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Magma

 Nucleus
 compute the nucleus of a magma

 Calling Sequence Nucleus( m )

Parameters

 m - Array representing the Cayley table of a finite magma

Description

 • The nucleus of a magma is the set of its members that associate with every member of the magma. That is, an element x is in the nucleus if all of the equations (xy)z = x(yz), (yx)z = y(xz) and (yz)x = y(zx) are satisfied, for all y and z from the magma.
 • The Nucleus command returns the nucleus of the magma m as a set.

Examples

 > with( Magma ):
 > 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)
 > Nucleus( m );
 $\left\{{1}{,}{2}{,}{3}\right\}$ (2)
 > m := << 1, 1, 1, 1, 1; 1, 1, 1, 1, 1 ; 1, 1, 1, 1, 1 ; 1, 1, 1, 2, 4 ; 1, 2, 2, 4, 5 >>;
 ${m}{≔}\left[\begin{array}{ccccc}{1}& {1}& {1}& {1}& {1}\\ {1}& {1}& {1}& {1}& {1}\\ {1}& {1}& {1}& {1}& {1}\\ {1}& {1}& {1}& {2}& {4}\\ {1}& {2}& {2}& {4}& {5}\end{array}\right]$ (3)
 > Nucleus( m );
 $\left\{{1}{,}{2}{,}{3}\right\}$ (4)
 > m := << 1, 1, 1, 1, 1 ; 1, 1, 1, 1, 1 ; 1, 1, 1, 1, 1 ; 1, 1, 1, 2, 4 ; 1, 2, 1, 4, 5 >>;
 ${m}{≔}\left[\begin{array}{ccccc}{1}& {1}& {1}& {1}& {1}\\ {1}& {1}& {1}& {1}& {1}\\ {1}& {1}& {1}& {1}& {1}\\ {1}& {1}& {1}& {2}& {4}\\ {1}& {2}& {1}& {4}& {5}\end{array}\right]$ (5)
 > Nucleus( m );
 $\left\{{1}{,}{2}{,}{3}\right\}$ (6)

Compatibility

 • The Magma[Nucleus] command was introduced in Maple 15.