Example of a commutative non-associative magma
|This article does not cite any references or sources. (April 2012)|
In mathematics, it can be shown that there exist magmas that are commutative but not associative. A simple example of such a magma is given by considering the children's game of rock, paper, scissors.
A commutative non-associative magma
- "paper beats rock";
- "scissors beat paper";
- "rock beats scissors";
- "rock ties with rock";
- "paper ties with paper";
- "scissors tie with scissors".
By definition, the magma is commutative, but it is also non-associative, as the following shows:
A commutative non-associative algebra
for . Vector addition and scalar multiplication are defined component-wise, and vectors are multiplied using the above rules for multiplying the elements and . The set
forms a basis for the algebra . As before, vector multiplication in is commutative, but not associative.