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?

A finite quasigroup is essentially a Latin square used as a "multiplication" table.  Consider for $n \gt 2$ a Latin square, and label the rows (resp. columns) with a permutation of the symbols not appearing in any row (resp. in any column).  This determines a quasigroup without identity, if the entries of the Latin square are considered the result of the binary operation on the row symbol and column symbol assigned to that entry.
The rows and columns may then be permuted to any common order you please, and for symbol set $\{1,2,3\}$ a specific example of the Latin square with rows and columns in the canonical order would be:
$$ \begin{bmatrix} 1 & 3 & 2 \\ 3 & 2 & 1 \\ 2 & 1 & 3 \end{bmatrix} $$
In this fashion $2 * 3 = 1$, but no element acts as a left (resp. a right) identity.
