A crossed set is a quandle Q such that, for all x and y in Q, we have x * y = x if, and  only if, y * x = y.

The IsCrossedSet command returns true if the given magma is a crossed set.  It returns false otherwise.

Note that it is not assumed that the magma m is a quandle, so this check is applied first.

$\mathrm{with}\left(\mathrm{Magma}\right)\:$

$m≔⟨⟨⟨1\|3\|2⟩\,⟨3\|2\|1⟩\,⟨2\|1\|3⟩⟩⟩$

$m≔\left[\begin{array}{ccc}1& 3& 2\\ 3& 2& 1\\ 2& 1& 3\end{array}\right]$

$\mathrm{IsCrossedSet}\left(m\right)$

$\mathrm{true}$

$m≔⟨⟨⟨1\|1\|1⟩\,⟨3\|2\|2⟩\,⟨2\|3\|3⟩⟩⟩$

$m≔\left[\begin{array}{ccc}1& 1& 1\\ 3& 2& 2\\ 2& 3& 3\end{array}\right]$

$\mathrm{IsCrossedSet}\left(m\right)$

$\mathrm{false}$

The Magma[IsCrossedSet] command was introduced in Maple 16.

For more information on Maple 16 changes, see Updates in Maple 16.