What is a monoid? |

#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!
If you like this content, you can support me on Patreon: www.patreon.com/user?u=86649007 .
0:00 Introduction
1:12 Associativity
3:40 Neutral element
5:28 Definition of a monoid
7:49 A few notes about inverses
#some1

[@kemsekov6331]

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

[@AllAnglesMath]

Thank you, glad you enjoyed it!

[@hyperduality2838]

@@AllAnglesMath Dude this is the Hegelian dialectic. Thesis (pile 1) is dual to anti-thesis (pile 2) synthesizes pile 3 (the solution or target) -- the time independent Hegelian dialectic. Binary implies duality. Being is dual to non being creates becoming -- Plato. The Hegelian dialectic is a monoid. Stack 1 (thesis) is dual to stack 2 (anti-thesis) = stack 3 (synthesis or emergence). "Always two there are" -- Yoda.

[@geraltofrivia9424]

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

[@xsytrance]

Thank you for the amazing explanation!

[@lorenkuhn3806]

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.

[@nickkma1006]

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.

[@AllAnglesMath]

Maybe one day I will make a video about some of my favorite books. I do not think that mathematics can capture the meaning of life, I just like to marvel at the patterns and connections. I do have books about other subjects ;-)

[@nickkma1006]

​@@AllAnglesMath My bad, I had poorly framed the " unearthing meaning of life ..." from what I had intended to say. Let me rephrase my thoughts. The patterns evolving from choice of your books in the video looks to me as if you were attempting to unearth objective reality of the physical reality in the language of mathematics, which exists independently of our subjective experiences of it. I think you have clarified your position on this topic already. Thanks for the reply 😃

[@joshuap1348]

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) :)

[@hyperduality2838]

Dude this is the Hegelian dialectic. Thesis (pile 1) is dual to anti-thesis (pile 2) synthesizes pile 3 (the solution or target) -- the time independent Hegelian dialectic. Binary implies duality. Being is dual to non being creates becoming -- Plato. The Hegelian dialectic is a monoid. Stack 1 (thesis) is dual to stack 2 (anti-thesis) = stack 3 (synthesis or emergence). "Always two there are" -- Yoda.

[@BCarli1395]

Very interesting and clearly presented. Thank you.

[@AllAnglesMath]

Thank you so much! If you're interested in monoids, we also have a more elaborate video with examples from computer science.

[@karlrombauts4909]

This is a fantastic explanation!

[@westhamdd84]

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

[@hyperduality2838]

Dude this is the Hegelian dialectic. Thesis (pile 1) is dual to anti-thesis (pile 2) synthesizes pile 3 (the solution or target) -- the time independent Hegelian dialectic. Binary implies duality. Being is dual to non being creates becoming -- Plato. The Hegelian dialectic is a monoid. Stack 1 (thesis) is dual to stack 2 (anti-thesis) = stack 3 (synthesis or emergence). "Always two there are" -- Yoda.

[@AllAnglesMath]

I have a series on group theory coming up, so stay tuned!

[@hyperduality2838]

@@AllAnglesMath Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Summations (integration, syntropy) are dual to differences (differentiation, entropy). The integers or real numbers are self dual, watch the following:- kzbin.info/www/bejne/d6mzqJuAia2rick Injective is dual to surjective synthesizes bijective or isomorphism. Positive is dual to negative. Sine is dual to co-sine. The word co means mutual or duality!

[@cesarmexica]

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

[@AllAnglesMath]

Welcome, I hope you will enjoy it here ;-)

[@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.

[@AllAnglesMath]

Really good points. Typically, subtraction is defined in terms of inverses, not the other way around. So something like X - Y would be defined as *adding* X to the inverse of Y. So you still have only a single binary operation, and it's still associative.

[@juliavixen176]

@@AllAnglesMath I think of negative numbers (and the set of integers Z) as vectors. Specifically, a natural number scalar (from N) times either a (+1) or (-1) unit vector. Geometrically, this is like a 180° rotation (like how the imaginary unit vector is like a 90° rotation.) This is why (-1)•(-1)=(+1) Anyway, negative numbers are not numbers in the same way that imaginary (complex) numbers aren't numbers. The integers are one dimensional numbers, and the complex numbers are two dimensional. The scalars are zero dimensional points... or, by definition, the count of the elements within a set... recursive succession from the empty set.

[@AllAnglesMath]

@@juliavixen176 It's interesting that you look at negative numbers that way. It separates the "sign" part from the "absolute value" part. It's like a precursor to polar coordinates. Really interesting. Some of your ideas occur in our videos on complex numbers, by the way.

[@jcloewe8692]

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

[@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. :-)

[@AllAnglesMath]

Good point. And love those Life Gliders in your logo ;-)

[@negoitamihai8251]

cool! thanks for your insights!

[@annvanaelst8214]

duidelijke uitleg!

[@georgelaing2578]

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!!

[@quarkquark1]

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.

[@paulzupan3732]

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.

[@Nimblewright1992]

The monoid is the combination of a binary operation, a set it operates on, and an empty element that is part of that set. For instance: stacking, the set of all possible piles of books and the empty pile is a monoid. As is multiplication, the set of all rational numbers and the number 1.

[@Nimblewright1992]

Or in other terms, that I’ve also heard: stacks of books are a monoid under stacking, with the empty stack as the neutral element; or numbers are a monoid under multiplication with 1 as the neutral element

[@paulzupan3732]

@@Nimblewright1992 This is exactly the answer I was looking for, thank you so much!

[@Darrida]

Bourbaki abolished the term semigroup for taste reason.

[@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)

[@AllAnglesMath]

My native language is Dutch.

[@05degrees]

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

[@AllAnglesMath]

Great idea. Any feedback so far?

[@05degrees]

@@AllAnglesMath Uh, probably they’ve forgotten or admired it silently. 😑 😶

[@adamsmith275]

...I am one of those!... (although I am not your friend!...) I was wondering if there is a didactic sequence... or these videos... (their themes...) are independent...

[@AllAnglesMath]

@@adamsmith275 The didactic sequence is explained in our first video: kzbin.info/www/bejne/bKrapZKond19nLs

[@snk-js]

why books don't simply do that

[@scollyer.tuition]

I wish that they did. In fact, I monoid that they don't.

[@Ryans_Science]

i own 2 of those six books

[@AllAnglesMath]

Which ones? What did you think of them?

[@Ryans_Science]

@@AllAnglesMath haha, the Conway one, and the Penrose one. I saved up all my money for that Conway one coz the dude's amazing. I also have his "On Numbers and Games". I would tell you what I think about them but I never seem to have the time to get into them (sigh)

[@AllAnglesMath]

@@Ryans_Science I know that feeling. Lots of untouched books in my closet too. And I agree that Conway was amazing. He will be missed.

[@Ryans_Science]

@@AllAnglesMath for sure. He's become one with the force like Obi Wan Kenobi now.

[@jonathandawson3091]

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

[@AllAnglesMath]

You're absolutely right, I did not spot this when making the video. Addition of equations is a monoid, but subtraction isn't because it's not associative. Thank you for the correction!

[@jonathandawson3091]

Very good video, I learned what a monoid is now and will never forget.

[@tombouie]

Thks & en.wikipedia.org/wiki/Algebra#Groups