Math 2.0

Group theory and generalizations Definition

Commutative magma

Builds on MagmaCommutativity

Definition

Let (M,⋅)(M, \cdot) be a magma. It is a commutative magma if x⋅y=y⋅xx \cdot y = y \cdot x for all x,y∈Mx, y \in M, that is, if the operation has the commutativity property.

Remarks

Example: ℚ\Q with x∘y=(x+y)/2x \circ y = (x + y)/2. The value is rational, and (x+y)/2=(y+x)/2(x + y)/2 = (y + x)/2. The magma is not associative: (0∘4)∘8=5(0 \circ 4) \circ 8 = 5 but 0∘(4∘8)=30 \circ (4 \circ 8) = 3.

Non-example: (ℤ,−)(\Z, -), since 1−0=11 - 0 = 1 but 0−1=−10 - 1 = -1.

In Mathlib this structure is the class CommMagma.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Commutativity, concept, no evidence levelCommutativityconceptMagma, concept, no evidence levelMagmaconceptCommutative magma, definition, no evidence levelCommutative magmadefinition
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  1. Commutativity concept
  2. Magma concept
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Further reading

  1. Web https://en.wikipedia.org/wiki/Commutative_magma
About this page
TagME71ZQE49J
URIm2://ME71ZQE49J
Kinddefinition
WritingW0 stub
MSC20N02
DefinesCommutative magma
UsesMagma, Commutativity
Changed2 November 2026

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