What is a groupoid?

1. Jun 28, 2011

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:

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.

2. Jun 28, 2011

micromass

Staff Emeritus

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.

3. Jun 28, 2011

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

4. Jun 28, 2011

micromass

Staff Emeritus
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.

5. Jun 28, 2011