Examples of quasigroups with no identity elements

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.