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