Can you give me some concrete examples of magmas?

Solution 1:

A ("strict") magma you've probably heard of is the vector cross-product in $\Bbb R^3$: $$ (a, b, c)\times(x, y, z) = (bz - cy, cx - az, ay - bx) $$ $\Bbb R^3$ is closed under this operation, but it has neither associativity, commutativity, identity nor divisibility.

Kind of in the same way that any square, any rectangle and any parallelogram fulfills the criteria of a trapezoid, and thus are trapezoids, we say that any group, monoid or semigroup is also a magma. All we demand from the structure in order to call it a magma is that it is closed under the binary operation.

And just as any trapezoid in which all angles happens to be right still is a trapezoid even though most people would call it a rectangle, so too will any closed / total algebraic structure with associativity, identity and divisibility be a magma, even though most people would call it a group.

Solution 2:

A magma is just a set $X$ together with a binary operation on $X$, i.e. a function $X\times X\to X$. Any such function will do!

For example, we could define a binary operation on $X=\mathbb R$ by

$$x\cdot y = xy+x^2-y.$$

Solution 3:

My favorite example: the operation $*$ on the odd integers with $a * b = (3a + b) / 2^k$ where $2^k$ is the highest power of $2$ dividing $3a + b$. With this notation, the Collatz conjecture can be restated:

For all odd integers $k$, does there exists an $n$ such that $$ \underbrace{\Big(\big(((k * 1) * 1) * \cdots \big) * 1\Big) * 1}_{n \text{ ones}} = 1? $$

Perhaps this sheds little light on actually solving the problem, but at least it provides a framework where the conjecture makes sense. The unpredictable, nonassociative behavior of this operation $*$ is one way of understanding why the Collatz problem is so hard.