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
coefficiants 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 fi nitely presented algebras, see Chapter Finitely Presented Algebras.
[Top] [Previous] [Up] [Next] [Index]
GAP 4 manual