Monoid

A monoid is a mathematical structure consisting of a set of elements and an associative binary operation on those elements, subject to the constraints that the set must be closed under the binary operation and must possess an identity element.

In Category Theory, a monoidal category is a one-element category. The elements of the monoid are represented as the Hom-Set of endomorphisms on that element. A category may also possess a monoid: an operation on its elements which forms the strucure of a monoid.

Special Types of Monoids

Groups

A group is special type of monoid such that every element has an inverse under the binary operation. Because groups are much more pervasive in mathematics and physics than monoids, monoids are often described as a generalization of groups.

Monads

For a category \(C\), a monoid in its category of endofunctors \(F: C \to C\) is a monad.