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Quellcode-Bibliothek algfp.gd
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Columbo aufrufen.gd Download desUnknown {[0] [0] [0]}Datei anzeigen
#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Alexander Hulpke.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the declarations for finitely presented algebras
##
#############################################################################
##
#C IsElementOfFpAlgebra
##
DeclareCategory( "IsElementOfFpAlgebra", IsRingElement );
#############################################################################
##
#C IsElementOfFpAlgebraCollection
##
DeclareCategoryCollections( "IsElementOfFpAlgebra" );
#############################################################################
##
#C IsElementOfFpAlgebraFamily
##
DeclareCategoryFamily( "IsElementOfFpAlgebra" );
#############################################################################
##
#C IsSubalgebraFpAlgebra
##
DeclareCategory( "IsSubalgebraFpAlgebra", IsAlgebra );
#############################################################################
##
#M IsSubalgebraFpAlgebra( <D> ) . for alg. that is coll. of f.p. alg. elms.
##
InstallTrueMethod( IsSubalgebraFpAlgebra,
IsAlgebra and IsElementOfFpAlgebraCollection );
#############################################################################
##
#P IsFullFpAlgebra( <A> )
##
## A f.~p. algebra is given by generators which are arithmetic expressions
## in terms of a set of generators $X$ of an f.~p. algebra that was
## constructed as a quotient of a free algebra.
##
## A *full f.~p. algebra* is a f.~p. algebra that contains $X$.
#T or better postulate ``contains this free algebra''?
## (So a full f.~p. algebra need *not* contain the whole family of its
## elements.)
##
DeclareProperty( "IsFullFpAlgebra",
IsFLMLOR and IsElementOfFpAlgebraCollection );
#T or do we need `IsAlgebraOfFamily', as in the case of f.p. groups?
#############################################################################
##
#O ElementOfFpAlgebra( <Fam>, <elm> )
##
DeclareOperation( "ElementOfFpAlgebra",
[ IsElementOfFpAlgebraFamily, IsRingElement ] );
############################################################################
##
#O MappedExpression( <expr>, <gens1>, <gens2> )
##
## For an arithmetic expression <expr> in terms of the generators <gens1>,
## `MappedExpression' returns the corresponding expression in terms of
## <gens2>.
##
## Note that it is expected that one can raise elements in <gens2> to the
## zero-th power.
##
DeclareOperation( "MappedExpression",
[ IsElementOfFpAlgebra, IsHomogeneousList, IsHomogeneousList ] );
#############################################################################
##
#F FactorFreeAlgebraByRelators( <F>, <rels> ) . . . factor of free algebra
##
## is an f.p.~algebra $A$ isomorphic to the factor of the free (associative)
## algebra(-with-one) <F> by the two-sided ideasl spanned by the relators
## in the list <rels>.
##
## If <F> is an algebra-with-one then the generators in the list
## `GeneratorsOfAlgebraWithOne( $A$ )' correspond to the generators in the
## list `GeneratorsOfAlgebraWithOne( <F> )'.
## Otherwise the generators in the list `GeneratorsOfAlgebra( $A$ )'
## correspond to the generators in the list `GeneratorsOfAlgebra( <F> )'.
##
DeclareGlobalFunction( "FactorFreeAlgebraByRelators" );
#############################################################################
##
#A FreeGeneratorsOfFpAlgebra( <A> )
##
## is the list of underlying free generators corresponding to the generators
## of the finitely presented algebra <A>.
##
DeclareAttribute( "FreeGeneratorsOfFpAlgebra",
IsSubalgebraFpAlgebra and IsFullFpAlgebra );
############################################################################
##
#A RelatorsOfFpAlgebra( <A> )
##
## is the list of relators of the finitely presented algebra <A>,
## each relator being an expression in terms of the free generators
## provided by `FreeGeneratorsOfFpAlgebra( <A> )'.
##
DeclareAttribute( "RelatorsOfFpAlgebra",
IsSubalgebraFpAlgebra and IsFullFpAlgebra );
#############################################################################
##
#A FreeAlgebraOfFpAlgebra( <A> )
##
## is the underlying free algebra of the finitely presented algebra <A>.
## This is the algebra generated by the free generators provided by
## `FreeGeneratorsOfFpAlgebra( <A> )',
## with coefficient domain `LeftActingDomain( <A> )'.
##
DeclareAttribute( "FreeAlgebraOfFpAlgebra",
IsSubalgebraFpAlgebra and IsFullFpAlgebra );
#############################################################################
##
#P IsNormalForm( <elm> )
##
DeclareProperty( "IsNormalForm", IsObject );
#############################################################################
##
#A NiceNormalFormByExtRepFunction( <Fam> )
##
## `NiceNormalFormByExtRepFunction( <Fam> )' is a function that can be
## applied to the family <Fam> and the external representation of an element
## $e$ of <Fam>;
## This call returns the element of <Fam> that is equal to $e$ and in normal
## form.
##
## If the family <Fam> knows a nice normal form for its elements then the
## elements can be always constructed as normalized elements by
## `NormalizedObjByExtRep'.
##
## (Perhaps a normal form that is expensive to compute will not be regarded
## as a nice normal form.)
##
DeclareAttribute( "NiceNormalFormByExtRepFunction", IsFamily );
#############################################################################
##
#A NiceAlgebraMonomorphism( <A> )
##
## <#GAPDoc Label="NiceAlgebraMonomorphism">
## <ManSection>
## <Attr Name="NiceAlgebraMonomorphism" Arg='A'/>
## <Description>
## If <A>A</A> is an associative algebra with one, returns
## an isomorphism from <A>A</A> onto a matrix algebra
## (see <Ref Attr="IsomorphismMatrixAlgebra"/> for an example).
## If <A>A</A> is a finitely presented Lie algebra, returns an isomorphism
## from <A>A</A> onto a Lie algebra defined by a structure constants table
## (see <Ref Sect="Finitely Presented Lie Algebras"/> for an example).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
## Let <A> be a subspace or subalgebra of a f.p.~algebra.
##
## The `NiceAlgebraMonomorphism' value of the algebra stored in
## the `wholeFamily' component of the elements family of <A>
## is used to define the `\<' relation of algebra elements.
#T use it also for a ``nice normal form''!
##
## If a f.p.~algebra <A> knows the value of `NiceAlgebraMonomorphism'
## then it can be handled via the mechanism of nice bases
## (see~"IsFpAlgebraElementsSpace").
##
## `NiceAlgebraMonomorphism' is inherited to subalgebras and subspaces.
## (If one knows that <A> contains the source then one should set
## `GeneratorsOfLeftModule' for <A>,
## and also one can set `NiceFreeLeftModule' for <A> to the module
## of the basis `basisimages'.)
##
DeclareAttribute( "NiceAlgebraMonomorphism", IsSubalgebraFpAlgebra );
InstallSubsetMaintenance( NiceAlgebraMonomorphism,
IsFreeLeftModule and HasNiceAlgebraMonomorphism, IsFreeLeftModule );
#############################################################################
##
#F IsFpAlgebraElementsSpace( <V> )
##
## If an $F$-vector space <V> is in the filter `IsFpAlgebraElementsSpace'
## then this expresses that <V> consists of elements in a f.p.~algebra,
## and that <V> is handled via the mechanism of nice bases (see~"...")
## in the following way.
## Let $f$ be the `NiceAlgebraMonomorphism' value for <V>
## (see~"NiceAlgebraMonomorphism").
## The `NiceVector' value of $v \in <V>$ is defined as
## $`ImagesRepresentative'( f, v )$;
## the `NiceFreeLeftModuleInfo' value of <V> is irrelevant.
##
## So it is assumed that methods for computing the `NiceAlgebraMonomorphism'
## value are known.
##
DeclareHandlingByNiceBasis( "IsFpAlgebraElementsSpace",
"for free left modules of f.p. algebra elements" );
#############################################################################
##
#F FpAlgebraByGeneralizedCartanMatrix( <F>, <A> )
##
## is a finitely presented associative algebra over the field <F>,
## defined by the generalized Cartan matrix <A>.
##
DeclareGlobalFunction( "FpAlgebraByGeneralizedCartanMatrix" );
[ 0.108Quellennavigators
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2026-03-28
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