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While I’ve never taken an actual abstract algebra course, there are some things I know about the typical curriculum structure:

  • First, define an algebraic structure.
  • Explain groups.
  • Everything else.

But we seem to skip the most fundamental algebraic structure:

  • The magma

A magma is perhaps the simplest thing you could explain, way simpler than groups:

“A magma is a set equipped with one binary operation which is closed by definition.

That’s all there is to the definition of a magma!

Some well-known magmas are:

  • Integers over addition, subtraction, multiplication
  • Real numbers over addition, subtraction multiplication, division
  • Complex numbers over every arithmetic operation

Why did I never hear about a magma ever before while still being well into groups?

This diagram (source: Wikipedia: Magma (algebra)) can show how they are relevant in the structure of the algebraic structures, magmas to groups:

Magma diagram

Isn’t this a nice visual to explain how all the algebraic structures between magmas and groups are related?

PS: I find the name “magma” kind of interesting; why does it have the same name as molten natural material from which igneous rocks are formed? That makes them even more mysterious.

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Magmas say so very little about the structure of the operation that there is almost nothing useful you can say about a thing just knowing it's a magma. It doesn't properly get interesting until you do more with it, and you can tell that by the examples you've given: they are, sure, magmas, but they are all also commutative rings, which have two binary operations, each with their own structure well beyond what magmas give, and two of them are even fields, so there's still more to say about the properties.

A lot of mathematics is reduction of assumptions: you delete rules and see what interesting facts the remaining rules give you. The magma has had so many rules deleted that there's hardly anything left to be interesting, which is why it is not usually used on its own.

My own abstract algebra course mentioned magmas but only in passing: it's a thing with a binary operation, that's what it's called when we know nothing at all about the operation.

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My first Abstract Algebra course I embarked upon back in the day (far too young to appreciate it) started with semigroups, and investigated a few important ones e.g. the "left semigroup" and the "right semigroup" based on a reading of Whitelaw,'s 1978 An Introduction to Abstract Algebra, a kindheartedly slow and detailed approach to everything basic up to rings and fields.

It is sort of assumed that, unless you are going to do some fairly abstruse work at the sharp end, algebraic structures that are not associative come with their own baggage, so naturally you start any abstract algebra course with associativity.

Seth Warner sort of takes this approach in his Modern Algebra (1964), still not really surpassed in the thoroughness of its approach (although feel free to challenge that, of course). He does set one or two exercises exploring some of the properties of some of the more truly abstract constructs, and they are well worth exploring, by and large. I confess I only ever got a quarter of the way into it, so I am not familiar with everything in it.

Oh, and it's actually easier to start with groups and work down, because it is so easy to apply them to number sets. Does you no harm to learn about a group in your teens (that's what my fiercely experimental approach did for me back in the 70's) so that when you encounter abst alg at undergrad level you're more ready for it.

Do they teach group theory or even more than the basic level set theory in teenage education nowadays? I suspect they probably don't. Please correct me if I am wrong.

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    $\begingroup$ The UK Further Maths A-level syllabus can include a very light introduction to group theory. See eg pp 77-84 of this pdf ocr.org.uk/Images/… of an optional paper at one exam board's FM A-level $\endgroup$
    – AakashM
    Feb 10 at 15:09
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    $\begingroup$ This answer is better than mine. $\endgroup$ Feb 10 at 16:04
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    $\begingroup$ In France, the first explicit mention of groups (and of abstract algebra) is during bachelor, after the baccalauréat $\endgroup$
    – Didier
    Feb 10 at 21:54
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    $\begingroup$ @DanUznanski I thought they were both good in different ways, and that this was one of the times math SE lived up to its potential. $\endgroup$
    – MJD
    Feb 11 at 14:21
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The single most important of the structures in that diagram, by a long mile, is the group; monoid is a distant second. The absolute least important is the magma, which is a non-structure. Indeed, probably the most important function of the magma and the unital magma is to complete that diagram. When (if) you start to see a lot of those group-like structures, it can be helpful to conceptualize them as this way to understand the relationships between them.

That said, most people that learn algebra don't eventually go on to use many other group-like structures, but they almost certainly will encounter one of groups, rings, fields, modules, vector spaces, algebras, etc., i.e. algebraic structures which are a group and then also something else. So, groups are a natural place to start, at least on your first brush with algebra.

I liken it to studying calculus the first time, you don't really construct the real numbers from nothing, you just take them as given. Then, when you revisit the subject (as advanced calculus or analysis), you're a bit more careful to build up to the real numbers. Mathematically, it's not really necessary to do so; real analysis is the same whether the real numbers are constructed or given. Pedagogically, however, it makes sense to go deeper into that foundation when coming back for a more careful pass of the subject, to get a better feel for what exactly that primary structure of study is.

The same goes for algebra: when it's time to dig deep into the variety of group-like structures, then it makes sense to start from magmas and construct that whole diagram.

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In addition to other good answers: I think it is not actually mathematically clarifying to try to classify mathematical ideas by "their axioms", "ordering" them by whether they have more or fewer axioms, etc.

Although tastes differ, by this year, my own perception is that "axioms" are generally an attempt to most-efficiently/minimally characterize things we've already seen... to reason more clearly about them.

E.g., if we had never seen symmetry groups, groups of permutations, etc., why would we have any motivation to "define groups"? Or, an on-going pseudo-debate: "do rings have units, or not?!" (In my opinion, many do, but not all. One has to specify, in context. Don't call rings-without-units "rngs"... just toooo cute. :)

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Bourbaki introduce Magma in their books.

We sometimes call such a structure a "groupoid", but the term groupoid is also used with a different meaning, especially in geometry. Maybe this was the reason why the term "magma" was invented, in order to keep the term "grupoid" for geometry.

Why didn't you learn about Magmas? Maybe this was simply a didactical decision. Would it be important to provide every possible structure with a separate name? The style of how mathematics is done depends on the mathematicians, and in this case, the historical development (e.g. preferring category theory over universal algebra) was different than assumed by Bourbaki or other authors.

Teaching something always requires choices to be made.

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  • $\begingroup$ You write "groupoid" and "grupoid". Is it different in both areas? Or is it just a typo? $\endgroup$ Feb 11 at 17:59
  • $\begingroup$ J.A. Green, in his Sets and Groups (1965) briefly discusses "gruppoids" (which is terminology borrowed I believe from the German "gruppe") which are magmas which are specifically closed. The most basic magma may not even be closed. $\endgroup$ Feb 12 at 7:10

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