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.
Regardless of the "ladder" (or any other attempts to organize algebraic structures), what I find interesting (and somewhat unexpected) is that each particular structure exhibits so many features exclusive to it and such a rich behavior that is not found in any other structures - even closely related ones (like, for example, commutative vs. non-commutative rings) - that these attempts of organizing them and of some kind generalization seem to have not much value. It is only category theory that has managed to bring in something of a common viewpoint on many mathematical constructs (and not just those in algebra).
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.
For me the first example where I really got why algebraic structures was useful this video on using abstract algebra in analytics[0].
This helped me grasp something that I had read from Alexander Stepanov[1] that I hadn't fully understood before (not being familiar with the algebraic terminology):
> I suddenly realized that the ability to add numbers in parallel depends on the fact that addition is associative...In other words, I realized that a parallel reduction algorithm is associated with a semigroup structure type. That is the fundamental point: algorithms are defined on algebraic structures.
I think the use case of building infrastructure for parallel/distributed computation as described above is a nice, concrete example of why using abstract algebra in our programs can be useful. It certainly isn't the only use case though. Other things include managing complex control flow, or passing an implicit context through a computational pipeline.
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…
Why not call the DNA to RNA enzyme reverse transcriptase and the RNA to DNA one transcriptase?
> 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 the definition doesn't work even for commutative rings; as I mention, it says that the only ℤ-module structure on ℤ/2ℤ is the trivial one, which is not the usual understanding of the term. I agree that, if you switch A and R in Hom(A, Z(R)), then an element of the Hom space Hom_{ring}(R, Z(A)) makes A into an R-algebra, but I would argue it's not the only way; there's a map Hom_{ring}(R, Z(A)) -> Hom_{ring}(R, End_{gp}(A)), but it need not be surjective if the rings aren't assumed unital. Consider, for example, a polynomial ring R = k[t] and its ideal A = tR, which has a natural structure of an R-algebra.
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…
We suffer here from lack of classical education. Greek and Latin would probably help.
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.
This is a really cool presentation where the authors "step up the ladder" to design a really elegant API for animations as semirings (where * is used to sequence animations, and + for animations running in parallel), and then go on to implement it in Swift: https://bkase.github.io/slides/algebra-driven-design/
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".
If someone came up to me and said, "there are integer fingers in one hand" I would be the opposite of confused.
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".
If someone came up to me and said, "there are integer fingers in one hand" I would be the opposite of confused.
It's the kind of "technically correct" that is literally useless.