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The Magma package is a collection of routines for working with small, finite magmas. A magma (sometimes also called a groupoid or a general binary system) is simply a set, together with a binary operation defined on that set.

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The following is a list of the commands in the main Magma package.

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A magma is a set together with a binary operation. Since, by transport of structure, every finite magma of order n is isomorphic to a magma on the set of integers from 1 to n, magmas are represented uniformly throughout the package by their Cayley tables as Arrays with entries from 1 to n. Thus, the (i,j) entry of the Cayley table is the result of multiplying the elements i and j.

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Arrays representing Cayley tables of magmas are stored in row-major order (C_order) and have datatype integer[4].

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Each command in the Magma package can be accessed by using either the long form or the short form of the command name in the command calling sequence. For example, if M is a magma you may use either Magma:-IsAssociative(M) or with(Magma); then IsAssociative(M).

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

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$\mathrm{Enumerate}\left(4\,'\mathrm{associative}'\,'\mathrm{commutative}'\right)$

$58$

(1)

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$\mathrm{Enumerate}\left(4\,'\mathrm{associative}'\,'\mathrm{commutative}'\,'\mathrm{identity}'\right)$

$19$

(2)

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$\mathrm{Q4}≔\mathrm{Enumerate}\left(4\,'\mathrm{quandle}'\,'\mathrm{output}'='\mathrm{list}'\right)$

$\mathrm{Q4}≔\left[\left[\begin{array}{cccc}1& 1& 1& 1\\ 2& 2& 2& 2\\ 3& 3& 3& 3\\ 4& 4& 4& 4\end{array}\right]\,\left[\begin{array}{cccc}1& 1& 1& 1\\ 2& 2& 2& 2\\ 3& 4& 3& 3\\ 4& 3& 4& 4\end{array}\right]\,\left[\begin{array}{cccc}1& 1& 1& 1\\ 2& 2& 2& 2\\ 4& 4& 3& 3\\ 3& 3& 4& 4\end{array}\right]\,\left[\begin{array}{cccc}1& 1& 1& 1\\ 2& 2& 4& 3\\ 3& 4& 3& 2\\ 4& 3& 2& 4\end{array}\right]\,\left[\begin{array}{cccc}1& 1& 1& 1\\ 3& 2& 2& 2\\ 4& 3& 3& 3\\ 2& 4& 4& 4\end{array}\right]\,\left[\begin{array}{cccc}1& 1& 2& 2\\ 2& 2& 1& 1\\ 4& 4& 3& 3\\ 3& 3& 4& 4\end{array}\right]\,\left[\begin{array}{cccc}1& 3& 4& 2\\ 4& 2& 1& 3\\ 2& 4& 3& 1\\ 3& 1& 2& 4\end{array}\right]\right]$

(3)

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$\mathrm{select}\left(\mathrm{IsAssociative}\,\mathrm{Q4}\right)$

$\left[\left[\begin{array}{cccc}1& 1& 1& 1\\ 2& 2& 2& 2\\ 3& 3& 3& 3\\ 4& 4& 4& 4\end{array}\right]\right]$

(4)

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The Magma package was introduced in Maple 15.

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