List of common semigroups (which are not groups)? Is there a list of common finite semigroups?  I'm assuming a sensible classification as one has with finite simple groups is impossible, but it would be nice to see a list of common small examples, however incomplete.
For example, if one is testing a simple proposition about finite semigroups against some examples, what does one start with?
 A: Common semigroups of low order which are not groups.
The semigroup $N_2 = \{a,0\}$ where $0$ is a zero and $a^2 = 0$.
The monoid $U_1 = \{1, 0\}$ under usual multiplication of integers.
Let $I$ and $J$ be two nonempty sets. Define an operation on $I \times J$ by setting, for every $(i,j), (i',j') \in I \times J$,
$$
 (i,j)(i',j') = (i, j')
$$
This defines a semigroup of order $|I||J|$, usually denoted by $B(I, J)$.
Second part of your question.
Let $B_2$ be the set of $2 \times 2$ matrices with $0$-$1$ entries and at most one nonzero entry. Equipped with the usual multiplication of matrices, $B_2$ is a semigroup.
$$
 B_2 = \left\{  
 \begin{pmatrix} 1&0 \\ 0&0 \end{pmatrix}, 
 \begin{pmatrix} 0&1 \\ 0&0 \end{pmatrix}, 
 \begin{pmatrix} 0&0 \\ 1&0 \end{pmatrix}, 
 \begin{pmatrix} 0&0 \\ 0&1 \end{pmatrix}, 
 \begin{pmatrix} 0&0 \\ 0&0 \end{pmatrix}
 \right\}
$$
Setting $a=\left(
\begin{smallmatrix}
 0\ 1\\
 0\ 0
\end{smallmatrix}
\right)
$
and
$
b=\left( 
\begin{smallmatrix}
 0\ 0\\
 1\ 0
\end{smallmatrix}
\right)
$, one gets 
$ab=\left( 
\begin{smallmatrix}
 1\ 0\\
 0\ 0
\end{smallmatrix}
\right)
$,
$ba=\left( 
\begin{smallmatrix}
 0\ 0\\
 0\ 1
\end{smallmatrix}
\right)
$
and
$0=\left( 
\begin{smallmatrix}
 0\ 0\\
 0\ 0
\end{smallmatrix}
\right)$. Thus $B_2 = \{a, b, ab, ba, 0\}$. Furthermore, the relations $aa=bb=0$, $aba=a$ and $bab=b$ suffice to recover completely the multiplication in $B_2$. If one adds the identity matrix, one gets the 6-element monoid $B^1_2$. The semigroups $B_2$ and $B_2^1$ are nicknamed the universal counterexamples because they provide many counterexamples in semigroup theory.
