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 Cayley tables:
$$\begin{array}{c|ccccccc}
\ast & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ \hline
1 & 1 & 3 & 2 & 5 & 4 & 7 & 6\\
2 & 3 & 2 & 1 & 6 & 7 & 4 & 5\\
3 & 2 & 1 & 3 & 7 & 6 & 5 & 4\\
4 & 5 & 6 & 7 & 4 & 1 & 2 & 3\\
5 & 4 & 7 & 6 & 1 & 5 & 3 & 2\\
6 & 7 & 4 & 5 & 2 & 3 & 6 & 1\\
7 & 6 & 5 & 4 & 3 & 2 & 1 & 7
\end{array}
\qquad\qquad
\begin{array}{c|ccccccc}
\circ & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ \hline
1 & 1 & 7 & 6 & 5 & 4 & 3 & 2\\
2 & 7 & 2 & 5 & 6 & 3 & 4 & 1\\
3 & 6 & 5 & 3 & 7 & 2 & 1 & 4\\
4 & 5 & 6 & 7 & 4 & 1 & 2 & 3\\
5 & 4 & 3 & 2 & 1 & 5 & 7 & 6\\
6 & 3 & 4 & 1 & 2 & 7 & 6 & 5\\
7 & 2 & 1 & 4 & 3 & 6 & 5 & 7
\end{array}
$$
show two different Steiner quasigroups of order $7$. None of them have left nor right identities. Such quasigroups of order $m>1$ exist for $m\equiv_6 1$ or $m\equiv_6 3$.
Another example of finite quasigroup (order $4$) with no element $e$ such that
$$ex=x=xe$$
for all $x$ is:
$$\begin{array}{c|cccc}
\bullet & 1 & 2 & 3 & 4\\ \hline
1 & 3 & 2 & 1 & 4\\
2 & 2 & 1 & 4 & 3\\
3 & 1 & 4 & 3 & 2\\
4 & 4 & 3 & 2 & 1\\
\end{array}
$$
of course a latin square as well.

Last but not least. In part two of this article you'll see a classical example of quasigroup over an infinite set, with no identity, having its roots and motivations not in abstract algebra but in geometry. The first time I saw this one I thought it was pretty different from any other quasigroup I had seen before and definitely not trivial...hope you like it!

I would like to make clear that we call quasigroup a groupoid (magma) with left and right division and cancellation laws, while we call loop a quasigroup with identity element.
