Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Ladder of Algebraic Structures

jwkennington.com

Post reply on HN
Print view View original

21–30 of 59 posts

Re: Ladder of Algebraic Structures

#21

Post by moron4hire »

fyp wrote: post #10

Most algebraic structures are best understood by which axioms it satisfies. For example basically every subset of axioms of an abelian group is useful enough to have a name. Wiki has a really nice table: Semigroupoid Small Category Groupoid Magma Quasigroup Unital Magma Loop Semigroup Inverse Semigroup Monoid Commutative monoid Group Abelian group https://en.wikipedia.org/wiki/Abelian_group

Aside: "wiki" is a term referring to a general class of software. The name of the crowd-sourced, free encyclopedia is "Wikipedia", as it is built with wiki software.

In other words, Wikipedia is a member of the set of wikis. You wouldn't call "5" just "integer", e.g. it would be confusing to say "there are integer fingers on one hand".

Re: Ladder of Algebraic Structures

#22

Post by joppy »

JadeNB wrote: post #19
foxes wrote: post #7

Out of clarity this is an "algebra over a field" vs a more general concept of an algebra over a ring. More generally an algebra A, over a ring R, an R-algebra , is a ring A equipped with a map Hom(A,Z(R)). Algebra over a field is a special case. Here's a "fun" object for you to consider: https://en.wikipedia.org/wiki/Field_with_one_element

> More generally an algebra A, over a ring R, an R-algebra, is a ring A equipped with a map Hom(A,Z(R)). I don't think that's the usual definition of an algebra. For example, it would mean that there is no difference between an algebra over a non-commutative ring and over its centre, which seems weird; and it clashes with the usual habit to regard every non-0 commutative ring as a non-trivial ℤ-module, whereas, for e…

I think that usually when people say “algebra over a ring” they assume that ring to be commutative, so that the word “bilinear” in “bilinear multiplication” is useful. It’s possible to define an algebra over a non-commutative ring as a bimodule (rather than left module or right module) equipped with a bilinear multiplication, but I have rarely seen this used.

The definition the parent poster used (or intended to use, but wrote the wrong way around, I believe) was that an algebra over a non-commutative ring is just an algebra over its commutative centre. (In which case, we’re still really just talking about algebras over commutative rings).

Re: Ladder of Algebraic Structures

#23

Post by RossBencina »

billfruit wrote: post #14

How does geometric spaces like affine, projective etc come into this taxonomy.

A projective space is defined as a quotient of a vector space under the equivalence relation x ~ y (exists k =/= 0 such that x = ky).

https://en.wikipedia.org/wiki/Projective_space#Definition

Re: Ladder of Algebraic Structures

#24

Post by ivanbakel »

Sharlin wrote: post #20

Why is "commutative +" a step up rather than a step to the right? I guess there should be Abelian groups and commutative rings somewhere between groups and modules.

Probably because the diagram originated in a Vector Spaces book, and commutativity is viewed more as a valuable property than a structural constraint.

Do physicists have any use for non-commutative algebra? It already seems pretty niche in mathematics.

Re: Ladder of Algebraic Structures

#25

Post by sergius »

ivanbakel wrote: post #24
Sharlin wrote: post #20

Why is "commutative +" a step up rather than a step to the right? I guess there should be Abelian groups and commutative rings somewhere between groups and modules.

Probably because the diagram originated in a Vector Spaces book, and commutativity is viewed more as a valuable property than a structural constraint. Do physicists have any use for non-commutative algebra? It already seems pretty niche in mathematics.

Rotation group maybe? And those gauge symmetries... heck, just look into quantum mechanics operators.

Re: Ladder of Algebraic Structures

#28

Post by Aardwolf »

fyp wrote: post #10

Most algebraic structures are best understood by which axioms it satisfies. For example basically every subset of axioms of an abelian group is useful enough to have a name. Wiki has a really nice table: Semigroupoid Small Category Groupoid Magma Quasigroup Unital Magma Loop Semigroup Inverse Semigroup Monoid Commutative monoid Group Abelian group https://en.wikipedia.org/wiki/Abelian_group

This is one of those things were better naming would make mathematics easier, imho. The words are just so random and inconsistent.

Example: Commutative and Abelian are synonyms, but there's "Commutative monoid" and "Abelian group". Why not use same adjective. But of course also the random bag of words that have nothing to do with the concept, like magma.

Re: Ladder of Algebraic Structures

#29

Post by heinrichhartman »

Categories are Algebraic structures, that are related to this hierarchy:

- Monoids are Categories with a single object.

- Algebras (Non-commutative, Associative) are k-linear Categories, with a single object.

- Any object X in a k-linear category comes with an algebra: R = End(X) = Hom(X,X).

- Any other objects comes with an R-module: Hom(R, X)

- In some cases, we can use this to describe the category as a category of R modules: https://en.wikipedia.org/wiki/Gabriel%E2%80%93Popescu_theore...

Re: Ladder of Algebraic Structures

#30

Post by kortex »

Aardwolf wrote: post #28
fyp wrote: post #10

Most algebraic structures are best understood by which axioms it satisfies. For example basically every subset of axioms of an abelian group is useful enough to have a name. Wiki has a really nice table: Semigroupoid Small Category Groupoid Magma Quasigroup Unital Magma Loop Semigroup Inverse Semigroup Monoid Commutative monoid Group Abelian group https://en.wikipedia.org/wiki/Abelian_group

This is one of those things were better naming would make mathematics easier, imho. The words are just so random and inconsistent. Example: Commutative and Abelian are synonyms, but there's "Commutative monoid" and "Abelian group". Why not use same adjective. But of course also the random bag of words that have nothing to do with the concept, like magma.

Could take a page out of the biologist's book. "what's this thing?" Transcriptase - enzyme (-ase) which transcribes - DNA to RNA. "What about this" Reverse transcriptase - does the reverse of transcriptase.

Angiotensin-converting enzyme - does exactly what it says on the tin. You can lex it even further:

- Angio - heart (from ango, vessel)

- Tens - from hypertension, vis tendo, tendere, to stretch.

- (-in) - suffix associated with polypeptides:

- Convertere - turn around, from:

- Con - with

- Vert - turn

- En - inside

- Zyme - from zume/zymē - leavened, loosely, biological thing which causes leavening

It just makes so much sense! Lexemes are so cool. Like digging into linguistic source code.

Post reply on HN