A loop is a quasigroup with an identity element.

The IsLoop command returns true if the given magma is a loop, and returns false otherwise.

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

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

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

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

$\mathrm{true}$

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

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

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

$\mathrm{false}$

The Magma[IsLoop] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.