And the next part of the course discusses various reduction structures, including reduction rings, modules over reduction rings, polynomial rings over reduction rings, and further quotients, products, and sums of reduction rings. In each of the structures, we emphasize that establishing a critical-pair completion procedure can be carried over to other algebraic structures.

## Recommended Prerequisites for Well-Founded Confluence

I recommend the prerequisite course Ordered Sets. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free articles on introduction to proofs.

## Course Content

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### David A. Smith

#### Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.