61.3 Natural Embeddings related to Magma Rings

Neither the coefficient ring R nor the magma M are regarded as subsets of the magma ring RM, so one has to use embeddings (see Embedding) explicitly whenever one needs for example the magma ring element corresponding to a given magma element. Here is an example.

gap> f:= Rationals;;  g:= SymmetricGroup( 3 );;
gap> fg:= FreeMagmaRing( f, g );
<algebra-with-one over Rationals, with 2 generators>
gap> Dimension( fg );
6
gap> gens:= GeneratorsOfAlgebraWithOne( fg );
[ (1)*(1,2,3), (1)*(1,2) ]
gap> ( 3*gens[1] - 2*gens[2] ) * ( gens[1] + gens[2] );
(-2)*()+(3)*(2,3)+(3)*(1,3,2)+(-2)*(1,3)
gap> One( fg );
(1)*()
gap> emb:= Embedding( g, fg );;
gap> elm:= (1,2,3)^emb;  elm in fg;
(1)*(1,2,3)
true
gap> new:= elm + One( fg );
(1)*()+(1)*(1,2,3)
gap> new^2;
(1)*()+(2)*(1,2,3)+(1)*(1,3,2)
gap> emb2:= Embedding( f, fg );;
gap> elm:= One( f )^emb2;  elm in fg;
(1)*()
true

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