Heap

Remarks

Example: a group $G$ with $[x, y, z] = x y^{-1} z$. For clause 1, all three expressions equal $a b^{-1} c d^{-1} e$; the third one because $(d c^{-1} b)^{-1} = b^{-1} c d^{-1}$. For clause 2, $a a^{-1} b = b$ and $b a^{-1} a = b$.

Non-example: $\Z$ with $[x, y, z] = x + y + z$. Clause 1 holds, since all three expressions equal $a + b + c + d + e$. Clause 2 fails: $[0, 1, 1] = 2 \ne 0$.

Conversely, let $H$ be a nonempty heap and fix $e \in H$. Then $x \cdot y = [x, e, y]$ makes $H$ a group with identity $e$ and inverse $x^{-1} = [e, x, e]$, and the heap operation is recovered as $[x, y, z] = x y^{-1} z$. Different choices of $e$ give isomorphic groups. Some authors allow the empty heap, which corresponds to no group.