A left quasigroup is a magma in which the rows of the Cayley table are permutations of the elements.

The IsLeftQuasigroup command returns true if the given magma is a left quasigroup, and returns false otherwise.

A right quasigroup is a magma in which the columns of the Cayley table are permutations of the elements.

The IsRightQuasigroup command returns true if the given magma is a right quasigroup, and returns false otherwise.

A quasigroup is a magma whose Cayley table is a Latin square.  That is, each row and column is a permutation of the integers from 1 to n, where n is the order of the magma.  Thus, a magma is a quasigroup if, and only if, it is both a left quasigroup and a right quasigroup.

The IsQuasigroup command returns true if the given magma is a quasigroup, and returns false otherwise.

The Magma[IsQuasigroup] command is thread-safe as of Maple 16.

The Magma[IsLeftQuasigroup] and Magma[IsRightQuasigroup] commands are thread-safe as of Maple 17.

For more information on thread safety, see index/threadsafe.

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

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

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

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

$\mathrm{true}$

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

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

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

$\mathrm{false}$

The Magma[IsQuasigroup], Magma[IsLeftQuasigroup] and Magma[IsRightQuasigroup] commands were introduced in Maple 15.

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