Abbreviation: Sgrp

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

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