32.3 Magmas Defined by Multiplication Tables

For a square matrix A with n rows such that all entries of A are in the range [ 1 \.\. n ], MagmaByMultiplicationTable returns a magma M with multiplication * defined by A. That is, M consists of the elements m₁, m₂, ¼, m_n, and m_i \* m_j = m_A[i][j].

The ordering of elements is defined by m₁ < m₂ < ¼ < m_n, so m_i can be accessed as MagmaElement( M, i ), see MagmaElement.

The only differences between MagmaByMultiplicationTable and MagmaWithOneByMultiplicationTable are that the latter returns a magma-with-one (see MagmaWithOne) if the magma described by the matrix A has an identity, and returns fail if not.

The only differences between MagmaByMultiplicationTable and MagmaWithInversesByMultiplicationTable are that the latter returns a magma-with-inverses (see MagmaWithInverses) if each element in the magma described by the matrix A has an inverse, and returns fail if not.

For a magma M and a positive integer i, MagmaElement returns the i-th element of M, w.r.t. the ordering <. If M has less than i elements then fail is returned.

For a list elms of elements that form a magma M, MultiplicationTable returns a square matrix A of positive integers such that A[i][j] = k holds if and only if elms[i] * elms[j] = elms[k]. This matrix can be used to construct a magma isomorphic to M, using MagmaByMultiplicationTable.

gap> l:= [ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) ];;
gap> a:= MultiplicationTable( l );
[ [ 1, 2, 3, 4 ], [ 2, 1, 4, 3 ], [ 3, 4, 1, 2 ], [ 4, 3, 2, 1 ] ]
gap> m:= MagmaByMultiplicationTable( a );
<magma with 4 generators>
gap> One( m );
m1
gap> elm:= MagmaElement( m, 2 );  One( elm );  elm^2;
m2
m1
m1
gap> Inverse( elm );
m2
gap> AsGroup( m );
<group of size 4 with 2 generators>
gap> a:= [ [ 1, 2 ], [ 2, 2 ] ];
[ [ 1, 2 ], [ 2, 2 ] ]
gap> m:= MagmaByMultiplicationTable( a );
<magma with 2 generators>
gap> One( m );  Inverse( MagmaElement( m, 2 ) );     
m1
fail

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