Monoid

It is a category with only one object. It is all the same than a group as a category , but you don't need to have inverses. It is the simplest category at all.

A monoid is a triple $(M,\cdot ,e)$ where:

1. $M$ is a set,
2. $\cdot$ is a binary operation (called multiplication or composition) mapping $M\times M\to M$,
3. $e$ is an element of $M$ (called the identity),
   satisfying the following axioms:
4. Associativity:
   For all $a,b,c\in M$,$$(a\cdot b)\cdot c=a\cdot (b\cdot c).$$
5. Identity:
   For all $a\in M$,$$e\cdot a=a\cdot e=a.$$

Examples of Monoids:

1. Natural numbers under addition:
   $(\mathbb{N},+,0)$ where $\mathbb{N}=\{0,1,2,\dots \}$, $+$ is addition, and $0$ is the identity.
2. Natural numbers under multiplication:
   $(\mathbb{N},\times ,1)$. It is a commutative, cancellative monoid with a single unit ($1$); its group of fractions is ${\mathbb{Q}}^{+}$. By the fundamental theorem of arithmetic it is the free commutative monoid on the primes: every $n\in \mathbb{N}$ is a unique product of primes, $n=\prod _p{p}^{a_p}$. (If $0\in \mathbb{N}$, then $0$ is an absorbing element.)
3. Strings under concatenation:
   $({\mathrm{\Sigma }}^{\ast },\cdot ,ϵ)$ where ${\mathrm{\Sigma }}^{\ast }$ is the set of all strings over an alphabet $\mathrm{\Sigma }$, $\cdot$ is concatenation, and $ϵ$ is the empty string.
4. Non-zero real numbers under multiplication:
   $({\mathbb{R}}^{\ast },\times ,1)$ where ${\mathbb{R}}^{\ast }=\mathbb{R}\setminus \{0\}$ and $\times$ is multiplication.
5. Functions from a set to itself under composition:
   $(X^X,\circ ,{\text{id}}_X)$ where $X^X$ is the set of all functions $X\to X$, $\circ$ is function composition, and ${\text{id}}_X$ is the identity function.

Related: group as a category