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I have never taken a formal course in Abstract Algebra (yet), but I am interested in learning more about the subject, as I know it is extremely important in Modern Mathematics and a powerful tool beyond (like in, say, Physics).

However, in my limited exposure to the subject, I have found the taxonomy of the various Algebraic Structures very difficult to follow. For example, in basic Group Theory, there are groups, but also semigroups, monoids, semilattices, quasigroups, loops, and Abelian groups. I have no difficulty in understanding the definitions of each, but understanding (or maybe more accurately, remembering) the interrelationships between them is confusing and difficult. I suppose this is largely because many of the names of these structures (like many names in math, unfortunately) are very non-descriptive.

I have seen some authors create a flowchart-like graphic to help illustrate these relationships (e.g. a monoid is a semigroup with identity, or equivalently a group without inverses, etc.), and while helpful, they are always rather limited in scope and always deal with a linear progression of the structure hierarchy and do not include structures that "branch out" so to speak from the main hierarchy (e.g. quasigroups and semilattices which are special cases of magmas and semigroups respectively, but have no direct inclusive relationship to, say, monoids)

So my question is: Are there any robust graphics illustrating the interrelationships between the various algebraic structures?

Personally, I'm mainly interested in such graphics for Groups and Rings (especially Rings), but I leave this question open to answers for more advanced structures too (like modules).

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    $\begingroup$ It isn't actually very important to get a grip on every permutation of axioms there are (although I understand the urge to want to categorize them). For a person learning abstract algebra, it would seem the most logical thing to do (the thing done in almost every course on algebra) is to learn about groups, then about fields and rings. After that, one can start to specialize in their structures of interest (say, down through semigroups and into loops, or down from rings into semirings or nearrings.) $\endgroup$
    – rschwieb
    Jul 17, 2017 at 20:31
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    $\begingroup$ IMO it's a particular form of self-torture to think too much about all the permutations at once, like in this book. Yes, individually these specialized general structures can have nice applications, and they are interesting to work in. But I think it would be bad advice for a student to drive around in that wilderness before learning about groups and fields. $\endgroup$
    – rschwieb
    Jul 17, 2017 at 20:34
  • $\begingroup$ So in summary let me applaud your efforts thinking about it, and say the charts are interesting to look at. I don't think you wasted your time, but you might be on the verge of wasting your time :) $\endgroup$
    – rschwieb
    Jul 17, 2017 at 20:35
  • $\begingroup$ @rschwieb Thanks for your feedback! Honestly, I'm not too concerned about understanding the taxonomy to the nth degree. At least, not for getting started in the subject. The question was mostly motivated by my curiosity and not by my anxiety. But one serious reason I wanted to understand it was because, for example, the definition of a vector space contains many axioms, but its definition can be significantly shortened and simplified by employing the concepts of groups, rings, and fields. And describing things in terms of simpler things is one of the most powerful things math can do. $\endgroup$
    – WB-man
    Jul 17, 2017 at 21:22
  • $\begingroup$ Matthias Vallentin's Abstract Algebra Cheatsheet is a nice graphical summary of a few important structures. $\endgroup$
    – Mars
    May 2, 2021 at 20:41

2 Answers 2

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Here is my own attempt at organizing the interrelationships for Groups and Rings. But as I said, I am an amateur in the subject at best, and so this may be incorrect in some places. If so, I would be really grateful to any corrections anyone can give me.

The way I've chosen to organize the structures is as a downwardly flowing web, whose nodes are the various algebraic structures, and whose directional edges denote the axioms that need to be added to the upstream structure in order to produce the downstream structure. This is done in a symmetric way, so that the axiom together with its upstream structure is both necessary and sufficient for producing the next downstream structure.

Groups

Taxonomy of Group-like Structures

Rings

Note: I know there are different methods to define Rings (e.g. whether it includes a multiplicative identity or not). In the following graphic I adopt the convention where a ring does contain $1$, and those that don't are called "rngs".

Also, because some of the ring-like structures (e.g. near-rings) branch out early in the hierarchy before reaching an "officially" named structure like semirings, I had to invent some of my own names for various primitive ringoids to provide the nodes for the branching. Such homemade names I have enclosed in "quotes" to distinguish them from the more standard names.

Taxonomy of Ring-like Structures

These graphics were generated using draw.io

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  • $\begingroup$ That's truly comprehensive! Thank you a lot. As for a newbie in pure mathematics that would be certainly useful if not mandatory. Are there any studies (I mean publications) that explicitly devoted to categorizing different structures used all over the place? I know universal algebra is what this study all about. But still I'm curious if there are articles with exactly this formulation of problem, namely, categorizing existing/explored algebraic structures. $\endgroup$
    – Timur Fayzrakhmanov
    Sep 29, 2019 at 13:28
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    $\begingroup$ @TimurFayzrakhmanov That I don't know. That's a good question, though. You might try posting it yourself as an official question on the site. As for me, I had never seen any references (whether professional or casual) that dealt with broadly categorizing and interrelating these structures. Hence why I felt compelled to try to do it myself. $\endgroup$
    – WB-man
    Sep 30, 2019 at 18:25
  • $\begingroup$ Is it right to say, that in order to get (or jump in) to some structure downwards we don't need to traverse through ALL the arrows and accumulate ALL the axioms that come along. That is to say a Set is an Abelian Group, it is enough to show the Set has a binary operation that is commutative. Am I right? I'm asking just because recently I've faced with the definition that breaks your graph.. $\endgroup$
    – Timur Fayzrakhmanov
    Oct 2, 2019 at 15:20
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    $\begingroup$ @TimurFayzrakhmanov No, to reach abelian group from set, you really do need to accumulate all the axioms in between. A set with a commutative binary operation that is NOT associative (for example) would not be an abelian group. That said, there is one "skip" in the Ring diagram where we skip from a division ring to a finite field merely by adding the finiteness axiom. But that's okay, because it turns out a finite division ring automatically acquires commutativity of multiplication. $\endgroup$
    – WB-man
    Oct 3, 2019 at 17:47
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When I started with Ring Theory, I saw a graphic in some script (which I don't remember yet) that helped me preparing for my exam on a course titled "Introduction to Algebra" which was basically Group and Ring Theory. I extended this graphic by adding some more properties (concerning $R[X]$). I have this graphic still hand-written so I converted it (the program I used - DrawExpress - could sadly not manage Latex-Expressions). Maybe it helps you to structure the interrelationships between the properties of rings. Ring Interrelationships

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