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 <math>\mathbb{N}</math> and addition is written as <math>+</math> the magma is written as <math>(\mathbb{N} , +)</math>. 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 <math>\mathbb{Z}</math> and multiplication (in abstract mathematics) is written as <math>\cdot</math> the magma is written as <math>(\mathbb{Z}, \cdot)</math>. 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 <math>\mathbb{R}^*</math>. It can be shown that the <math>(\mathbb{R}^*, \div)</math> is a magma.