Monoids | Group theory episode 1

#monoids #grouptheory #computerscience #abstractalgebra

What are monoids, why are they so abstract, and how do you calculate an empty product? In this video, we explain the exact purpose for each of the rules of monoids, with specific focus on the neutral element. We show many examples such as string concatenation and computer programs. And we already foreshadow the videos about group theory by looking at inverses.

We only have a single link for you today, and it may look familiar:
This is the very first video we ever made, for the 2021 Summer of Math Exposition. It explains monoids by stacking books on top of each other, which is really just a fancy way of concatenating strings vertically instead of horizontally ;-)

0:00 Introduction
1:30 What is a monoid?
3:36 Example: blocks
4:58 Homomorphisms
6:30 Elements are active
7:42 Example: Computer programs
11:09 Associativity
12:36 Commutativity
13:51 Why these specific rules?
16:13 The neutral element
18:30 Abstraction = power
19:30 Inverses

This video is published under a CC Attribution license

[Yeb.]

Finally, someone who ACTUALLY explains group theory

[deltapi]

I wish all teaching material had this pace.

[alegian]

I'm a computer science student, so I can't help but appreciate the "simple computer" example. So simple yet raises soooo many questions

[deltapi]

that pause at 4:22 .. well done. You know what you are doing.

[cdellio]

I can't believe your videos haven't garnered more attention. The animation is simple and concise -- as it should be. Your explanation style is likewise, and exemplary. Keep it up, please.

[joe_hoeller_chicago]

Your channel is badass, this is amazing!

[thejoe]

This is quite frankly the best video and explanation on Monoids I have seen on the Internet. It has finally helped me understand this concept and link it to related mathematical fields, like vector spaces. I would really like to see a video explaining Monads, since they have always been a mistery to me, but I suspect they are very closely related and maybe one of the examples in this video was also a monad in addition to being a monoid? Thanks!

[chineduecheruo]

Best explanation of Monoids, ive heard! Thanks.

[genecat]

I am nearly retired now and your explanations bring me back to a time when I first recognized something tantalizing about mathematics. I've always had the appetite to go deeper but was never exposed to this kind of lucidity that makes it so natural and satisfying, had I been, I'm pretty sure it would have changed many of the choices I made. Powerful stuff here.

[beatboxanimationnandan]

Now I can digest group theory
Great 🎉🎉

[jamiepond]

Absolutely outstanding work. Please, for the goodness of all of us on this planet, keep it up.

[EricKolotyluk]

The best explanation of Monads and Monoids I have ever seen. Way to go!

[pra.]

Very nice! I wish my abstract algebra class covered Monoids, the computer applications are especially cool

[lih]

Much appreciation for these videos. I never knew group theory was so beautiful and made so much sense!

[SciStone]

man i love your content, great work! The programming examples for the neutral element of multiplication blew my mind a little, didnt really think about that before.

[christosgeorgiadis]

Great work, thank you!

I've watched many an exposition of group theory and have never come across such a clear way of motivating the subject.

[kquat]

Excellent presentation. Concise and great clarity.

[pmmeurcatpics]

You've got me so hyped up for the future videos! Really wish I could see them earlier but oh well, I am but a poor student for now:(

[TrollFunMineMafia]

i can't wait for the video on groups, this series is gonna be so awesome, thank you so much :>

[jaopredoramires]

Now i finally understand why we think of Monoids/Monads when programming, thank you