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

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]

Subclasses

Bands

Commutative semigroups

Monoids

Semigroups with zero

Superclasses

Groupoids

Partial semigroups

References