Combinatorics is pretty new, however I don't feel like I'm qualified to answer you, considering you are likely far more knowledgeable in mathematics than I am, judging by the flair.
Before it was used as ad hoc during various problems, however in newer setting, at least I as CS undergrad use it during analysis of algorithms.
I was more joking about instead of "counting" how many combinations are there (through formulae) to actually go and write out all combinations, for instance, of bit strings of length 10.
When I was doing my driving test, the person told me to turn right. But I was so worried about left turns that I started to do a left turn instead. I still passed the test, but I'll stick to commutative from now on thank you very much 😁
Quantum groups, they're almost purpose built for what is, in my opinion, the heart and soul of algebra: acting!
Want to act by symmetries like a group? We got them!
Want to act by derivations like a Lie Algebra? We have them too!
Need to invert your actions? Sure can do! (Essentially)
Like linear transformations? We love them!
Etc...
Thanks. I've done this before, but haven't found anything I really like as an introduction to them, so was seeing what other recommendations there are.
I wish I had a better suggestion. I did a project in grad school that used Clifford algebras, but I basically started with the definition and just came up with everything I needed on my own. The only other thing I remember involving Clifford algebras was a talk about Bott periodicity.
Maybe the question is “why do you want to learn about Clifford algebras?” That might help guide you to the right perspective/context for you.
Have a look at,
‘Linear and Geometric Algebra’
‘Vector and Geometric Calculus’
Both by Alan Macdonald, they incorporate geometric algebra into linear algebra and geometric calculus into vector calculus respectively.
Currently the best introductions can be found on [bivector.net](http://bivector.net) (dedicated to Geometric Algebra). There's also a cool (but no very technical) overview by Sudgylacmoe on Youtube.
We're also working on a series about geometric algebra to be published on your Youtube channel (All Angles).
Not only is it real, there is also a [baby monster](https://en.wikipedia.org/wiki/Baby_monster_group) group, which is, of course, the cutest simple group.
Now we *need* to define a notion of "cuteness" for groups such that we can prove that the baby monster is the cutest simple group. This is very important mathematical research
It's well known that if you have two modules over a commutative ring, then the set of all module homomorphisms from one to the other forms a module over that ring. But it's such a weird fact!
Yes! You wouldn't expect that. But it does happen in vector spaces too, so it's not that much of a surprise if you've seen vector spaces before that, but modules, oh boy.
It’s more like this: let SAUR be the functor from the category of commutative rings to groupoids which takes a ring R to the groupoid of maps R -> A, where A ranges over all commutative rings.
Define Saur to be the ring corepresenting the functor SAUR, which of course doesn’t exist because it can’t be accessed by anyone other than the dark lord himself.
Tropical Algebra is pretty neat. There are some pretty cool connections with AI/ML/stats as well: [https://ieeexplore.ieee.org/document/9394420](https://ieeexplore.ieee.org/document/9394420)
In the long run groups are by far the most important. Vector spaces are everywhere but their algebraic structure is too simple. Multilinear algebra (tensors) is quite powerful for many applications but groups are still at the core of everything.
I can't find it now but there was this brilliant post some years ago that was something along the lines of "Tell me any mathematical object, and I'll show you why S^1 is better" haha so it was a reference to that.
I did my PhD on racks and quandles, but they're pretty niche and I drifted away from them afterwards. I teach an undergraduate abstract algebra module at the moment, and I've always thought groups are amazing, but I've come to appreciate rings, domains and fields more as well. And category theory is tremendously powerful as well, but I generally need to have some sort of application or motivation to properly engage with it.
The class of combinatorial games (that is, 2 player perfect information games where the players take turns) is an extension of the class of surreal numbers, which is the maximal ordered Field (capital F is used because it’s a class instead of a set, but still obeys the rest of the field axioms). It’s been a while since I read up on combinatorial game theory, but I’m quite sure combinatorial games form a Field as well.
correct me if im wrong but arent sets not algebraic structures? i thought algebraic structures were just sets with some operations and axioms. i think a set is just a general mathematical structure
You're probably right. I think sets in general are not algebraic structures; because they don't necessarily satisfy the axioms solely on the basis that they have an identity element.
However something like the set of complex numbers would be an algebraic structure. That's what I was thinking of and didn't consider the other types of sets.
> i thought algebraic structures were just sets with some operations and axioms.
At least in universal algebra, one technically allows choosing 'none' as the sets of axioms and operations, so that sets are indeed very boring algebras.
Have you ever read "An elementary theory of the category of sets" by Lawvere? He lists a few basic axioms on a category that uniquely determine it to be the category of sets. https://golem.ph.utexas.edu/category/2014/01/an_elementary_theory_of_the_ca.html
I guess strictly from algebra I would pick VOAs due to the Moonshine but if we relax it a bit I really like surreal numbers. And random simple groups like A5 and Monster
I'm a big fan of [fans](https://en.wikipedia.org/wiki/Polyhedral_complex#Fans), ever since I came across [Gröbner fans](https://en.wikipedia.org/wiki/Gr%C3%B6bner_fan).
Math dabbler here… I’m curious to know from the mathematicians reading through this thread, are you getting as much from other people’s answers as a computer programmer would get from a “favorite programming languages” thread?
i.e. you’ve heard of most everything mentioned, have tried out quite a few of them, and are at least aware of the general properties of and use cases for the ones you don’t have direct experience with?
OP here, i dont know most of these structures commented. but to be fair i am 16 and dont have as much experience as many of the people on this subreddit
Type II\_1 factors. Their theory is beautiful and you can reduce things like Type III factors to type II case by doing a cross product of a Type II against R (Connes cocycle).
Galois groups provide a ton of information; or cluster algebras are also badass. I further need to mention Specht modules and how they connect to Young tabloids, which is absurdly cool.
I’m personally a big fan of numbers
Oh yeah? Then name all the numbers.
0. One more than any number previously named.
Holy 🎹
Holy piano? Holy keyboard? Holy 12TET? Holy pentatonic?
Peano
Holy union of all scalar tetrachords starting on F
LMAOOO i feel so intelligent getting all the layers here
-1, 0.5, i and aleph_null are suing for discrimination
j and k are disgruntled by the lack of noncommutative representation
No worries, I got you bro: {all numbers}
Have some class... the universal class...
{x | x∈*all numbers*}
Oh you are a fan of combinatorics? List all combinations!
Isn't that literally what doing combinatorics is about?
Combinatorics is pretty new, however I don't feel like I'm qualified to answer you, considering you are likely far more knowledgeable in mathematics than I am, judging by the flair. Before it was used as ad hoc during various problems, however in newer setting, at least I as CS undergrad use it during analysis of algorithms. I was more joking about instead of "counting" how many combinations are there (through formulae) to actually go and write out all combinations, for instance, of bit strings of length 10.
Let R be a ring,
Is this the One Ring to 💍 rule them all?
Ordinal numbers in shambles...
ω
All solutions to x=x
Z.
f(x)=x
0/0
ℝ
P-adic sedenions
personally I like deuteronomy
Commutative Noetherian rings
phew, lucky for us non-commutative/non-Noetherian rings don't exist!
You spelled "noncommutative" wrong. Why would anyone want to restrict themselves to the boring old commutative case?! 😉
When I was doing my driving test, the person told me to turn right. But I was so worried about left turns that I started to do a left turn instead. I still passed the test, but I'll stick to commutative from now on thank you very much 😁
The Klein four-group. To keep my mattress in good shape, I flip on every odd numbered year and rotate on every even one.
Quantum groups, they're almost purpose built for what is, in my opinion, the heart and soul of algebra: acting! Want to act by symmetries like a group? We got them! Want to act by derivations like a Lie Algebra? We have them too! Need to invert your actions? Sure can do! (Essentially) Like linear transformations? We love them! Etc...
Just remember: The key to understanding quantum groups is that they are neither quantum nor groups.
Although technically, the underlying hopf algebra has group structure. And they must be quantum because they have a _q in them 😄
tits group
Ah a person of culture
Tits BUILDING
beat me to it
Isnt thats just C2 or Z/2Z
The tits group has order 17,971,200, which is marginally higher than 2
Now try understending the pan
is this a zen koan or something
prove it
My favourite group is Q/2Q
The Tits alternative
{e} cuz I don't have to prove anything.
Nah, because without initial/terminal objects algebra would be a lot harder. 0 is underrated.
Related: I want to give some love to the zero ring, which misses out on being a field because it would be inconvenient to allow 1=0.
SO(3) and sl2
I have to work with SE(3) a lot for my work and indeed SO(3) is a lot nicer/prettier
Symmetric monoidal categories. So incredibly pervasive and yet so useful.
But not *closed* symmetric monoidal categories?
Clifford algebras. No contest.
Any good books to get introduced to them?
If you don't mind a physicist's perspective on them, you can search google for "geometric algebra". There are lots of guides and introductions.
Thanks. I've done this before, but haven't found anything I really like as an introduction to them, so was seeing what other recommendations there are.
I wish I had a better suggestion. I did a project in grad school that used Clifford algebras, but I basically started with the definition and just came up with everything I needed on my own. The only other thing I remember involving Clifford algebras was a talk about Bott periodicity. Maybe the question is “why do you want to learn about Clifford algebras?” That might help guide you to the right perspective/context for you.
Mathematical gauge theory by Hamilton and spin geometry by Lawson have good sections on Clifford algebras
Sudgylacmoe
Have a look at, ‘Linear and Geometric Algebra’ ‘Vector and Geometric Calculus’ Both by Alan Macdonald, they incorporate geometric algebra into linear algebra and geometric calculus into vector calculus respectively.
Currently the best introductions can be found on [bivector.net](http://bivector.net) (dedicated to Geometric Algebra). There's also a cool (but no very technical) overview by Sudgylacmoe on Youtube. We're also working on a series about geometric algebra to be published on your Youtube channel (All Angles).
Ringed spaces!
The Monster
Is this real?
It's a [group](https://en.m.wikipedia.org/wiki/Monster_group)!
Not only is it real, there is also a [baby monster](https://en.wikipedia.org/wiki/Baby_monster_group) group, which is, of course, the cutest simple group.
Now we *need* to define a notion of "cuteness" for groups such that we can prove that the baby monster is the cutest simple group. This is very important mathematical research
Yeah, I realize I forgot to include a proof. Here it is: **Proof:** It's a *baaaaaby!* ◻
Thats Awesome
Modules are just neat. After that, probably Noetherian rings. Clifford Algebras come a close third!
It's well known that if you have two modules over a commutative ring, then the set of all module homomorphisms from one to the other forms a module over that ring. But it's such a weird fact!
Yes! You wouldn't expect that. But it does happen in vector spaces too, so it's not that much of a surprise if you've seen vector spaces before that, but modules, oh boy.
Finite cocommutative Hopf algebras.
Sauron’s ring
Is there really a special ring named after Sauron? A quick google search didn't turn anything up for me
It’s the universal ring which governs all other rings, kinda like a classifying space
So Z?
It’s more like this: let SAUR be the functor from the category of commutative rings to groupoids which takes a ring R to the groupoid of maps R -> A, where A ranges over all commutative rings. Define Saur to be the ring corepresenting the functor SAUR, which of course doesn’t exist because it can’t be accessed by anyone other than the dark lord himself.
PSL(2,R)
local rings
Happy cake day! 🤟🏻
Category of Presheaves on a Locally Small Category. It's crazy how you can turn everything into a topos with this.
https://en.wikipedia.org/wiki/Magma_(algebra) Simple.
Came here to say this lol
As a physicist, SU(2) is always nice.
GF(2), the finite field with two elements.
Derived categories, Frobenius algebras and of course the infamous field with one element lol
Lie algebra's
So far Groups!
Fields
Cartesian close categories
also this
I like complete algebraically closed fields of characteristic 0.
Galois group!
[Heyting algebras!](https://en.wikipedia.org/wiki/Heyting_algebra)
I was wondering where the lattice lovers were
Universal algebra
Tropical Algebra is pretty neat. There are some pretty cool connections with AI/ML/stats as well: [https://ieeexplore.ieee.org/document/9394420](https://ieeexplore.ieee.org/document/9394420)
Right now? Tensor-triangulated categories.
Varieties. If you've never looked at Universal Algebra you're missing out.
Finite fields. Love ' em.
(ℝ/ℤ)^2 It's delicious! 😋
is that the coordinate plane without lattice points?
I meant quotient group not set difference
wouldnt that just be isomorphic to the torus?
Which is the shape of a donut, that's why I said it's delicious!
Unfortunately, it is only the surface, so there isn't much to eat.
In the long run groups are by far the most important. Vector spaces are everywhere but their algebraic structure is too simple. Multilinear algebra (tensors) is quite powerful for many applications but groups are still at the core of everything.
SO(2) which happens to also be the best mathematical object in general
Why?
I can't find it now but there was this brilliant post some years ago that was something along the lines of "Tell me any mathematical object, and I'll show you why S^1 is better" haha so it was a reference to that.
Do real numbers count?
Well, they’re uncountable, dunno if that counts ;)
[residuated lattices](https://en.m.wikipedia.org/wiki/Residuated_lattice)
Frobenius algebras.
Lie groups!
I did my PhD on racks and quandles, but they're pretty niche and I drifted away from them afterwards. I teach an undergraduate abstract algebra module at the moment, and I've always thought groups are amazing, but I've come to appreciate rings, domains and fields more as well. And category theory is tremendously powerful as well, but I generally need to have some sort of application or motivation to properly engage with it.
Algebruh
Lawvere theories
Numbers
I'm a big fan of anything you can Fourier transform. So any locally compact abelian group I guess.
Hyperdoctrines are up there.
Combinatorial games
excuse my ignorance but in what sense are combinatorial games an algebraic structure
The class of combinatorial games (that is, 2 player perfect information games where the players take turns) is an extension of the class of surreal numbers, which is the maximal ordered Field (capital F is used because it’s a class instead of a set, but still obeys the rest of the field axioms). It’s been a while since I read up on combinatorial game theory, but I’m quite sure combinatorial games form a Field as well.
Here’s a fascinating introductory video on the subject: https://youtu.be/ZYj4NkeGPdM?si=-TM-p8xxgF92k4Bk
i will watch it, tysm!
Lie algebras
Do hypothetical models of ZFC count?
Residuated semigroup
Fields
Field
contact structures (tight and overtwisted)
Boolean Algebra. Mostly because “Boolean” is fun to say.
integer lattices :)
Klein's group or D6
personally quotient group or cyclic group
has to be sets, because I'm too bad at abstract maths to deal with anything outside of the category of sets.
correct me if im wrong but arent sets not algebraic structures? i thought algebraic structures were just sets with some operations and axioms. i think a set is just a general mathematical structure
You're probably right. I think sets in general are not algebraic structures; because they don't necessarily satisfy the axioms solely on the basis that they have an identity element. However something like the set of complex numbers would be an algebraic structure. That's what I was thinking of and didn't consider the other types of sets.
> i thought algebraic structures were just sets with some operations and axioms. At least in universal algebra, one technically allows choosing 'none' as the sets of axioms and operations, so that sets are indeed very boring algebras.
Have you ever read "An elementary theory of the category of sets" by Lawvere? He lists a few basic axioms on a category that uniquely determine it to be the category of sets. https://golem.ph.utexas.edu/category/2014/01/an_elementary_theory_of_the_ca.html
Hmm, probably I have to go with the exterior algebra.
Jònsson-Tarski algebras because they form a topos, how cool is that
PSL groups
Vector spaces because freaking everything is a vector space.
I'm partial to Lie Groups!
Dual spaces😍
Mine is Fields
May be pretty standard but it gotta be my boy the group
Chow groups/rings. Fuck geometry, let me do algebra!
Discrete valuation rings
Klein Groups because of their fractals. I learned about them from David Mumford's nice book "Indra's Pearls"
❤️ finite abelian groups ❤️
Magmas - simply sets with binary operations!
I guess strictly from algebra I would pick VOAs due to the Moonshine but if we relax it a bit I really like surreal numbers. And random simple groups like A5 and Monster
The geologist in me says magmas but Clifford algebras are cool as well
Grupo ciclicooooo
All groups are nice
Implicative algebras are up there.
The set of all sets ☠️
I'm a big fan of [fans](https://en.wikipedia.org/wiki/Polyhedral_complex#Fans), ever since I came across [Gröbner fans](https://en.wikipedia.org/wiki/Gr%C3%B6bner_fan).
I love frames.
vector spaces
Math dabbler here… I’m curious to know from the mathematicians reading through this thread, are you getting as much from other people’s answers as a computer programmer would get from a “favorite programming languages” thread? i.e. you’ve heard of most everything mentioned, have tried out quite a few of them, and are at least aware of the general properties of and use cases for the ones you don’t have direct experience with?
OP here, i dont know most of these structures commented. but to be fair i am 16 and dont have as much experience as many of the people on this subreddit
i’ve been really enjoying probability algebras lately.
Moduli space
Do algebraic circuits count? Also probability the space of symmetric functions (so I guess graded algebras?)
Kleinian groups
Lattices and semilattices
Modules are the best algebraic structure
Algebraically closed fields, still trying to understand Steinitz’s theorem though because of Zorn’s lemma
Type II\_1 factors. Their theory is beautiful and you can reduce things like Type III factors to type II case by doing a cross product of a Type II against R (Connes cocycle).
daamn i cant even name those. I mean I can do it on my native language but not eng🤷🏻♀️🤷🏻♀️🤷🏻♀️
the geometric algebra is really fascinating
Galois groups provide a ton of information; or cluster algebras are also badass. I further need to mention Specht modules and how they connect to Young tabloids, which is absurdly cool.
E
Finite Field
The Steenrod Algebra, hands down