Group theory and generalizations Definition

Magma

Definition

A magma is a pair $(M, \cdot)$ , where $M$ is a set and $\cdot \colon M \times M \to M$ is a binary operation on $M$ . A homomorphism of magmas $(M, \cdot)$ and $(N, *)$ is a map $h \colon M \to N$ with $h(x \cdot y) = h(x) * h(y)$ for all $x, y \in M$ .

Remarks

Example: $(\Z, -)$ is a magma, since the difference of two integers is an integer. It is not associative: $(1 - 1) - 1 = -1$ but $1 - (1 - 1) = 1$ .

Non-example: $(\N, -)$ is not a magma, since $1 - 2 \notin \N$ , so subtraction is not a binary operation on $\N$ .

The empty set with the empty operation is a magma. The word groupoid is an older name for a magma.

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Binary operation concept
Element definition
Set definition
Magma definition

Tag	`K5G3VB3DEZ`
URI	`m2://K5G3VB3DEZ`
Kind	definition
Writing	W0 stub
MSC	20N02, 08A62
Defines	Magma
Uses	Set, Binary operation
Changed	2 November 2026

Magma

Definition

Remarks

Reading path

Further reading

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