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.