IsLeftQuasigroup - Maple Help

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Magma

 IsQuasigroup
 test whether a finite magma is a quasigroup
 IsLeftQuasigroup
 test whether a finite magma is a left quasigroup
 IsRightQuasigroup
 test whether a finite magma is a right quasigroup

 Calling Sequence IsQuasigroup( m ) IsLeftQuasigroup( m ) IsRightQuasigroup( m )

Parameters

 m - Array representing the Cayley table of a finite magma

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Magma}\right):$
 > $m≔⟨⟨⟨1|2|3⟩,⟨2|3|1⟩,⟨3|1|2⟩⟩⟩$
 ${m}{≔}\left[\begin{array}{ccc}{1}& {2}& {3}\\ {2}& {3}& {1}\\ {3}& {1}& {2}\end{array}\right]$ (1)
 > $\mathrm{IsQuasigroup}\left(m\right)$
 ${\mathrm{true}}$ (2)
 > $m≔⟨⟨⟨1|2|3⟩,⟨2|3|3⟩,⟨3|1|2⟩⟩⟩$
 ${m}{≔}\left[\begin{array}{ccc}{1}& {2}& {3}\\ {2}& {3}& {3}\\ {3}& {1}& {2}\end{array}\right]$ (3)
 > $\mathrm{IsQuasigroup}\left(m\right)$
 ${\mathrm{false}}$ (4)

Compatibility

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