Commutative 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 $\cdot :M\times M\to M$ 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

$(x,y,z)=xr+yp+zs$,

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.