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 $\cdot : M \times M \to M$ 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

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

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.