Monoid

A monoid is a semigroup $(M,\cdot)$ together with an element $e\in M$ (called an identity element) such that for every $a\in M$, $e\cdot a = a \quad\text{and}\quad a\cdot e = a.$

Monoids generalize groups by dropping the requirement that elements have inverses. Many monoids arise from composition of endomorphisms (self-maps).

Examples:

• $(\mathbb{N},+,0)$ is a monoid.
• $(\mathbb{N},\times,1)$ is a monoid.
• For any set $X$, the set of all functions $X\to X$ is a monoid under composition, with identity $\mathrm{id}_X$.
• The set of all $n\times n$ real matrices is a monoid under multiplication, with identity matrix $I_n$; it is not a group because not every matrix is invertible.