# Monoid¶

A monoid is an algebraic structure with an identity value and an associative binary operation.

## Backlinks¶

- Monoidal Identity
- The monoidal identity is the identity value for a particular monoid. A monoid must have an identity value and an associative binary operation. The identity value m is a value which makes no change to any other value n in that monoid, when n is combined with m using that binary operation.

- A category with a single object is a monoid
- Any category with a single object is a monoid. The object in a monoidal category can have any number of morphisms greater than 1 - the requirement for an identity morphism isn't relaxed.

- A monoid has an associative binary operation
- Any monoid has an associative binary operation that takes any two elements in the associated set and produces an element in that set.

- The composition of morphisms in a monoidal category corresponds to an associative binary operation
- For a monoidal category M each morphism in M(m, m) corresponds to the operation defined for the associated monoidal set on one of the elements.

- The identity morphism in a monoidal category corresponds to the unit element
- The unit element defined for a set-theory monoid corresponds to the identity morphism within a monoidal category.

- A set-theory monoid and a category-theory monoid are the same thing
- The set theory definition and the category theory definition of monoids are different views over the same concept. Category theory defines a monoid as any category with a single object. Set theory defines a monoid as a set with an associative binary operation and a unit element.

- A monoid has a unit element
- A monoid has a unit element u which interacts with the associative binary operation defined for its set, \\otimes, in the following way