Ladder of Algebraic Structures(jwkennington.com) |
Ladder of Algebraic Structures(jwkennington.com) |
This is aimed primarily at mathematical physics, and is intended as a quick reference -- it's obviously incomplete and isn't a substitute for Hungerford, Lang, or [insert favorite algebra book].
I hope you find it as helpful as I did in making it!
Semigroupoid
Small Category
Groupoid
Magma
Quasigroup
Unital Magma
Loop
Semigroup
Inverse Semigroup
Monoid
Commutative monoid
Group
Abelian group
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.
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.
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.)
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".
The resulting grid can be factored around set, bag, list and binary tree, with empty/non-empty variants.
Then there is interaction of the structures with binary operators on the data elements themselves, giving a nice analysis of map, filter, fold (reduce) in functional programming.
A.Bunkenburg, The Boom Hierarchy
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.49....
There is a rich literature if you chase the references back and forth, starting from Bird-Meertens Formalism (Squiggol), Hoogendijk, through Backhouse and Malcolm, to Meijer and McBride.
For example, I can see the use of commutativity (ab = ba) and anticommutativity (ab = -ba), but I'm not sure what I'd do with the negation of commutativity (ab ≠ ba).
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.
[0] https://www.infoq.com/presentations/abstract-algebra-analyti...
(How this is implemented in SageMath.)
commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields
- 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...
Also, the label "algebra" is vague here, and refers to an "algebra over a field", but sometimes it refers to an "algebra over a ring".
Is "the theory of everything'' merely the ultimate ensemble theory?
https://arxiv.org/abs/gr-qc/9704009
and a sketchy one for physics in his paper:
The Mathematical Universe
It's a really unique book -- was pleasantly surprised with it. It's probably the most lucid introduction to category theory I've read.
Do physicists have any use for non-commutative algebra? It already seems pretty niche in mathematics.
1. Isn't this more like a tree, where only one path is shown?
2. Is it possible to find a pattern and extend the ladder in the most logical way?
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 example, the only homomorphism ℤ/2ℤ → ℤ is the trivial one.
I would expect rather the datum of an R-algebra structure on a ring A to be a ring homomorphism R → End_{gp}(A). EDIT: Now that I think of it, maybe got your A and R mixed up and meant the more restrictive definition, whereby the ring homomorphism I mention is supposed to factor through R → Z(A) → End_{gp}(A)? I'd call this more restricted notion, at least over a unital ring R, a unital algebra A (but often people want implicitly to assume unital-ness).
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).
First, regarding the OP: Having spent many years studying algebra, I don't find the hierarchy of axioms to be very useful in thinking about these things. Sure, you can think of a field as a "commutative ring with inverses", but rings and fields present themselves so differently that this connection doesn't end up being all that useful. Fields are not rich enough on their own to support much interest. You'll find them mostly as building blocks rather than powerful tools in and of themselves. Ditto for modules and vector spaces. Sure, a module is "like a vector space but over a ring", but vector spaces are so boring by themselves that they show up mostly as scaffolding. The study of modules, on the other hand, is its own branch of mathematics. It's much more useful to think of them in terms of what you actually do with them.
Now, on to definitions. The following few paragraphs are all very small-minded and look far more complicated than they actually are. It all encodes pretty much what you'd expect.
If you want to define algebras over commutative rings, you need to start with left- and right-algebras. A left-algebra is an abelian group A equipped with a map \phi: R -> End(A). The abelian group structure defines the addition in the algebra, and the map defines the left-multiplication: if r \in R, and a \in A, then you define a times r as \phi(r)(a), where \phi(r) is an endomorphism on A.
A right-algebra is the same, only the map is from R to the opposite ring of End(A), where the opposite ring is the one you get by just reversing the multiplication. You need to do this because associativity demands that you compute ((a)r)s, where a \in A, r,s \in R, by first acting on a with r, then by s. But with the usual conventions of composition of functions, \phi(r) \circ \phi(s) means you first "do" s, then r. So you need to flip it. Working with left- and right-algebras is a pain in the butt because you have to carry around a ton of left-right nonsense.
A bialgebra (in the literature I read) is a an abelian group that is both a left- and right-algebra. A central bialgebra is one where the left and right multiplication are the same, which is not a given. Noncentral bialgebras are especially annoying, mostly because you have to figure out how to do pre-subscripts in LaTeX so you can write nonsense like "_R M_S".
Obviously, all of these things collapse if R is commutative. Noncommutative ring theory requires a special kind of patience. And don't even get me started on noncommutative geometry.
I agree. Too cute for its own sake.
1L * A = A = A * 1R