Abbreviation: CMon

A commutative monoid is a monoids $\mathbf{M}=\langle M,\cdot ,e\rangle $ such that

$\cdot $ is commutative: $x\cdot y=y\cdot x$

A commutative monoid is a structure $\mathbf{M}=\langle M,\cdot ,e\rangle $, where $\cdot $ is an infix binary operation, called the monoid product, and $e$ is a constant (nullary operation), called the identity element, such that

$\cdot $ is commutative: $x\cdot y=y\cdot x$

$\cdot $ is associative: $(x\cdot y)\cdot z=x\cdot (y\cdot z)$

$e$ is an identity for $\cdot $: $e\cdot x=x$

Let $\mathbf{M}$ and $\mathbf{N}$ be commutative monoids. A morphism from $\mathbf{M}$ to $\mathbf{N}$ is a function $h:M\to N$ that is a homomorphism:

$h(x\cdot y)=h(x)\cdot h(y)$, $h(e)=e$

Example 1: $\langle\mathbb{N},+,0\rangle$, the natural numbers, with addition and zero. The finitely generated free commutative monoids are direct products of this one.

$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &5\\ f(4)= &19\\ f(5)= &78\\ f(6)= &421\\ f(7)= &2637\\ \end{array}$

Abelian groups

Semilattices with identity

Commutative semigroups

Monoids