Understanding Groupoids: Axioms & Difference from Groups

[AdrianZ]

What is a groupoid? I'm reading a book about gyrovectors and in the first chapter it starts defining something which it calls it a groupoid but doesn't inform the reader about its axioms. so I felt uncomfortable to face a new mathematical definition without knowing the axioms that it should satisfy.
Here is what the book says:

Definition 2.1 (Binary Operations, Groupoids, and Groupoid
Automorphisms). A binary operation + in a set S is a function + :
S x S -> S. We use the notation a + b to denote +(a, b) for any a, b in S.
A groupoid (S, +) is a nonempty set, S, with a binary operation, +. An
automorphism phi of a groupoid (S, +) is a bijective (that is, one-to-one)
self-map of S, phi : S -> S, which preserves its groupoid operation, that is,
phi(a + b) = phi(a) + phi(b) for all a, b in S.

What axioms should a groupoid satisfy? and what's the difference between a groupoid and group? because the name apparently has been derived from the word group.

 

[micromass]

Hi AdrianZ!

A groupoid doesn't have any axioms. It's just any set with any binary operation.
So $(\mathbb{N},+)$, $(\mathbb{N},\cdot)$, $(\mathbb{Z},\cdot)$, etc are all groupoids.

The only requirement is that the operation must really be an operation. For example $(\mathbb{R},/)$ isn't a groupoid as 1/0 doesn't exist.

A more conventional name for a groupoid is a magma: http://en.wikipedia.org/wiki/Magma_(algebra)
The name magma is preferred, because groupoid often has another meaning in category theory and Lie algebras.

 

[AdrianZ]

So you mean the binary operation must be well-defined for any two elements in a groupoid? right?

 

[micromass]

So you mean the binary operation must be well-defined for any two elements in a groupoid? right?

Yes, the operation must be well-defined and the set must be closed under the operation. (for example $(\mathbb{N},-)$ is not a groupoid since 2-3 is not a natural number. That are the only requirements.

In other words, the operation

$$*:M\times M\rightarrow M$$


must be a function.

 

[AdrianZ]

Okay. Thanks.