Abbreviation: CSgrp

A \emph{commutative semigroup} is a semigroups $\mathbf{S}=\langle S,\cdot \rangle $ such that

$\cdot $ is commutative: $xy=yx$

A \emph{commutative semigroup} is a structure $\mathbf{S}=\langle S,\cdot \rangle $, where $\cdot $ is an infix binary operation, called the \emph{semigroup product}, such that

$\cdot $ is associative: $(xy)z=x(yz)$

$\cdot $ is commutative: $xy=yx$

Let $\mathbf{S}$ and $\mathbf{T}$ be commutative semigroups. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:Sarrow T$ that is a homomorphism:

$h(xy)=h(x)h(y)$

Example 1: $\langle \mathbb{N},+\rangle $, the natural numbers, with additition.

$\begin{array}{lr} Search for finite commutative semigroups

f(1)= &1
f(2)= &3
f(3)= &12
f(4)= &58
f(5)= &325
f(6)= &2143
f(7)= &17291
\end{array}$

Semilattices

Commutative monoids

Semigroups

Partial commutative semigroups