## Commutative semigroups

Abbreviation: CSgrp

### Definition

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

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

### Definition

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: $xy=yx$

##### Morphisms

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:

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

### Examples

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

### Properties

Classtype variety decidable in polynomial time decidable no no no no no no no no no no

### Finite members

$\begin{array}{lr} Search for finite commutative semigroups f(1)= &1 f(2)= &3 f(3)= &12 f(4)= &58 f(5)= &325 f(6)= &2143 f(7)= &17291 \end{array}$