Steiner quasigroup

Example

Let $S$ be a set with a Steiner triple system on it, that is, a Steiner system $S(2, 3, n)$. Its blocks are $3$-element subsets of $S$, and every two distinct points lie in exactly one block. The Steiner quasigroup of this system is $S$ with the binary operation $\cdot$ given by:

1. $x \cdot x = x$ for every $x \in S$;
2. $x \cdot y = z$ for $x \ne y$, where $\set{x, y, z}$ is the unique block that contains $x$ and $y$.

This operation is commutative and idempotent, and it satisfies $x \cdot (x \cdot y) = y$ for all $x, y$. For $x \ne y$ the last law holds because $x \cdot y = z$ and $x \cdot z = y$ come from the same block.

Check that $S$ is a quasigroup, that is, every equation $a \cdot y = b$ and $y \cdot a = b$ has exactly one solution $y$. The law $x \cdot (x \cdot y) = y$ says that the map $y \mapsto a \cdot y$ is its own inverse, so it is a bijection. Hence the equation $a \cdot y = b$ has the unique solution $y = a \cdot b$. By commutativity the equation $y \cdot a = b$ has the same unique solution.