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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

 > $\mathrm{with}\left(\mathrm{Magma}\right):$
 > $m≔⟨⟨⟨1|2|3⟩,⟨2|3|1⟩,⟨3|1|2⟩⟩⟩$
 ${m}{≔}\left[\begin{array}{ccc}{1}& {2}& {3}\\ {2}& {3}& {1}\\ {3}& {1}& {2}\end{array}\right]$ (1)
 > $\mathrm{Nucleus}\left(m\right)$
 $\left\{{1}{,}{2}{,}{3}\right\}$ (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)
 > $\mathrm{Nucleus}\left(m\right)$
 $\left\{{1}{,}{2}{,}{3}\right\}$ (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)
 > $\mathrm{Nucleus}\left(m\right)$
 $\left\{{1}{,}{2}{,}{3}\right\}$ (6)

Compatibility

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