Magma (algebra)

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In abstract algebra, a magma (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M \times M \rightarrow M. A binary operation is closed by definition, but no other axioms are imposed on the operation.

The term magma for this kind of structure was introduced by Nicolas Bourbaki. The term groupoid is an older, but still commonly used alternative which was introduced by Øystein Ore.

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Definition

A magma is a set M matched with an operation "\cdot" that sends any two elements a,b \in M to another element a \cdot b. The symbol "\cdot" is a general placeholder for a properly defined operation. To qualify as a magma, the set and operation (M,\cdot) must satisfy the following requirement (known as the magma axiom):

And in mathematical notation:

Etymology

In French, the word "magma" has multiple common meanings, one of them being "jumble". It is likely that the French Bourbaki group referred to sets with well-defined binary operations as magmas with the "jumble" definition in mind.

Types of magmas

Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what axioms one might require of the operation. Commonly studied types of magmas include

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