Magma (mathematics)

In mathematics, a magma is kind of algebraic structure. It is a set with a binary operation on that set.

A binary operation works by taking two elements from a set (that do not have to be different) and returning some other element of that set.

If we give the set a label (such as X) and the binary operation a label (such as •). Then we give the magma the label (X, •).

Examples

The natural numbers with addition form a magma. Because the set of natural numbers is written as $\mathbb{N}$ and addition is written as $+$ the magma is written as $(\mathbb{N},+)$. The name of the magma would be "The natural numbers under addition".

The integers with multiplication form a magma. Because the set of integers is written as $\mathbb{Z}$ and multiplication (in abstract mathematics) is written as $\cdot$ the magma is written as $(\mathbb{Z},\cdot )$. The name of the magma would be "The integers under multiplication".

The real numbers under division do not form a magma. This is because numbers cannot be divided by 0. A binary operation requires that any two elements can be taken from the set (in this case in order) to produce another element from the set. The real numbers without 0 is written as ${ℝ}^{*}$. It can be shown that the $({ℝ}^{*},÷)$ is a magma.