DirectProduct - Maple Help

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Magma

 DirectProduct
 compute the direct product of two magmas

 Calling Sequence DirectProduct( A, B )

Parameters

 A - Array representing the Cayley table of a finite magma B - Array representing the Cayley table of a finite magma

Description

 • The direct product of two magmas A and B is the set of pairs (a,b), with a in A and b in B, and with binary operation defined componentwise.
 • The DirectProduct( A, B ) command returns the Cayley table of the direct product of the magmas A and B.

Examples

 > $\mathrm{with}\left(\mathrm{Magma}\right):$
 > $A≔⟨⟨⟨1|2⟩,⟨2|1⟩⟩⟩$
 ${A}{≔}\left[\begin{array}{cc}{1}& {2}\\ {2}& {1}\end{array}\right]$ (1)
 > $B≔⟨⟨⟨1|1|2⟩,⟨2|3|2⟩,⟨3|2|1⟩⟩⟩$
 ${B}{≔}\left[\begin{array}{ccc}{1}& {1}& {2}\\ {2}& {3}& {2}\\ {3}& {2}& {1}\end{array}\right]$ (2)
 > $\mathrm{DirectProduct}\left(A,B\right)$
 $\left[\begin{array}{cccccc}{1}& {1}& {2}& {4}& {4}& {5}\\ {2}& {3}& {2}& {5}& {6}& {5}\\ {3}& {2}& {1}& {6}& {5}& {4}\\ {4}& {4}& {5}& {1}& {1}& {2}\\ {5}& {6}& {5}& {2}& {3}& {2}\\ {6}& {5}& {4}& {3}& {2}& {1}\end{array}\right]$ (3)
 > $\mathrm{AreIsomorphic}\left(\mathrm{DirectProduct}\left(A,B\right),\mathrm{DirectProduct}\left(B,A\right)\right)$
 ${\mathrm{true}}$ (4)
 > $C≔⟨⟨⟨1|2|3⟩,⟨2|3|1⟩,⟨3|1|2⟩⟩⟩$
 ${C}{≔}\left[\begin{array}{ccc}{1}& {2}& {3}\\ {2}& {3}& {1}\\ {3}& {1}& {2}\end{array}\right]$ (5)
 > $\mathrm{AreIsomorphic}\left(\mathrm{DirectProduct}\left(\mathrm{DirectProduct}\left(A,B\right),C\right),\mathrm{DirectProduct}\left(A,\mathrm{DirectProduct}\left(B,C\right)\right)\right)$
 ${\mathrm{true}}$ (6)

Compatibility

 • The Magma[DirectProduct] command was introduced in Maple 2016.