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 commutative non-associative magma derived from the rock, paper, scissors game.
Let formula_1 , standing for the "rock", "paper" and "scissors" gestures respectively, and consider the binary operation formula_2 derived from the rules of the game as follows:
This results in the Cayley table:
By definition, the magma formula_13 is commutative, but it is also non-associative, as shown by:
but
i.e.
Other examples.
The "mean" operation formula_17 on the rational numbers (or any commutative number system closed under division) is also commutative but not in general associative, e.g.
but
Generally, the mean operations studied in topology need not be associative.
The construction applied in the previous section to rock-paper-scissors applies readily to variants of the game with other numbers of gestures, as described in the section "Variations", as long as there are two players and the conditions are symmetric between them; more abstractly, it may be applied to any trichotomous binary relation (like "beats" in the game). The resulting magma will be associative if the relation is transitive and hence is a (strict) total order;
otherwise, if finite, it contains directed cycles (like rock-paper-scissors-rock) and the magma is non-associative. To see the latter, consider combining all the elements in a cycle in reverse order, i.e. so that each element combined beats the previous one;
the result is the last element combined, while associativity and commutativity would mean that the result only depended on the set of elements in the cycle.
The bottom row in the Karnaugh diagram above gives more example operations, defined on the integers (or any commutative ring).
Derived commutative non-associative algebras.
Using the rock-paper-scissors example, one can construct a commutative non-associative algebra over a field formula_20: take formula_21 to be the three-dimensional vector space over formula_20 whose elements are written in the form
for formula_24. Vector addition and scalar multiplication are defined component-wise, and vectors are multiplied using the above rules for multiplying the elements formula_25.
The set
forms a basis for the algebra formula_21. As before, vector multiplication in formula_21 is commutative, but not associative.
The same procedure may be used to derive from any commutative magma formula_30 a commutative algebra over formula_20 on formula_32, which will be non-associative if formula_30 is.