Given a magma M then the free magma ring (or magma ring for short) RM of M over a ring-with-one R is the set of finite sums åi Î I ri mi with ri Î R, and mi Î M. With the obvious addition and R-action from the left, RM is a free R-module with R-basis M, and with the usual convolution product, RM is a ring.
Typical examples of free magma rings are
The coefficient ring R and the magma M cannot be regarded as subsets of RM, hence the natural embeddings of R and M into RM must be handled via explicit embedding maps (see Natural Embeddings related to Magma Rings). Note that in a magma ring, the addition of elements is in general different from an addition that may be defined already for the elements of the magma; for example, the addition in the group ring of a matrix group does in general not coincide with the addition of matrices. Consider the following example.
gap> a:= Algebra( GF(2), [ [ [ Z(2) ] ] ] );; Size( a ); 2 gap> rm:= FreeMagmaRing( GF(2), a );; gap> emb:= Embedding( a, rm );; gap> z:= Zero( a );; o:= One( a );; gap> imz:= z ^ emb; IsZero( imz ); (Z(2)^0)*[ [ 0*Z(2) ] ] false gap> im1:= ( z + o ) ^ emb; (Z(2)^0)*[ [ Z(2)^0 ] ] gap> im2:= z ^ emb + o ^ emb; (Z(2)^0)*[ [ 0*Z(2) ] ]+(Z(2)^0)*[ [ Z(2)^0 ] ] gap> im1 = im2; false
FreeMagmaRing( R, M ) F
is a free magma ring over the ring R, free on the magma M.
GroupRing( R, G ) F
is the group ring of the group G, over the ring R.
IsFreeMagmaRing( D ) C
A domain lies in the category IsFreeMagmaRing if it has been
constructed as a free magma ring.
In particular, if D lies in this category then the operations
LeftActingDomain (see LeftActingDomain) and UnderlyingMagma
(see UnderlyingMagma) are applicable to D,
and yield the ring R and the magma M
such that D is the magma ring RM.
So being a magma ring in GAP includes the knowledge of the ring and the magma. Note that a magma ring RM may abstractly be generated as a magma ring by a magma different from the underlying magma M. For example, the group ring of the dihedral group of order 8 over the field with 3 elements is also spanned by a quaternion group of order 8 over the same field.
gap> d8:= DihedralGroup( 8 ); <pc group of size 8 with 3 generators> gap> rm:= FreeMagmaRing( GF(3), d8 ); <algebra-with-one over GF(3), with 3 generators> gap> emb:= Embedding( d8, rm );; gap> gens:= List( GeneratorsOfGroup( d8 ), x -> x^emb );; gap> x1:= gens[1] + gens[2];; gap> x2:= ( gens[1] - gens[2] ) * gens[3];; gap> x3:= gens[1] * gens[2] * ( One( rm ) - gens[3] );; gap> g1:= x1 - x2 + x3;; gap> g2:= x1 + x2;; gap> q8:= Group( g1, g2 );; gap> Size( q8 ); 8 gap> ForAny( [ d8, q8 ], IsAbelian ); false gap> List( [ d8, q8 ], g -> Number( AsList( g ), x -> Order( x ) = 2 ) ); [ 5, 1 ] gap> Dimension( Subspace( rm, q8 ) ); 8
IsFreeMagmaRingWithOne( obj ) C
IsGroupRing( obj ) P
A group ring is a magma ring where the underlying magma is a group.
UnderlyingMagma( RM ) A
AugmentationIdeal( RG ) A
is the augmentation ideal of the group ring RG, i.e., the kernel of the trivial representation of RG.
IsElementOfFreeMagmaRing( obj ) C
IsElementOfFreeMagmaRingCollection( obj ) C
IsElementOfFreeMagmaRingFamily( Fam ) C
Elements of families in this category have trivial normalisation, i.e.,
efficient methods for \= and \<.
In order to treat elements of free magma rings uniformly,
also without an external representation, the attributes
CoefficientsAndMagmaElements (see CoefficientsAndMagmaElements)
and ZeroCoefficient (see ZeroCoefficient)
were introduced that allow one to ``take an element of an arbitrary
magma ring into pieces''.
Conversely, for constructing magma ring elements from coefficients
and magma elements, ElementOfMagmaRing (see ElementOfMagmaRing)
can be used.
(Of course one can also embed each magma element into the magma ring,
see Natural Embeddings related to Magma Rings,
and then form the linear combination,
but many unnecessary intermediate elements are created this way.)
CoefficientsAndMagmaElements( elm ) A
is a list that contains at the odd positions the magma elements, and at the even positions their coefficients in the element elm.
ZeroCoefficient( elm ) A
For an element elm of a magma ring (modulo relations) RM,
ZeroCoefficient returns the zero element of the coefficient ring R.
ElementOfMagmaRing( Fam, zerocoeff, coeffs, mgmelms ) O
ElementOfMagmaRing returns the element åi=1n ci mi¢,
where coeffs = [ c1, c2, ¼, cn ] is a list of coefficients,
mgmelms = [ m1, m2, ¼, mn ] is a list of magma elements,
and mi¢ is the image of mi under an embedding
of a magma containing mi into a magma ring
whose elements lie in the family Fam.
zerocoeff must be the zero of the coefficient ring
containing the ci.
63.3 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
63.4 Magma Rings modulo Relations
A more general construction than that of free magma rings allows one to create rings that are not free R-modules on a given magma M but arise from the magma ring RM by factoring out certain identities. Examples for such structures are finitely presented (associative) algebras and free Lie algebras (see FreeLieAlgebra).
In GAP, the use of magma rings modulo relations is limited to
situations where a normal form of the elements is known and where
one wants to guarantee that all elements actually constructed are
in normal form.
(In particular, the computation of the normal form must be cheap.)
This is because the methods for comparing elements in magma rings
modulo relations via \= and \< just compare the involved
coefficients and magma elements,
and also the vector space functions regard those monomials as
linearly independent over the coefficients ring that actually occur
in the representation of an element of a magma ring modulo relations.
Thus only very special finitely presented algebras will be represented as magma rings modulo relations, in general finitely presented algebras are dealt with via the mechanism described in Chapter Finitely Presented Algebras.
IsElementOfMagmaRingModuloRelations( obj ) C
IsElementOfMagmaRingModuloRelationsCollection( obj ) C
This category is used, e. g., for elements of free Lie algebras.
IsElementOfMagmaRingModuloRelationsFamily( Fam ) C
NormalizedElementOfMagmaRingModuloRelations( F, descr ) O
Let F be a family of magma ring elements modulo relations,
and descr the description of an element in a magma ring modulo
relations.
NormalizedElementOfMagmaRingModuloRelations returns a description of
the same element,
but normalized w.r.t. the relations.
So two elements are equal if and only if the result of
NormalizedElementOfMagmaRingModuloRelations is equal for their internal
data, that is, CoefficientsAndMagmaElements will return the same
for the corresponding two elements.
NormalizedElementOfMagmaRingModuloRelations is allowed to return
descr itself, it need not make a copy.
This is the case for example in the case of free magma rings.
IsMagmaRingModuloRelations( obj ) C
A GAP object lies in the category IsMagmaRingModuloRelations
if it has been constructed as a magma ring modulo relations.
Each element of such a ring has a unique normal form,
so CoefficientsAndMagmaElements is well-defined for it.
This category is not inherited to factor structures, which are in general best described as finitely presented algebras, see Chapter Finitely Presented Algebras.
IsElementOfMagmaRingModuloSpanOfZeroFamily( Fam ) C
We need this for the normalization method, which takes a family as first argument.
IsMagmaRingModuloSpanOfZero( RM ) C
MagmaRingModuloSpanOfZero( R, M, z ) F
Let R be a ring, M a magma, and z an element of M with the
property that z * m = z for all m Î M.
The element z could be called a ``zero element'' of M,
but note that in general z cannot be obtained as Zero( m )
for each m Î M, so this situation does not match the definition of
Zero (see Zero).
MagmaRingModuloSpanOfZero returns the magma ring R M modulo
the relation given by the identification of z with zero.
This is an example of a magma ring modulo relations,
see Magma Rings modulo Relations.
63.6 Technical Details about the Implementation of Magma Rings
The family containing elements in the magma ring RM in fact contains all elements with coefficients in the family of elements of R and magma elements in the family of elements of M. So arithmetic operations with coefficients outside R or with magma elements outside M might create elements outside RM.
It should be mentioned that each call of FreeMagmaRing creates
a new family of elements,
so for example the elements of two group rings of permutation groups
over the same ring lie in different families and therefore are regarded
as different.
gap> g:= SymmetricGroup( 3 );; gap> h:= AlternatingGroup( 3 );; gap> IsSubset( g, h ); true gap> f:= GF(2);; gap> fg:= GroupRing( f, g ); <algebra-with-one over GF(2), with 2 generators> gap> fh:= GroupRing( f, h ); <algebra-with-one over GF(2), with 1 generators> gap> IsSubset( fg, fh ); false gap> o1:= One( fh ); o2:= One( fg ); o1 = o2; (Z(2)^0)*() (Z(2)^0)*() false gap> emb:= Embedding( g, fg );; gap> im:= Image( emb, h ); <group of size 3 with 1 generators> gap> IsSubset( fg, im ); true
There is no generic external representation for elements in an arbitrary free magma ring. For example, polynomials are elements of a free magma ring, and they have an external representation relying on the special form of the underlying monomials. On the other hand, elements in a group ring of a permutation group do not admit such an external representation.
For convenience, magma rings constructed with FreeAlgebra,
FreeAssociativeAlgebra, FreeAlgebraWithOne, and
FreeAssociativeAlgebraWithOne support an external representation of
their elements, which is defined as a list of length 2,
the first entry being the zero coefficient, the second being a list with
the external representations of the magma elements at the odd positions
and the corresponding coefficients at the even positions.
As the above examples show, there are several possible representations
of magma ring elements, the representations used for polynomials
(see Polynomials and Rational Functions)
as well as the default representation IsMagmaRingObjDefaultRep
of magma ring elements.
The latter simply stores the zero coefficient and a list containing
the coefficients of the element at the even positions
and the corresponding magma elements at the odd positions,
where the succession is compatible with the ordering of magma elements
via /.
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GAP 4 manual
May 2002