Example of a commutative non-associative magma

In mathematics, it can be shown that there exist magmas that are commutative but not associative. A simple example of such a magma is given by considering the children's game of rock, paper, scissors.

A commutative non-associative magma

Let $M\; :=\; \{\; r,\; p,\; s\; \}$ and consider the binary operation $cdot\; :\; M\; imes\; M\; o\; M$ defined, loosely inspired by the rock-paper-scissors game, as follows:

:$r\; cdot\; p\; =\; p\; cdot\; r\; =\; p$ "paper beats rock";:$p\; cdot\; s\; =\; s\; cdot\; p\; =\; s$ "scissors beat paper";:$r\; cdot\; s\; =\; s\; cdot\; r\; =\; r$ "rock beats scissors";:$r\; cdot\; r\; =\; r$ "rock ties with rock";:$p\; cdot\; p\; =\; p$ "paper ties with paper";:$s\; cdot\; s\; =\; s$ "scissors tie with scissors".

By definition, the magma $(M,\; cdot)$ is commutative, but it is also non-associative, as the following shows:

:$r\; cdot\; (p\; cdot\; s)\; =\; r\; cdot\; s\; =\; r$

but

:$(r\; cdot\; p)\; cdot\; s\; =\; p\; cdot\; s\; =\; s.$

A commutative non-associative algebra

Using the above example, one can construct a commutative non-associative algebra over a field $K$: take $A$ to be the three-dimensional vector space over $K$ whose elements are written in the form

:$(x,\; y,\; z)\; =\; x\; r\; +\; y\; p\; +\; z\; s$,

for $x,\; y,\; z\; in\; K$. Vector addition and scalar multiplication are defined component-wise, and vectors are multiplied using the above rules for multiplying the elements $r,\; p$ and $s$. The set

:$\{\; (1,\; 0,\; 0),\; (0,\; 1,\; 0),\; (0,\; 0,\; 1)\; \}$ i.e. $\{\; r,\; p,\; s\; \}$

forms a basis for the algebra $A$. As before, vector multiplication in $A$ is commutative, but not associative.