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Example: rock, paper, scissors

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

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

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

So that for example:

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

This results in the Cayley table:²

⋅ r p s r r p r p p p s s r s s {\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 ( M , ⋅ ) {\displaystyle (M,\cdot )} is commutative, but it is also non-associative,³ as shown by:

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

but

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

i.e.

r ⋅ ( p ⋅ s ) ≠ ( r ⋅ p ) ⋅ 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.⁴

Applications

The arithmetic mean, and generalized means of numbers or of higher-dimensional quantities, such as Frechet means, are often commutative but non-associative.⁵

Commutative but non-associative magmas may be used to analyze genetic recombination.⁶

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 /wiki/ArXiv_(identifier) ↩

2. 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 /wiki/ArXiv_(identifier) ↩

3. 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 /wiki/Doi_(identifier) ↩

4. 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 /wiki/Doi_(identifier) ↩

5. 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 /wiki/ArXiv_(identifier) ↩

6. 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 /wiki/Doi_(identifier) ↩