Examples of quasigroups with no identity elements If you scroll to the bottom of this page, there is a table claiming quasigroups have divisibility but not identity (in general).
What would be some examples of quasigroups without an identity element?

The only quasigroups of orders $1$ and $2$ are the (unique) groups of those orders. So, a minimal example must have order $\geq 3$; indeed, (up to isomorphism) there are five quasigroups of order $3$ but but only one of these (the unique group with three elements) has an identity.
Probably the most familiar of the four quasigroups of order $3$ without identity is subtraction of integers modulo $3$, which has multiplication table
$$
\begin{array}{c|ccc}
- & 0 & 1 & 2 \\
\hline
0 & 0 & 2 & 1 \\
1 & 1 & 0 & 2 \\
2 & 2 & 1 & 0
\end{array} .
$$
(Despite that this operation has no identity, $0$ is a right identity of $-$, that is, $x - 0 = x$ for all $x$.)
Multiplication tables for all five quasigroups (up to isomorphism) are given in this answer to a related question.
