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groupoid (Definition)

A groupoid $ G$ is a set together with a binary operation $ \cdot : G \times G \longrightarrow G$. The groupoid (or “magma”) is closed under the operation.

There is also a separate, category-theoretic definition of “groupoid.”



"groupoid" is owned by akrowne.
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See Also: semigroup, group, loop and quasigroup

Other names:  magma

Attachments:
equation (Definition) by pahio
absorbing element (Definition) by pahio
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Cross-references: operation, closed under, binary operation
There are 12 references to this entry.

This is version 7 of groupoid, born on 2002-02-02, modified 2002-11-06.
Object id is 1691, canonical name is Groupoid.
Accessed 8974 times total.

Classification:
AMS MSC: 20N02 (Group theory and generalizations :: Other generalizations of groups :: Sets with a single binary operation )
Pending Errata and Addenda
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Closed binary operation by ratboy on 2004-06-04 12:13:46
The remark about closure seems redundant. How could a set G fail to be closed under a binary operation on G? If, on the other hand, the intent is to define the relevant notion of closure, there ought to be some mention of the situation where it matters: a subset C of G is a subgroupoid of G if the restriction of the operation to CxC is a binary operation on C.




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disambiguation? by AxelBoldt on 2002-11-06 15:54:19
Since some groupoids of the first kind are also groupoids of the second kind, and vice versa, I think it would be beneficial to have two articles, one about groupoid (binary operation) and one about groupoid (category). That way, people could link to the article they want. Right now, the word "groupoid" cannot be used in any article without immediately giving the intended definition.
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