Table of Contents
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.
Basic results
Properties
| Classtype | variety |
|---|---|
| Equational theory | decidable in polynomial time |
| Quasiequational theory | decidable |
| First-order theory | |
| Locally finite | no |
| Residual size | |
| Congruence distributive | no |
| Congruence modular | no |
| Congruence n-permutable | no |
| Congruence regular | no |
| Congruence uniform | no |
| Congruence extension property | |
| Definable principal congruences | |
| Equationally def. pr. cong. | no |
| Amalgamation property | no |
| Strong amalgamation property | no |
| Epimorphisms are surjective | no |
Finite members
$\begin{array}{lr} Search for finite commutative semigroups
f(1)= &1
f(2)= &3
f(3)= &12
f(4)= &58
f(5)= &325
f(6)= &2143
f(7)= &17291
\end{array}$
