
Ladder of Algebraic Structures - JWKennington
https://jwkennington.com/blog/algebra-ladder/
======
JWKennington
I first encountered a diagram of algebraic structures at the end of
Jeevanjee's second chapter, "Vector Spaces", which elegantly summarizes the
high-level differences in structure between sets, vector spaces, and inner
product spaces. I've attempted to augment this map along two dimensions: a
structure dimension that aims to measure the number of attributes an algebraic
object has, and a specificity dimension that measures the number of
constraints placed on each attribute.

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!

------
davnn
There is also the abstract algebra cheatsheet [1]. Not my work, I have just
bookmarked it a couple of years ago.

[1] [https://github.com/mavam/abstract-algebra-
cheatsheet](https://github.com/mavam/abstract-algebra-cheatsheet)

------
fyp
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](https://en.wikipedia.org/wiki/Abelian_group)

~~~
7373737373
I'd like to seen an extension of this table with the negation of these axioms

~~~
hope-striker
What would you do with that?

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).

~~~
pfortuny
Nope: the negation is "there is a couple a,b such that ab!=ba", which means
just "strictly not commutative group": I do not think there is a relevant
theory to be done about them (otherwise, I guess it would have been done).

~~~
hope-striker
Ah, whoops. That seems even more useless, though.

------
Koshkin
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).

------
fermigier
Slightly related:
[http://nicolas.thiery.name/Talks/2018-10-08-CategoriesPyData...](http://nicolas.thiery.name/Talks/2018-10-08-CategoriesPyData.pdf)

(How this is implemented in SageMath.)

------
_hardwaregeek
Likewise this is a pretty useful chain of inclusions:

commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains
⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃
fields ⊃ finite fields

~~~
vortico
Was just about to post that. You can get very fine-grained from sets to any
mathematical structure based on sets, where the inclusion chain is a spectrum
of structureless to structured.

------
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...](https://en.wikipedia.org/wiki/Gabriel%E2%80%93Popescu_theorem)

------
_ouml_
See page 4 of [https://leanprover-community.github.io/papers/mathlib-
paper....](https://leanprover-community.github.io/papers/mathlib-paper.pdf)
for a part of the hierarchy of algebraic structures in the Lean theorem
prover. (If you give it a normed field, it will use this hierarchy to
automatically deduce that it is also a ring or a topological space, etc...)

------
hope-striker
Small note: people often (but don't always) assume that a ring is unital (has
an identity element), and that an algebra over a field is unital and
associative.

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".

------
h91wka
This diagram doesn't show semigroup and monoid. Although these structures
aren't used in physics much, I find them very useful for understanding groups.

------
foxes
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](https://en.wikipedia.org/wiki/Field_with_one_element)

~~~
JadeNB
> 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 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).

~~~
joppy
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).

~~~
JadeNB
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.

------
tom_mellior
Somewhat ironic that lattices are missing from this lattice of algebraic
structures :-) Though I guess they might not be as important in mathematical
physics as in some other areas.

------
mikhailfranco
Max Tegmark has a larger diagram for math in his paper:

 _Is "the theory of everything'' merely the ultimate ensemble theory?_

[https://arxiv.org/abs/gr-qc/9704009](https://arxiv.org/abs/gr-qc/9704009)

and a sketchy one for physics in his paper:

 _The Mathematical Universe_

[https://arxiv.org/abs/0704.0646](https://arxiv.org/abs/0704.0646)

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

~~~
vector_spaces
+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.

------
twic
Another attempt at that diagram, with more structures but less detail on how
they differ:

[http://us.metamath.org/mpegif/mmtopstr.html](http://us.metamath.org/mpegif/mmtopstr.html)

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

~~~
RossBencina
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](https://en.wikipedia.org/wiki/Projective_space#Definition)

------
Sharlin
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.

~~~
ivanbakel
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.

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

------
killjoywashere
Not often that something on HN causes me to hit print, but that figure is
worth printing and tucking into a book.

------
amelius
Questions:

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?

