Commutative magma

In mathematics, there exist magmas that are commutative but not associative . A simple example of such a magma may be derived from the children's game of rock, paper, scissors. Such magmas give rise to non-associative algebras.

A magma which is both commutative and associative is a commutative semigroup.

In the game of rock paper scissors, let ${\displaystyle M:=\{r,p,s\}}${\displaystyle M:=\{r,p,s\}} , standing for the "rock", "paper" and "scissors" gestures respectively, and consider the binary operation ${\displaystyle \cdot :M\times M\to M}${\displaystyle \cdot :M\times M\to M} derived from the rules of the game as follows:^[1]

For all ${\displaystyle x,y\in M}${\displaystyle x,y\in M}:

• If ${\displaystyle x\neq y}${\displaystyle x\neq y} and ${\displaystyle x}${\displaystyle x} beats ${\displaystyle y}${\displaystyle y} in the game, then ${\displaystyle x\cdot y=y\cdot x=x}${\displaystyle x\cdot y=y\cdot x=x}
• ${\displaystyle x\cdot x=x}${\displaystyle x\cdot x=x}     I.e. every ${\displaystyle x}${\displaystyle x} is idempotent.

So that for example:

• ${\displaystyle r\cdot p=p\cdot r=p}${\displaystyle r\cdot p=p\cdot r=p}   "paper beats rock";
• ${\displaystyle s\cdot s=s}${\displaystyle s\cdot s=s}   "scissors tie with scissors".

This results in the Cayley table:^[1]

${\displaystyle {\begin{array}{c|ccc}\cdot &r&p&s\\\hline r&r&p&r\\p&p&p&s\\s&r&s&s\end{array}}}${\displaystyle {\begin{array}{c|ccc}\cdot &r&p&s\\\hline r&r&p&r\\p&p&p&s\\s&r&s&s\end{array}}}

By definition, the magma ${\displaystyle (M,\cdot )}${\displaystyle (M,\cdot )} is commutative, but it is also non-associative,^[2] as shown by:

${\displaystyle r\cdot (p\cdot s)=r\cdot s=r}${\displaystyle r\cdot (p\cdot s)=r\cdot s=r}

but

${\displaystyle (r\cdot p)\cdot s=p\cdot s=s}${\displaystyle (r\cdot p)\cdot s=p\cdot s=s}

i.e.

${\displaystyle r\cdot (p\cdot s)\neq (r\cdot p)\cdot s}${\displaystyle r\cdot (p\cdot s)\neq (r\cdot p)\cdot s}

It is the simplest non-associative magma that is conservative, in the sense that the result of any magma operation is one of the two values given as arguments to the operation.^[2]

The arithmetic mean, and generalized means of numbers or of higher-dimensional quantities, such as Frechet means, are often commutative but non-associative.^[3]

Commutative but non-associative magmas may be used to analyze genetic recombination.^[4]

• Non-associative algebra