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#############################################################################
##
#W fixedrep.gd GAP4 Package `ResClasses' Stefan Kohl
##
## This file contains declarations needed for computing with unions of
## residue classes with distinguished ("fixed") representatives.
##
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##
#C IsUnionOfResidueClassesWithFixedRepresentatives
##
## The category of all unions of residue classes which are endowed with
## distinguished ('fixed') representatives.
##
DeclareCategory( "IsUnionOfResidueClassesWithFixedRepresentatives",
IsDomain and IsListOrCollection );
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##
#C IsUnionOfResidueClassesOfZWithFixedRepresentatives
#C IsUnionOfResidueClassesOfZ_piWithFixedRepresentatives
#C IsUnionOfResidueClassesOfGFqxWithFixedRepresentatives
#C IsUnionOfResidueClassesOfZorZ_piWithFixedRepresentatives
##
## The category of the unions of residue classes with fixed representatives
## of Z, Z_(pi) or GF(q)[x], respectively. The last-mentioned category
## is the union of the first-mentioned two categories.
##
DeclareCategory( "IsUnionOfResidueClassesOfZWithFixedRepresentatives",
IsDomain and IsListOrCollection );
DeclareCategory( "IsUnionOfResidueClassesOfZ_piWithFixedRepresentatives",
IsDomain and IsListOrCollection );
DeclareCategory( "IsUnionOfResidueClassesOfGFqxWithFixedRepresentatives",
IsDomain and IsListOrCollection );
DeclareCategory( "IsUnionOfResidueClassesOfZorZ_piWithFixedRepresentatives",
IsDomain and IsListOrCollection );
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##
#R IsUnionOfResidueClassesWithFixedRepsStandardRep
##
## The representation of unions of residue classes with fixed
## representatives -- of the integers, of semilocalizations Z_(pi)
## of the integers and of univariate polynomial rings GF(q)[x]
## over finite fields.
##
## The component <classes> is a list of residue classes, given as pairs
## ( <modulus>, <representative> ). The representatives are *not* reduced
## modulo the respective moduli, and the moduli may be different --
## in contrast to `IsResidueClassUnionResidueListRep' no common modulus is
## stored, and *unions which are equal as sets are distinguished* if their
## respective <classes> - components are distinct.
##
DeclareRepresentation( "IsUnionOfResidueClassesWithFixedRepsStandardRep",
IsComponentObjectRep and IsAttributeStoringRep,
[ "classes" ] );
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##
#O UnionOfResidueClassesWithFixedRepresentativesCons(<filter>,<R>,<classes>)
#F UnionOfResidueClassesWithFixedRepresentatives( <R>, <classes> )
#F UnionOfResidueClassesWithFixedRepresentatives( <classes> )
#F UnionOfResidueClassesWithFixedReps( <R>, <classes> )
#F UnionOfResidueClassesWithFixedReps( <classes> )
##
## The constructor for unions of residue classes with fixed representatives.
##
## Returns the union of the residue classes
##
## <classes>[i][2] ( mod <classes>[i][1] )
##
## of the ring <R>.
##
DeclareConstructor( "UnionOfResidueClassesWithFixedRepresentativesCons",
[ IsUnionOfResidueClassesWithFixedRepresentatives,
IsRing, IsList ] );
DeclareGlobalFunction( "UnionOfResidueClassesWithFixedRepresentatives" );
DeclareSynonym( "ResidueClassUnionWithFixedRepresentatives",
UnionOfResidueClassesWithFixedRepresentatives );
DeclareSynonym( "UnionOfResidueClassesWithFixedReps",
UnionOfResidueClassesWithFixedRepresentatives );
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##
#F ResidueClassWithFixedRepresentative( <R>, <m>, <r> ) rc. with fixed rep.
#F ResidueClassWithFixedRepresentative( <m>, <r> ) . . . . . . . . . (dito)
#F ResidueClassWithFixedRep( <R>, <m>, <r> ) . . . . . . . . . . . . (dito)
#F ResidueClassWithFixedRep( <m>, <r> ) . . . . . . . . . . . . . . (dito)
#P IsResidueClassWithFixedRepresentative( <obj> ) . corresponding property
#P IsResidueClassWithFixedRep( <obj> ) . . . . . . . . . . . . . . . (dito)
##
## Returns the residue class <r> ( mod <m> ) of the ring <R>,
## with the fixed representative <r>.
##
## If the argument <R> is not given, it defaults to `Integers'.
## Residue classes with fixed representatives have the property
## `IsResidueClassWithFixedRepresentative'.
##
DeclareGlobalFunction( "ResidueClassWithFixedRepresentative" );
DeclareSynonym( "ResidueClassWithFixedRep",
ResidueClassWithFixedRepresentative );
DeclareProperty( "IsResidueClassWithFixedRepresentative", IsObject );
DeclareSynonym( "IsResidueClassWithFixedRep",
IsResidueClassWithFixedRepresentative );
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##
#O Modulus( <U> ) . . modulus of a union of residue classes with fixed reps
##
DeclareOperation( "Modulus",
[ IsUnionOfResidueClassesWithFixedRepresentatives ] );
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##
#O Classes( <U> )
##
## Returns the list of pairs ( <representative>, <modulus> )
## as passed as argument <classes> when constructing <U> via
## `UnionOfResidueClassesWithFixedRepresentatives'.
##
DeclareOperation( "Classes",
[ IsUnionOfResidueClassesWithFixedRepresentatives ] );
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##
#O Modulus( <cl> ) . . . modulus of residue class with fixed representative
#O Residue( <cl> ) . . . residue of residue class with fixed representative
##
DeclareOperation( "Modulus", [ IsResidueClassWithFixedRepresentative ] );
DeclareOperation( "Residue", [ IsResidueClassWithFixedRepresentative ] );
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##
#F AllResidueClassesWithFixedRepresentativesModulo( [ <R>, ] <m> )
#F AllResidueClassesWithFixedRepsModulo( [ <R>, ] <m> )
##
## Returns a list of all residue classes (mod <m>) of the ring <R>,
## with fixed representatives.
##
## The fixed representatives are the same as those returned by the
## function `AllResidues' when called with the arguments <R> and <m>.
##
DeclareGlobalFunction( "AllResidueClassesWithFixedRepresentativesModulo" );
DeclareSynonym( "AllResidueClassesWithFixedRepsModulo",
AllResidueClassesWithFixedRepresentativesModulo );
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##
#O AsListOfClasses( <U> )
##
## Returns a list of the classes which form the union <U> of residue classes
## with fixed representatives.
##
DeclareOperation( "AsListOfClasses",
[ IsUnionOfResidueClassesWithFixedRepresentatives ] );
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##
#O Multiplicity( <n>, <U> ) . . . . . . . . . . multiplicity of <n> in <U>
#O Multiplicity( <cl>, <U> ) . . . . . . . . . . multiplicity of <cl> in <U>
##
## Returns the multiplicity of the ring element <n> / the residue class <cl>
## in the residue class union <U> with fixed representatives.
##
DeclareOperation( "Multiplicity",
[ IsObject,
IsUnionOfResidueClassesWithFixedRepresentatives ] );
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##
#P IsOverlappingFree( <U> )
##
## We call a residue class union <U> with fixed representatives
## *overlapping free* if and only if it consists of pairwisely disjoint
## residue classes.
##
DeclareProperty( "IsOverlappingFree",
IsUnionOfResidueClassesWithFixedRepresentatives );
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##
#O AsOrdinaryUnionOfResidueClasses( <U> )
##
## Returns the set-theoretic union of the residue classes in the union <U>
## of residue classes with fixed representatives. The returned object is
## an ordinary residue class union without fixed representatives which
## behaves like a subset of the underlying ring.
##
DeclareOperation( "AsOrdinaryUnionOfResidueClasses",
[ IsUnionOfResidueClassesWithFixedRepresentatives ] );
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##
#O RepresentativeStabilizingRefinement( <U>, <k> ) . . . . refinement of <U>
##
## Returns the representative stabilizing refinement of <U> into <k> parts.
##
## We define the *representative stabilizing refinement* of a residue class
## [r/m] of Z with fixed representative into k parts by [r/km] U [(r+m)/km]
## U ... U [(r+(k-1)m)/km].
##
## We extend this definition in a natural way to unions of residue classes.
##
## If the argument <k> is zero, it is tried to simplify <U> by iteratively
## uniting residue classes by the reverse of the process described above.
##
DeclareOperation( "RepresentativeStabilizingRefinement",
[ IsUnionOfResidueClassesWithFixedRepresentatives,
IsRingElement ] );
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##
#A Delta( <U> ) . . . . . . . . . . . . . . . . . . . . . invariant `Delta'
##
## For a residue class [r/m] with fixed representative we set
## Delta([r/m]) := r/m - 1/2 and extend this definition additively to unions
## of such residue classes.
##
## If no representatives are fixed, this definition is still unique (mod 1).
##
DeclareAttribute( "Delta", IsUnionOfResidueClassesWithFixedRepresentatives );
DeclareAttribute( "Delta", IsResidueClassUnion );
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##
#A Rho( <U> ) . . . . . . . . . . . . . . . . . . . . . . . invariant `Rho'
##
## For a residue class [r/m] with fixed representative and
## fixed orientation, i.e. fixed sign of m, we set
## Rho([r/m]) := exp(sgn(m)*Delta([r/m])/2) and extend this definition in
## the obvious way to unions of such residue classes.
##
## If no representatives and no orientation is fixed, this definition
## can still be made unique by restricting the exponent to the interval
## [0,1/2[.
##
DeclareAttribute( "Rho", IsUnionOfResidueClassesWithFixedRepresentatives );
DeclareAttribute( "Rho", IsResidueClassUnion );
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##
#E fixedrep.gd . . . . . . . . . . . . . . . . . . . . . . . . . . ends here
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