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#############################################################################
##
#W resclass.gd GAP4 Package `ResClasses' Stefan Kohl
##
## This file contains declarations needed for computing with unions of
## residue classes +/- finite sets ("residue class unions").
##
#############################################################################
#############################################################################
##
#C IsResidueClassUnion . . . . . . . . . . . . . . . . residue class unions
#C IsUnionOfResidueClasses
##
## The category of all unions of residue classes +/- finite sets, hence of
## all "residue class unions".
##
DeclareCategory( "IsResidueClassUnion", IsDomain and IsListOrCollection );
DeclareSynonym( "IsUnionOfResidueClasses", IsResidueClassUnion );
#############################################################################
##
#C IsResidueClassUnionOfZ . . . . . . . . . . . . residue class unions of Z
#C IsUnionOfResidueClassesOfZ ditto
#C IsResidueClassUnionOfZxZ " of Z^2
#C IsUnionOfResidueClassesOfZxZ ditto
#C IsResidueClassUnionOfZ_pi " of Z_(pi)
#C IsUnionOfResidueClassesOfZ_pi ditto
#C IsResidueClassUnionOfGFqx " of GF(q)[x]
#C IsUnionOfResidueClassesOfGFqx ditto
#C IsResidueClassUnionOfZorZ_pi " of Z or Z_(pi)
#C IsUnionOfResidueClassesOfZorZ_pi ditto
##
## The category of the residue class unions of Z, of Z^2, of a (semi-)
## localization Z_(pi) of Z or of a polynomial ring GF(q)[x] over a finite
## field, respectively. The last-mentioned category is the union of the
## first-mentioned and the third-mentioned one.
##
DeclareCategory( "IsResidueClassUnionOfZ",
IsDomain and IsListOrCollection );
DeclareSynonym( "IsUnionOfResidueClassesOfZ", IsResidueClassUnionOfZ );
DeclareCategory( "IsResidueClassUnionOfZxZ",
IsDomain and IsListOrCollection );
DeclareSynonym( "IsUnionOfResidueClassesOfZxZ", IsResidueClassUnionOfZxZ );
DeclareCategory( "IsResidueClassUnionOfZ_pi",
IsDomain and IsListOrCollection );
DeclareSynonym( "IsUnionOfResidueClassesOfZ_pi", IsResidueClassUnionOfZ_pi );
DeclareCategory( "IsResidueClassUnionOfGFqx",
IsDomain and IsListOrCollection );
DeclareSynonym( "IsUnionOfResidueClassesOfGFqx", IsResidueClassUnionOfGFqx );
DeclareCategory( "IsResidueClassUnionOfZorZ_pi",
IsDomain and IsListOrCollection );
DeclareSynonym( "IsUnionOfResidueClassesOfZorZ_pi",
IsResidueClassUnionOfZorZ_pi );
#############################################################################
##
#R IsResidueClassUnionResidueListRep . . representation by list of residues
##
## The representation of unions of residue classes of the ring Z of the
## integers, of Z^2, of semilocalizations Z_(pi) of the integers and of
## univariate polynomial rings GF(q)[x] over finite fields, by list of
## residues.
##
## Components:
##
## - <m>: the modulus
## - <r>: the list of class representatives
## - <included>: set of elements added to the union of residue classes
## - <excluded>: set of elements subtracted from " " "
##
## The representation is unique, i.e. two residue class unions are equal
## if and only if their stored representations are equal.
##
## This is achieved in the following way:
##
## - If the underlying ring R is Z, Z_(pi) or GF(q)[x], the modulus <m> is
## chosen to be multiplicatively minimal, and to be its own standard
## associate in R. If R is Z^2, the modulus <m> is a lattice, which is
## stored as an invertible 2x2 matrix whose rows are the spanning vectors.
## That matrix is chosen to be in Hermite normal form and to have the
## smallest possible determinant.
## - The set <included> contains only elements which are not congruent to
## one of the residues in <r> modulo <m>.
## - The set <excluded> contains only elements which are congruent to one
## of the residues in <r> modulo <m>.
##
DeclareRepresentation( "IsResidueClassUnionResidueListRep",
IsComponentObjectRep and IsAttributeStoringRep,
[ "m", "r", "included", "excluded" ] );
#############################################################################
##
#R IsResidueClassUnionClassListRep . . . . representation by list of classes
##
## The representation of unions of residue classes of the ring Z of the
## integers, of Z^2, of semilocalizations Z_(pi) of the integers and of
## univariate polynomial rings GF(q)[x] over finite fields, by normalized
## list of disjoint residue classes.
##
## Components:
##
## - <cls>: the list of residue classes, as pairs [ residue, modulus ]
## - <m>: the modulus
## - <included>: set of elements added to the union of residue classes
## - <excluded>: set of elements subtracted from " " "
##
## The representation is unique, i.e. two residue class unions are equal
## if and only if their stored representations are equal.
##
## This is achieved in the following way:
##
## - The list <cls> of residue classes is sorted, firstly by ascending
## moduli and secondly by ascending residues.
## - The list of residue classes is always as short as possible, and the
## list of their moduli is always lexicographically minimal, i.e. for
## example 0(3) U 1(6) U 5(6) is not represented as [[0,3],[1,6],[5,6]],
## but rather as [[1,2],[0,6]].
## - Residue classes with the same modulus are ordered by ascending residue.
## - The set <included> contains only elements which are neither in one of
## the residue classes in the list <cls> nor in <excluded>.
## - The set <excluded> contains only elements which are neither in one of
## the residue classes in the list <cls> nor in <included>.
##
## At present, this representation is only available for residue class
## unions of Z.
##
DeclareRepresentation( "IsResidueClassUnionClassListRep",
IsComponentObjectRep and IsAttributeStoringRep,
[ "cls", "m", "included", "excluded" ] );
#############################################################################
##
#O ClassListRep( <U> )
#O SparseRep( <U> )
#O SparseRepresentation( <U> )
#O ResidueListRep( <U> )
#O StandardRep( <U> )
#O StandardRepresentation( <U> )
##
## Conversion between the two representations of residue class unions:
## `IsResidueClassUnionResidueListRep' and `IsResidueClassUnionClassListRep'
##
## Remark: the declarations are for `IsObject' since `SparseRep' and
## `StandardRep' are used in RCWA for rcwa mappings as well.
##
DeclareOperation( "ClassListRep", [ IsObject] );
DeclareSynonym( "SparseRep", ClassListRep );
DeclareSynonym( "SparseRepresentation", ClassListRep );
DeclareOperation( "ResidueListRep", [ IsObject ] );
DeclareSynonym( "StandardRep", ResidueListRep );
DeclareSynonym( "StandardRepresentation", ResidueListRep );
#############################################################################
##
#R IsResidueClassUnionsIteratorRep . . . . . . . . . iterator representation
##
DeclareRepresentation( "IsResidueClassUnionsIteratorRep",
IsComponentObjectRep,
[ "U", "counter", "element", "classpos" ] );
#############################################################################
##
#O ResidueClassUnionCons( <filter>, <R>, <m>, <r>, <included>, <excluded> )
#O ResidueClassUnionCons( <filter>, <R>, <cls> <included>, <excluded> )
#F ResidueClassUnion( [ <R>, ] <m>, <r>, [ <included>, <excluded> ] )
#F ResidueClassUnion( [ <R>, ] <cls>, [ <included>, <excluded> ] )
#F ResidueClassUnionNC( [ <R>, ] <m>, <r>, [ <included>, <excluded> ] )
#F ResidueClassUnionNC ( [ <R>, ] <cls>, [ <included>, <excluded> ] )
##
## The constructor for *residue class unions*,
## i.e. set-theoretic unions of residue classes +/- finite sets of elements.
##
## Returns
## - the union of the residue classes <r>[i] ( mod <m> ), respectively
## - the union of the residue classes <cls>[i][1] ( mod <cls>[i][2] )
## of the ring <R>, plus a finite set <included> and minus a finite set
## <excluded> of elements of <R>.
##
DeclareConstructor( "ResidueClassUnionCons",
[ IsResidueClassUnion, IsRing, IsRingElement,
IsList, IsList, IsList ] );
DeclareConstructor( "ResidueClassUnionCons",
[ IsResidueClassUnion, IsRing,
IsList, IsList, IsList ] );
DeclareConstructor( "ResidueClassUnionCons",
[ IsResidueClassUnion, IsRowModule, IsMatrix,
IsList, IsList, IsList ] );
DeclareConstructor( "ResidueClassUnionCons",
[ IsResidueClassUnion, IsRowModule,
IsList, IsList, IsList ] );
DeclareGlobalFunction( "ResidueClassUnion" );
DeclareGlobalFunction( "ResidueClassUnionNC" );
#############################################################################
##
#F ResidueClass( <R>, <m>, <r> ) . . . . . . . . . . . residue class of <R>
#F ResidueClass( <m>, <r> ) . . . . . . . . . residue class of the integers
#F ResidueClass( <r>, <m> ) . . . . . . . . . . . . . . . . . . . ( ditto )
#F ResidueClassNC( <R>, <m>, <r> ) . . . . . . . . . . residue class of <R>
#F ResidueClassNC( <m>, <r> ) . . . . . . . . residue class of the integers
#F ResidueClassNC( <r>, <m> ) . . . . . . . . . . . . . . . . . . ( ditto )
#P IsResidueClass( <obj> ) . . . . . . . . . . <obj> is single residue class
##
## Returns the residue class <r> ( mod <m> ) of the ring <R>, respectively
## the residue class <r> ( mod <m> ) of the default ring of <m> and <r>.
##
## In the two-argument case, if <r> and <m> are integers, they must be
## nonnegative, and <r> must lie in the range [0..<m>-1]. This is used to
## decide which argument is <m> and which is <r>. For other rings, similar
## criteria are used.
##
## Residue classes have the property `IsResidueClass'.
##
DeclareGlobalFunction( "ResidueClass" );
DeclareGlobalFunction( "ResidueClassNC" );
DeclareProperty( "IsResidueClass", IsObject );
#############################################################################
##
#F ResidueClassUnionViewingFormat( format ) . short <--> long viewing format
##
DeclareGlobalFunction( "ResidueClassUnionViewingFormat" );
#############################################################################
##
#O Modulus( <U> ) . . . . . . . . . . . . . modulus of a residue class union
##
DeclareOperation( "Modulus", [ IsResidueClassUnion ] );
DeclareSynonym( "Mod", Modulus );
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##
#O Residues( <U> ) . . . . . . . . . . . . . residues of residue class union
#O Residue( <cl> ) . . . . . . . . . . . . . . . . residue of residue class
#O Classes( <U> ) . . . . . . . residue classes forming residue class union
##
DeclareOperation( "Residues", [ IsResidueClassUnion ] );
DeclareOperation( "Residue", [ IsResidueClass ] );
DeclareOperation( "Classes", [ IsResidueClassUnion ] );
#############################################################################
##
#O IncludedElements( <U> ) . included single elements of residue class union
#O ExcludedElements( <U> ) . excluded single elements of residue class union
#O IncludedLines( <U> ) . . . . included lines of residue class union of Z^2
#O ExcludedLines( <U> ) . . . . excluded lines of residue class union of Z^2
##
DeclareOperation( "IncludedElements", [ IsResidueClassUnion ] );
DeclareOperation( "ExcludedElements", [ IsResidueClassUnion ] );
DeclareOperation( "IncludedLines", [ IsResidueClassUnionOfZxZ ] );
DeclareOperation( "ExcludedLines", [ IsResidueClassUnionOfZxZ ] );
#############################################################################
##
#F ResidueClassUnionsFamily( <R> [ , <fixedreps> ] )
##
## Returns the family of all residue class unions of <R>.
##
## The optional argument <fixedreps> is a boolean which determines whether
## the function returns the family of "usual" residue class unions of <R>
## (<fixedreps> = `false' or not given) or the family of unions of residue
## classes with fixed representatives (<fixedreps> = `true').
##
DeclareGlobalFunction( "ResidueClassUnionsFamily" );
#############################################################################
##
#F ZResidueClassUnionsFamily( <fixedreps> )
#F Z_piResidueClassUnionsFamily( <R> [ , <fixedreps> ] )
#F GFqxResidueClassUnionsFamily( <R> [ , <fixedreps> ] )
##
## Returns the family of unions of residue classes of Z,
## of a ring <R> = Z_(pi) or of a ring <R> = GF(q)[x], respectively.
##
## For the meaning of the optional argument <fixedreps>,
## see `ResidueClassUnionsFamily'.
##
DeclareGlobalFunction( "ZResidueClassUnionsFamily" );
DeclareGlobalFunction( "Z_piResidueClassUnionsFamily" );
DeclareGlobalFunction( "GFqxResidueClassUnionsFamily" );
#############################################################################
##
#A UnderlyingRing( <fam> ) . . . . . . . underlying ring of the family <fam>
#A UnderlyingIndeterminate( <fam> ) . . indet. of underlying polynomial ring
##
## The underlying ring / the indeterminate of the underlying polynomial ring
## of the family <fam>.
##
DeclareAttribute( "UnderlyingRing", IsFamily );
DeclareAttribute( "UnderlyingIndeterminate", IsFamily );
#############################################################################
##
#P IsZxZ . . . . . . . . . . . . . . . . . . . . . Z^2 = Z x Z = Integers^2
##
DeclareProperty( "IsZxZ", IsObject );
#############################################################################
##
#O AllResidues( <R>, <m> ) . . . . the residues (mod <m>) in canonical order
#F AllResidueClassesModulo( [ <R>, ] <m> ) . . the residue classes (mod <m>)
#O NumberOfResidues( <R>, <m> ) . . . . . . the number of residues (mod <m>)
#O NrResidues( <R>, <m> ) . . . . . . . . . . . . . . . . . . . . - ditto -
##
## Returns
## - a sorted list of all residues, resp.
## - a sorted list of all residue classes, resp.
## - the number of residues
## modulo <m> in the ring <R>.
##
DeclareOperation( "AllResidues", [ IsRing, IsRingElement ] );
DeclareOperation( "AllResidues", [ IsRowModule, IsMatrix ] );
DeclareOperation( "NumberOfResidues", [ IsRing, IsRingElement ] );
DeclareOperation( "NumberOfResidues", [ IsRowModule, IsMatrix ] );
DeclareSynonym( "NrResidues", NumberOfResidues );
DeclareGlobalFunction( "AllResidueClassesModulo" );
#############################################################################
##
#F All2x2IntegerMatricesInHNFWithDeterminantUpTo( <maxdet> )
##
## Returns a list of all 2 x 2 integer matrices in Hermite normal form
## whose determinant is less than or equal to <maxdet>.
##
DeclareGlobalFunction( "All2x2IntegerMatricesInHNFWithDeterminantUpTo" );
#############################################################################
##
#O IsSublattice( <L1>, <L2> )
#O Superlattices( <L> )
##
## Lattices are represented by integer matrices in Hermite normal form.
##
DeclareOperation( "IsSublattice", [ IsMatrix, IsMatrix ] );
DeclareOperation( "Superlattices", [ IsMatrix ] );
#############################################################################
##
#A SizeOfSmallestResidueClassRing( <R> )
##
DeclareAttribute( "SizeOfSmallestResidueClassRing", IsRing );
#############################################################################
##
#F RingToString( <R> ) . . . how the ring <R> is printed by `View'/`Display'
##
## The return value of this function determines the way the ring <R> is
## printed by the methods for `View'/`Display' for residue class unions.
##
DeclareGlobalFunction( "RingToString" );
#############################################################################
##
#O AsUnionOfFewClasses( <U> ) . . . . . write <U> as a union of few classes
##
## Returns a partition of <U> into 'few' residue classes, up to the finite
## sets of included / excluded elements.
##
DeclareOperation( "AsUnionOfFewClasses", [ IsResidueClassUnion ] );
#############################################################################
##
#O SplittedClass( <cl>, <t> ) . . partition of <cl> into <t> residue classes
##
## Returns a partition of the residue class <cl> into <t> residue classes
## with the same modulus.
##
DeclareOperation( "SplittedClass", [ IsResidueClassUnion, IsRingElement ] );
DeclareOperation( "SplittedClass", [ IsResidueClassUnion, IsRowVector ] );
#############################################################################
##
#O PartitionsIntoResidueClasses( <R>, <length> )
#O PartitionsIntoResidueClasses( <R>, <length>, <primes> )
##
## Returns a list of all partitions of the ring <R> into <length> residue
## classes.
##
DeclareOperation( "PartitionsIntoResidueClasses", [ IsRing, IsPosInt ] );
DeclareOperation( "PartitionsIntoResidueClasses", [ IsRing, IsPosInt,
IsList ] );
#############################################################################
##
#O RandomPartitionIntoResidueClasses( <R>, <length>, <primes> )
##
## Returns a random partition of the ring <R> into <length> residue classes
## whose moduli have only prime factors in <primes>.
##
DeclareOperation( "RandomPartitionIntoResidueClasses",
[ IsRing, IsPosInt, IsList ] );
#############################################################################
##
#O CoverByResidueClasses( <S>, <moduli> ) . . cover of S by residue classes
#O CoversByResidueClasses( <S>, <moduli> )
##
## One, respectively all, covers of the set <S> by residue classes with
## moduli <moduli>.
## If no such cover exists, the former operation returns `fail'.
##
DeclareOperation( "CoverByResidueClasses", [ IsListOrCollection, IsList ] );
DeclareOperation( "CoversByResidueClasses", [ IsListOrCollection, IsList ] );
#############################################################################
##
#A Density( <S> ) . . natural density of the set <S> in the underlying ring
##
DeclareAttribute( "Density", IsResidueClassUnion );
#############################################################################
##
#F DisplayAsGrid( <U> ) . display the residue class union <U> as ASCII grid
##
DeclareGlobalFunction( "DisplayAsGrid" );
#############################################################################
##
#E resclass.gd . . . . . . . . . . . . . . . . . . . . . . . . . . ends here
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