Does every non-empty quasigroup have a left or right identity?

Question

I know that some quasigroups are not loops, meaning they don't have a two-sided identity. But are there non-empty quasigroups that don't even have one-sided identities?

Answer 1

A quasigroup may be identified with a latin square by viewing the latin square as its "multiplication" table.

The task is then just to exhibit a latin square of size $n\times n$ with symbols $1,\ldots,n$ in which no row or column repeats the symbols in that order. To do so one may modify a latin square by permuting the rows and columns as one wishes.

For example:

$$ \begin{array}{c|ccc} * & 1 & 2 & 3 \\ \hline 1 & 1 & 3 & 2 \\ 2 & 3 & 2 & 1 \\ 3 & 2 & 1 & 3 \end{array} $$