Ladder of Algebraic Structures

193 点作者 JWKennington大约 5 年前

19 条评论

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大约 5 年前

There is also the abstract algebra cheatsheet [1]. Not my work, I have just bookmarked it a couple of years ago.[1] <a href="https://github.com/mavam/abstract-algebra-cheatsheet" rel="nofollow">https://github.com/mavam/abstract-algebra-cheatsheet</a>

fyp大约 5 年前

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:SemigroupoidSmall CategoryGroupoidMagmaQuasigroupUnital MagmaLoopSemigroupInverse SemigroupMonoidCommutative monoidGroupAbelian group<a href="https://en.wikipedia.org/wiki/Abelian_group" rel="nofollow">https://en.wikipedia.org/wiki/Abelian_group</a>

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Koshkin大约 5 年前

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大约 5 年前

Slightly related: <a href="http://nicolas.thiery.name/Talks/2018-10-08-CategoriesPyData.pdf" rel="nofollow">http://nicolas.thiery.name/Talks/2018-10-08-CategoriesPyData...</a>(How this is implemented in SageMath.)

_hardwaregeek大约 5 年前

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

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heinrichhartman大约 5 年前

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: <a href="https://en.wikipedia.org/wiki/Gabriel%E2%80%93Popescu_theorem" rel="nofollow">https://en.wikipedia.org/wiki/Gabriel%E2%80%93Popescu_theore...</a>

_ouml_大约 5 年前

See page 4 of <a href="https://leanprover-community.github.io/papers/mathlib-paper.pdf" rel="nofollow">https://leanprover-community.github.io/papers/mathlib-paper....</a> 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大约 5 年前

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大约 5 年前

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大约 5 年前

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:<a href="https://en.wikipedia.org/wiki/Field_with_one_element" rel="nofollow">https://en.wikipedia.org/wiki/Field_with_one_element</a>

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tom_mellior大约 5 年前

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大约 5 年前

Max Tegmark has a larger diagram for math in his paper:Is "the theory of everything'' merely the ultimate ensemble theory?<a href="https://arxiv.org/abs/gr-qc/9704009" rel="nofollow">https://arxiv.org/abs/gr-qc/9704009</a>and a sketchy one for physics in his paper:The Mathematical Universe<a href="https://arxiv.org/abs/0704.0646" rel="nofollow">https://arxiv.org/abs/0704.0646</a>

senderista大约 5 年前

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

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twic大约 5 年前

Another attempt at that diagram, with more structures but less detail on how they differ:<a href="http://us.metamath.org/mpegif/mmtopstr.html" rel="nofollow">http://us.metamath.org/mpegif/mmtopstr.html</a>

billfruit大约 5 年前

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

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Sharlin大约 5 年前

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.

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killjoywashere大约 5 年前

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

amelius大约 5 年前

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?