Example of a commutative non-associative magma

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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[edit]

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[edit]

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.