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

edit

In the game of rock paper scissors, let   , standing for the "rock", "paper" and "scissors" gestures respectively, and consider the binary operation   derived from the rules of the game as follows:[1]

For all  :
  • If   and   beats   in the game, then  
  •       I.e. every   is idempotent.
So that for example:
  •     "paper beats rock";
  •     "scissors tie with scissors".

This results in the Cayley table:[1]

 

By definition, the magma   is commutative, but it is also non-associative,[2] as shown by:

 

but

 

i.e.

 

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

edit

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

edit
  1. ^ a b 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. ^ a b 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