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A Brief Guide to a Few Algebraic Structures

argumatronic.com

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Re: A Brief Guide to a Few Algebraic Structures

#11

Post by lonelappde »

nilkn wrote: post #3

> 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).

It's "ring without i". i is of course "identity", which is conventionally denoted e, because i is usually the square root of unity. This works pretty well if you pronounce i as in French so it's a homophonic with English e. Isn't math a wonderful language of use for computer programming? It's so international, breaking the hegemony of English.

Re: A Brief Guide to a Few Algebraic Structures

#12

Post by pvg »

msclrhd wrote: post #4
nilkn wrote: post #3

> 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).

I pronounce it /r@N/ -- i.e. like 'ring' but with the 'a' in comma, or the o in button. Although if it was a Rng formed by a pseudo-random sequence, then R-N-G would be apt ;).

If you want to use them, HN will take most funny squiggles: \ˈrəŋ\. It draws the line at emojis.

Re: A Brief Guide to a Few Algebraic Structures

#14

Post by earthicus »

wsxcde wrote: post #13

Is there a good textbook that covers these? I am primarily interested in a computer science perspective to these algebraic structures.

Many of them feature prominently in Stepanov's books, Elements of Programming and From Mathematics to Generic Programming. 'Elements' is a technical monograph, while the other is a more lightweight read, halfway between semi-pop-science and a textbook.

There's also an old textbook by Birkhoff & Bartee called 'Modern Applied Algebra' that i like. It's not super duper computer science-y but might be worth browsing through the table of contents at least.

Edit: I snapped some photos of the table of contents since you can't see it on the amazon page:

https://imgur.com/a/RFIf4CD

Re: A Brief Guide to a Few Algebraic Structures

#15

Post by gtycomb »

wsxcde wrote: post #13

Is there a good textbook that covers these? I am primarily interested in a computer science perspective to these algebraic structures.

'Topics In Algebra' by I.N.Herstein. Back in college years this was my first book. Despite many other books (which might look easier) I always came back here because of the care it takes to walk through well crafted progression of concepts. (Not a 'computer science' book but you will need something like this for the Algebra part)

Re: A Brief Guide to a Few Algebraic Structures

#16

Post by dllthomas »

mlevental wrote: post #9
nilkn wrote: post #3

> 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).

it's a ring without (i)dentity. makes perfect sense to me. Besides this is a mathematical object not a programming thing - math has its own culture and conventions and pronunciation isn't a cultural value.

At least, not pronunciation that's unambiguous whilst you're not also writing on a blackboard.

Re: A Brief Guide to a Few Algebraic Structures

#18

Post by nandemo »

wsxcde wrote: post #13

Is there a good textbook that covers these? I am primarily interested in a computer science perspective to these algebraic structures.

My favourite abstract algebra textbook is Fraleigh's A First Course in Abstract Algebra. Unlike the article here, which just dumps you a bunch of definitions, the book introduces each structure with proper motivation. It's not a CS approach though.

https://www.amazon.com/First-Course-Abstract-Algebra-7th/dp/... (the 1-star reviews apply to the Kindle version, not the contents. Just get the paperback and you'll be fine)

If you want a CS approach I suggest learning basic Haskell then tackling the fantastic Typeclassopedia. The downside is you'll be missing on structures/theorems that are super useful in math but not that useful in programming.

https://wiki.haskell.org/Typeclassopedia

Re: A Brief Guide to a Few Algebraic Structures

#19

Post by nilkn »

amatecha wrote: post #10
nilkn wrote: post #3

> 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).

Yeh, she calls it out right in the definition: "If we may speak frankly, the rig and rng nomenclatures are abominations. Nevertheless, you may see them sometimes, but we will speak of them no more." :)

Sure, but I also wanted to suggest a simple descriptive name that sidesteps the problem.

Re: A Brief Guide to a Few Algebraic Structures

#20

Post by _hardwaregeek »

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
Found here: https://en.wikipedia.org/wiki/Integral_domain
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