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Ladder of Algebraic Structures
- JWKennington 6y agoI 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!
- h91wka 6y agoThis diagram doesn't show semigroup and monoid. Although these structures aren't used in physics much, I find them very useful for understanding groups.
- _hardwaregeek 6y agoLikewise 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 6y agoWas 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.
- _ouml_ 6y agoSee page 4 of https://leanprover-community.github.io/papers/mathlib-paper.pdf 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...)
- senderista 6y agoRobert Geroch's _Mathematical Physics_ is organized around algebraic structures, motivated by category theory.
- vector_spaces 6y ago+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.
- foxes 6y agoOut 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 6y ago> 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 6y agoI 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 6y agoBut 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.
- fermigier 6y agoSlightly related: http://nicolas.thiery.name/Talks/2018-10-08-CategoriesPyData.pdf http://nicolas.thiery.name/Talks/2018-10-08-CategoriesPyData... (How this is implemented in SageMath.)
- fyp 6y agoMost 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 6y agoI'd like to seen an extension of this table with the negation of these axioms
- hope-striker 6y agoWhat 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 6y agoNope: 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 6y agoAh, whoops. That seems even more useless, though.
- mikhailfranco 6y agoNon-commutative also means that if there are units, there may be different left and right units: 1L * A = A = A * 1R
- QuesnayJr 6y ago
- hope-striker 6y agoSmall 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".
- billfruit 6y agoHow does geometric spaces like affine, projective etc come into this taxonomy.
- RossBencina 6y agoA 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
- tom_mellior 6y agoSomewhat 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.
- davnn 6y agoThere 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
- twic 6y agoAnother 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
- Sharlin 6y agoWhy 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 6y agoProbably 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.
- deleted 6y ago[deleted]
- heinrichhartman 6y agoCategories 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_theorem https://en.wikipedia.org/wiki/Gabriel%E2%80%93Popescu_theore...
- amelius 6y agoQuestions: 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?
- Koshkin 6y agoRegardless 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).
- mikhailfranco 6y agoMax 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
- killjoywashere 6y agoNot often that something on HN causes me to hit print, but that figure is worth printing and tucking into a book.