Monoid

Symbol-free definition
A monoid is a set equipped with a binary operation that is associative and has a neutral element (or identity element).

Definition with symbols
A monoid is a set $$S$$ together witha binary operation $$*$$ and an element $$e \in S$$ such that:


 * $$(a * b) * c = a * (b * c) \forall a,b,c \in S$$ (associativity)
 * $$a * e = e * a = a \forall a \in S$$ (neutral element or identity element)

Note that monoid differs from the group in the sense of there being no guarantee for the existence of inverses. Thus, every group can be viewed as a monoid but not vice versa.

Facts in the definition
The neutral element (also called identity element) in a monoid is unique.

Occurrence of monoids
Monoids typically occur as collections of transformations which are closed under composition. For instance, the endomorphisms of a group form a monoid under composition: the identity element here is the identity map and the multiplication is the usual composition.

The invertible elements within the monoid form a group. If we are looking at a monoid of transformations, the invertible elements sitting there give transformations that have inverse transformations, or are reversible.

Stronger structures

 * Weaker than::Group is a monoid where every element has an inverse with respect to the identity element
 * Weaker than::Inverse monoid is a monoid where every element has an inverse in a somewhat weaker sense

Weaker structures

 * Stronger than::Semigroup is a set with an associative binary operation -- there may or may not be a neutral element
 * Stronger than::Magma is simply a set with a binary operation

Notation for the monoid operations
We typically omit the multiplication symbol when referring to the monoid operation. We also omit parentheses on account of associativity. (refer associative binary operation).

The identity element is usually denoted as $$e$$.

Mathjourneys links

 * Group Theory: A First Journey is an article making a reference to monoids in an attempt to motivate and build up to the definition of group