## Semigroups

Abbreviation: Sgrp

### Definition

A semigroup is a structure $\mathbf{S}=\langle S,\cdot \rangle$, where $\cdot$ is an infix binary operation, called the 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.

### Properties

Classtype variety decidable in polynomial time undecidable undecidable no unbounded no no no no no no no no 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]