Abbreviation: Sgrp

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: $(xy)z=x(yz)$.

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:

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

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.

$\begin{array}{lr} f(1)= &1 f(2)= &5 f(3)= &24 f(4)= &188 f(5)= &1915 f(6)= &28634 f(7)= &1627672 f(8)= &3684030417 f(9)= &105\,978\,177\,936\,292 \end{array}$

[http://oeis.org/A027851 Semigroups in the Encyclopedia of Integer Sequences]

Bands

Commutative semigroups

Monoids

Semigroups with zero

Groupoids

Partial semigroups