Commutative magma

In mathematics, there exist magmas that are commutative but not associative. A simple example of such a magma may be derived from the children's game of rock, paper, scissors. Such magmas give rise to non-associative algebras.

A magma which is both commutative and associative is a commutative semigroup.

In the game of rock paper scissors, let ${\displaystyle M:=\{r,p,s\}}$ , standing for the "rock", "paper" and "scissors" gestures respectively, and consider the binary operation ${\displaystyle \cdot :M\times M\to M}$ derived from the rules of the game as follows:

This results in the Cayley table:

By definition, the magma ${\displaystyle (M,\cdot )}$ is commutative, but it is also non-associative, as shown by:

but

i.e.

It is the simplest non-associative magma that is conservative, in the sense that the result of any magma operation is one of the two values given as arguments to the operation.