What is a monoid? | #SoME1

#abstractalgebra #monoids #some1

Monoids are everywhere in mathematics, but what are they? And why are they so useful? This video uses a simple example to show you exactly what the 4 rules of monoids are all about. I made this video for the 2021 Summer of Math Exposition. Enjoy!

0:00 Introduction
1:12 Associativity
3:40 Neutral element
5:28 Definition of a monoid
7:49 A few notes about inverses

#some1

[kemsekov]

This is the most clear and intuitive explanation I've found so far. Thanks!

[geraltofrivia]

This is absolutely great. Thank you for the great content and please PLEASE 🙏 do other videos

[xsytrance]

Thank you for the amazing explanation!

[nickkma]

You have an amazing collection of books, Can you do tour of your library? The pattern that evolves out from the choice of books shows your desire to unearth the whole meaning of life with mathematics.

[joshuap]

Currently, I still don't quite get it but you've explained it the best of all videos I've seen and I have a slight understanding. Thank you for taking the time to help others in sharing your wonderful gift(s) :)

[cannot-handle-handles]

Great video! Would have been even better if you explicitly showed that the neutral element is both left-neutral and right-neutral. :-)

[lorenkuhn]

Amazing! I knew monoids, but finally I really understand them.

I know it's a longer way and you are doing more 'basic' videos for now (which I am looking forward to), but I hope you will make a similar video for monads sometime.

[westhamdd]

You should make more of these. Either deeper into Monoids or do some on other structures.

[MrRedstonefreedom]

Thanks for the concrete example! I especially liked the antimatter books.

I wonder why you can't just express inverses as mere subtraction (taking away books). Probably because if you must subtract, you then lose things like associativity. If you lose associativity, suddenly you can't do things like arbitrary division of computation in the case of distributed networks.

I wonder if there is an "associativizer" pattern, that can take such a need (for lack of direct inverses, but need to correct for elements), and shim a level of indirection to regenerate associativity while affording an ability to apply inverses without direct ones.

The books may be a good example to think about this more. It may be simply impossible, consequently constraining you to segregate your phases of computation such that you don't "taint" your associative & thus parallelizable computation with non-associative operations.

Great stuff! Generated plenty food for thought with just a simple & meaty example.

[paulzupan]

Something that I’m still confused about is, what exactly is the monoid in this scenario? What I’m asking is, is the operation itself the monoid? Is the monoid a certain set of data? Monoid is a noun, so what kind of thing is a monoid? With dependency injection in OOP, dependency injection refers to the act of passing an instance of an object as a parameter to a method rather than instantiating the object inside the method, thereby separating creation from use. That action is what dependency injection is. I’m clear on what the word monoid refers to.

[jcloewe]

Great explanation! Reminded me of my discrete math prof Kim Factor

[cesarmexica]

A round of applause for this man!! subscribed to your chanel!

[georgelaing]

In older texts, the primary structure
was the module, a concept closely
related (but not identical) to monoid.
I hope to follow more of your videos!!

[chinpokomon_]

good explanation. you also remind me of a high school classmate of mine, are you sure English is your native? (it is very good, I am just curious)

[BCarli]

Very interesting and clearly presented. Thank you.

[degrees]

📚Shared this gem to the less-mathematical friends, let’s see how it’ll go!

[quarkquark]

Thanks for this! I almost clicked off after you'd given the definition (which is what I came for), but I'm glad I didn't.

That was enough to give me the mathematical intuition as a group without inverses, but now I also understand (what I presume is) the programming motivation as the extension of a binary operation.

[Darrida]

Bourbaki abolished the term semigroup for taste reason.

[jonathandawson]

Subtract equations is not monoid lol (based on the definitions provided in the video)