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.

Let $M:=\{r,p,s\}$ and consider the binary operation $\cdot :M\times M\to M$ defined 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 defintion, the magma $(M,\cdot )$ is commutative, but it is non-associative, as the following shows:

$r\cdot (p\cdot s)=r\cdot s=r\neq s=p\cdot s=(r\cdot p)\cdot s.$