Interesting seeing this defined from a functional programming/category theory perspective. Personally I like examples for my algebraic structures along with some simple proofs why they're that structure. What was extremely useful for my Algebra class was a Wikipedia note detailing the hierarchy of rings via set inclusion:
commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields
> Rng
I really think names without a fairly obvious way of saying them out-loud should just be avoided/deprecated. I can't pronounce this as "ring" (because then you'd think I was referring to a ring, not a rng). Do I just spell it out? R-N-G? I'd rather just call it a non-unital ring. That involves more letters but is much more descriptive (to a mathematician, anyway).
That is beautifully written and a nice recap on those things having studied maths a long long time ago. The Heything Algebra is quite interesting. I learned about that more recently on some lectures about logic and category theory online (I forgot the location now).
I really like this, but one change I'd like to see: given how many structures are defined in terms of simpler structures (which are defined in terms of yet simpler ones), it would be nice if a structure listed all of the laws, not just the ones that simpler structures don't. That way, you would have to jump around the page to get a complete picture of a single structure. Maybe have them be a different color, so you can see what is new vs. what is inherited?
Oh wow, this looks great. I had started working on my own FP glossary, but this has a ton of information and is clear enough that I'm sure many will find it very very helpful.
I'm learning category theory to understand haskell better.
I'm finding category theory to be almost a theory about program structure in the context of composition. It's giving me a whole new perspective on one of the least concrete things about programming namely design.
How relevant is abstract algebra to programming? Will it change my perspective on everything related to programming? How much of a mind bender is it compared to category theory?
I was wondering about something. If I have the sum x+x+x...+x (n times), then that is the same as x * n. If I have the product x * x * x ... * x (n times) then that is the same as x ^ n.
What is it called when this is generalized? E.g. call + op1, call * op2, call ^ op3. What would op0 be? And what would op0.5 be?
How does the unit element for these operations behave?
And the rules for associativity, commutativity, for increasing order of the operation?
Is there a good textbook that covers these? I am primarily interested in a computer science perspective to these algebraic structures.
A lambda-cube like visualization with the various properties (associativity etc.) as the axes could be helpful.
The Wikipedia page on Algebraic structures is rather more detail than I usually want, but the page "Outline of algebraic structures"[1] is a great quick reference.
[1] https://en.wikipedia.org/wiki/Outline_of_algebraic_structure...