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Ladder of Algebraic Structures

jwkennington.com

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Re: Ladder of Algebraic Structures

#31

Post by QuesnayJr »

kortex wrote: post #30
Aardwolf wrote: post #28

Earlier quoted context omitted.

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 a…

How did they get people to agree to it? Mathematical terminology is a crime, but the problem is that it's very hard to get people to coordinate on different terminology.

Re: Ladder of Algebraic Structures

#32

Post by QuesnayJr »

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.

Commutative and abelian aren't really synonyms. "Abelian" is reserved for objects that have a certain amount of rigidity. Commutative monoids are squishy, while abelian groups very rigid. Another place you'll see the name "abelian" is "abelian Lie algebras", which are also rigid. "Abelian categories" axiomatize the kind of rigidity abelian groups have.

Magmas are usually called "groupoids", but there's another generalization of group also called "groupoids". I'm actually not sure they really deserve a short name, rather than just "set with a binary operation", since there isn't much you can say about them in that generality that you can't generalize to "set with two binary operations", "set with a binary and a trinary operation", etc. The argument for a name is it gives you something to modify, since there are interesting special cases such as "medial groupoids". (An example of a medial groupoid is the real numbers with the "average of two numbers" operation.)

Re: Ladder of Algebraic Structures

#33

Post by QuesnayJr »

7373737373 wrote: post #11
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

I'd like to seen an extension of this table with the negation of these axioms

I think negations usually don't prove interesting, and mathematicians essentially use "non-" to mean "not necessarily". So the theory of "noncommutative rings" includes the theory of "commutative rings" as an easy special case.

Re: Ladder of Algebraic Structures

#34

Post by QuesnayJr »

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.

All of quantum mechanics is non-commutative algebra. The commutative relation [x, p] = i*hbar gives you the Weyl algebra, for example.

Re: Ladder of Algebraic Structures

#35

Post by whytaka »

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

What I would love are examples of how they are useful.

Re: Ladder of Algebraic Structures

#36

Post by vector_spaces »

senderista wrote: post #5

Robert Geroch's _Mathematical Physics_ is organized around algebraic structures, motivated by category theory.

+1 for Geroch's book

It's a really unique book -- was pleasantly surprised with it. It's probably the most lucid introduction to category theory I've read.

Re: Ladder of Algebraic Structures

#37

Post by gugagore »

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.

https://en.wikipedia.org/wiki/Rng_(algebra) is a small example of trying to use more consistent names, but it's too punny for my taste... (Rng is a ring without an identity element)

Re: Ladder of Algebraic Structures

#39

Post by rbonvall »

gugagore wrote: post #37
Aardwolf wrote: post #28

Earlier quoted context omitted.

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.

https://en.wikipedia.org/wiki/Rng_(algebra) is a small example of trying to use more consistent names, but it's too punny for my taste... (Rng is a ring without an identity element)

There's also a rig (a ring without "n"egatives): https://ncatlab.org/nlab/show/rig

I agree. Too cute for its own sake.

Re: Ladder of Algebraic Structures

#40

Post by [deleted] »

7373737373 wrote: post #11
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

I'd like to seen an extension of this table with the negation of these axioms

[deleted]
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