Table of Contents
Semigroups
Abbreviation: Sgrp
Definition
A \emph{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: $(xy)z=x(yz)$.
Morphisms
Let $\mathbf{S}$ and $\mathbf{T}$ be semigroups. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\to T$ that is a homomorphism:
$h(xy)=h(x)h(y)$
Examples
Example 1: $\langle X^{X},\circ \rangle $, the collection of functions on a sets $X$, with composition.
Example 1: $\langle \Sigma ^{+},\cdot \rangle $, the collection of nonempty strings over $\Sigma $, with concatenation.
Basic results
Properties
| Classtype | variety |
|---|---|
| Equational theory | decidable in polynomial time |
| Quasiequational theory | undecidable |
| First-order theory | undecidable |
| Locally finite | no |
| Residual size | unbounded |
| 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
$\begin{array}{lr}
f(1)= &1
f(2)= &5
f(3)= &24
f(4)= &188
f(5)= &1915
f(6)= &28634
f(7)= &1627672
f(8)= &3684030417
f(9)= &105\,978\,177\,936\,292
\end{array}$
[http://oeis.org/A027851 Semigroups in the Encyclopedia of Integer Sequences]
