# Monoid

algebraic structure with an associative operation and an identity element

In abstract algebra, a monoid is a set of elements along with an operation that has two key properties:

1. The operation can combine the elements associatively; e.g. ${\displaystyle (A+B)+C=A+(B+C)}$
2. There exists an identity element; e.g. ${\displaystyle 1\times X=X}$, or ${\displaystyle 0+X=X}$

The operation need not have the commutative property.

In computing science common monoids include addition, multiplication, or, and. These properties are useful for various problems e.g. they allow a large set of data to be divided, processed in parallel and combined. As each part produces a monoid, the final combined result will be the same. This also works with more complex monoids, such as a map from words to the number of times they appear in a document.

Character strings also form a monoid over concatenation with the empty string "" as the identity element. Examples of the two properties are "abc" + ("def" + "ghi") = ("abc" + "def") + "ghi" and "abc" + "" = ""+ "abc" = "abc". This is true even though "abc" + "def" ≠ "def" + "abc".