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
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
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
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).