Ladder of Algebraic Structures
jwkennington.com
Ladder of Algebraic Structures
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Re: Ladder of Algebraic Structures
#2I 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!
Re: Ladder of Algebraic Structures
#3Likewise 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
Re: Ladder of Algebraic Structures
#4See page 4 of https://leanprover-community.github.io/papers/mathlib-paper.... 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...)
Re: Ladder of Algebraic Structures
#5Robert Geroch's _Mathematical Physics_ is organized around algebraic structures, motivated by category theory.
Re: Ladder of Algebraic Structures
#6Likewise 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
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.
Re: Ladder of Algebraic Structures
#7Out 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:
Re: Ladder of Algebraic Structures
#8This diagram doesn't show semigroup and monoid. Although these structures aren't used in physics much, I find them very useful for understanding groups.
Re: Ladder of Algebraic Structures
#9Slightly related: http://nicolas.thiery.name/Talks/2018-10-08-CategoriesPyData...
(How this is implemented in SageMath.)
Re: Ladder of Algebraic Structures
#10Most 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