Magma (Algebraic)

magma

A magma is a set $G$ together with a binary operation:

μ : G \times G \to G; μ (a, b) \mapsto μ (a, b)

Some Examples

Take the set $Z = {1, 2, 3, 4, \dots}$ which has some example binary operations:

$μ (a, b) = a + b$
$μ (a, b) = a b$
$μ (a, b) = min {a, b}$
$μ (a, b) = max {a, b}$
$μ (a, b) = a$ ("projection")
$μ (a, b) = a^{b}$
$μ (a, b) = 2^{a b}$
...

Two Possible Properties

We may want to ask: Is the operation $μ$ (for $a, b, c \in G$ )

associative: $μ (a, μ (b, c)) = μ (μ (a, b), c)$ (think of $a (b c) = (a b) c$ from the integers, parenthesis don't matter, here the operations can be applied in whatever order as long as it's in-order))
commutativity: $μ (a, b) = μ (b, a)$ ( $a b = b a$ ) (order doesn't matter)

Looking at the examples above:

Both (commutative and associative)
Both
Both (equivalent to $min {a, b, c}$ )
Both (same as (3))
Not Commutative (since $μ (2, 1) = 2 \neq 1 = μ (1, 2)$ ), but is Associative
Not Commutative (since $μ (1, 2) = 1^{2} \neq 2^{1} = μ (2, 1)$ ), and Not Associative ( $a^{b^{c}}$ vs. $(a^{b})^{c} \equiv a^{b c}$ , just choose an example where $b^{c} \neq b c$ )
Commutative but not associative

It's actually not too uncommon to find operations that aren't commutative, but the associativity is an important property, since there's only a few operations that really aren't associative (ex: the cross product $\times$ in $R^{3}$ )