<s>[INST] specific magma examples Give an example of a magma $S$ such that $S$ has a zero and $S$ has a left zero divisor that is not a right zero divisor
an example of a magma with an identity such that there is an element with exactly $2$ left inverses but only one right inverse
For the first, I was thinking $\{0,a,b\}$ where $ab=0$, $ba=a$, $b^2=b$, $a^2=a$. 
Thanks
 [/INST] Your answer to the first question may be right (thanks @celtschk for commenting):
$$
\begin{array}{c|ccc}
\cdot & 0 & a & b\\
\hline
0 & 0 & 0 & 0\\
a & 0 & a & 0\\
b & 0 & a & b
\end{array}
$$
Here $0$ denotes an absorbing element, $a$ is only a left zero divisor, while $b$ is only a right zero divisor.

Second question: here they ask for an identity, let  us call it $1$. From your comment

@drhab a^2=a, ab=b, ba=1, b^2=a?

I get
$$
\begin{array}{c|ccc}
\bullet & 1 & a & b\\
\hline
1 & 1 & a & b\\
a & a & a & b\\
b & b & 1 & a
\end{array}
$$
but to obtain

an element with exactly 2 left inverses but only one right inverse

we must have, in the Cayley table, $1$ appearing exactly twice in the column labeled by the element and exactly once in the line labeled by the same element, for example:
$$
\begin{array}{c|ccc}
\bullet & 1 & a & b\\
\hline
1 & 1 & a & b\\
a & a & 1 & b\\
b & b & 1 & a
\end{array}
$$
In this last table (I got this from the previous changing only one cell, $a\bullet a$), $1$ is the identity, $a$ and $b$ are both left inverses of $a$, while only $a$ is a right inverse of $a$.
With four or more elements in the underlying set we could get more various examples, like:
$$
\begin{array}{c|cccccc}
* & 1 & 2 & 3 & 4 & 5 & 6\\
\hline
1 & \color{blue}{1} & \color{blue}{2} & \color{blue}{3} & \color{blue}{4} & \color{blue}{5} & \color{blue}{6}\\
2 & \color{blue}{2} & 3 & 2 & 3 & 2 & 3\\
3 & \color{blue}{3} & 3 & 2 & 3 & 2 & \color{red}{1}\\
4 & \color{blue}{4} & 3 & 2 & 3 & 2 & \color{red}{1}\\
5 & \color{blue}{5} & 3 & 2 & 3 & 2 & 3\\
6 & \color{blue}{6} & 3 & 2 & 3 & \color{red}{1} & 3\\
\end{array}
$$
Here, $1$ is the identity (in blue), $3$ and $4$ are left inverses of $6$ and $5$ is right inverse of $6$ (in red).
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