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.

Example: rock, paper, scissors

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

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

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

So that for example:

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

This results in the Cayley table:^[1]

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

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

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

but

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

i.e.

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

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]

Applications

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]

References

1. Aten, Charlotte (2020), "Multiplayer rock-paper-scissors", Algebra Universalis, 81 (3): Paper No. 40, 31, arXiv:1903.07252, doi:10.1007/s00012-020-00667-5, MR 4123817
2. Beaudry, Martin; Dubé, Danny; Dubé, Maxime; Latendresse, Mario; Tesson, Pascal (2014), "Conservative groupoids recognize only regular languages", Information and Computation, 239: 13–28, doi:10.1016/j.ic.2014.08.005, MR 3281897
3. Ginestet, Cedric E.; Simmons, Andrew; Kolaczyk, Eric D. (2012), "Weighted Frechet means as convex combinations in metric spaces: properties and generalized median inequalities", Statistics & Probability Letters, 82 (10): 1859–1863, arXiv:1204.2194, doi:10.1016/j.spl.2012.06.001, MR 2956628
4. Etherington, I. M. H. (1941), "Non-associative algebra and the symbolism of genetics", Proceedings of the Royal Society of Edinburgh, Section B: Biology, 61 (1): 24–42, doi:10.1017/s0080455x00011334