Understanding Groupoids: Axioms & Difference from Groups

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In summary, a groupoid is a nonempty set with a binary operation. It satisfies some axioms, but the difference between a groupoid and group is that a groupoid has a more conventional name.
  • #1
AdrianZ
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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.
 
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  • #2
Hi AdrianZ! :smile:

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

The only requirement is that the operation must really be an operation. For example [itex](\mathbb{R},/)[/itex] 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
So you mean the binary operation must be well-defined for any two elements in a groupoid? right?
 
  • #4
AdrianZ said:
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 [itex](\mathbb{N},-)[/itex] is not a groupoid since 2-3 is not a natural number. That are the only requirements.

In other words, the operation

[tex]*:M\times M\rightarrow M[/tex]


must be a function.
 
  • #5
Okay. Thanks.
 

1. What is a groupoid?

A groupoid is a mathematical structure that is similar to a group, but with more flexibility and generalization. It consists of a set of elements and a binary operation that satisfies certain axioms, allowing for the combination of any two elements in the set.

2. How is a groupoid different from a group?

The main difference between a groupoid and a group is that a groupoid does not necessarily have an identity element or inverses for every element. Instead, the binary operation in a groupoid only needs to be partially defined, meaning it can be applied to some but not all pairs of elements in the set.

3. What are the axioms that define a groupoid?

The axioms for a groupoid include closure, associativity, and existence of inverses. Closure means that the binary operation must produce an element within the set when applied to any two elements. Associativity means that the order of the operations does not matter. And existence of inverses means that for every element, there exists another element that when combined with it, produces the identity element.

4. How are groupoids used in mathematics?

Groupoids have applications in various areas of mathematics, such as topology, algebraic geometry, and functional analysis. They are particularly useful for studying symmetries and transformations of objects, as well as for understanding the structure of more complex mathematical objects.

5. Are there any real-life examples of groupoids?

Yes, groupoids can be found in many real-world systems, such as the symmetries of crystals, the movements of particles in physics, and the transformations of geometric shapes in computer graphics. They can also be used to model social networks and interactions between individuals in a group.

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