Monoid
In abstract algebra, a monoid is a set of elements along with an operation that has two key properties:
- The operation can combine the elements associatively; e.g. [math]\displaystyle{ (A + B) + C = A + (B + C) }[/math]
- There exists an identity element; e.g. [math]\displaystyle{ 1 \times X = X }[/math], or [math]\displaystyle{ 0 + X = X }[/math]
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".
Monoid Media
Algebraic structures between magmas and groups. For example, monoids are semigroups with identity.
Example monoid homomorphism x ↦ 2x from (N, +, 0) to (N, ×, 1). It is injective, but not surjective.
