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:
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.
Best Answer
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.