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##
#W rcwamono.gd GAP4 Package `RCWA' Stefan Kohl
##
## This file contains declarations of functions, operations etc. for
## computing with rcwa monoids.
##
## See the definitions given in the file rcwamap.gd.
##
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##
#C IsRcwaMonoidOverZ . . . . . . . . . . . . . . . . . . rcwa monoids over Z
#C IsRcwaMonoidOverZxZ . . . . . . . . . . . . . . . . rcwa monoids over Z^2
#C IsRcwaMonoidOverZ_pi . . . . . . . . . . . . . rcwa monoids over Z_(pi)
#C IsRcwaMonoidOverGFqx . . . . . . . . . . . . rcwa monoids over GF(q)[x]
#C IsRcwaMonoidOverZOrZ_pi . . . . . . . . . . rcwa monoids over Z or Z_(pi)
##
## The category of all rcwa monoids over Z, over Z^2, over semilocalizations
## of Z or over polynomial rings in one variable over a finite field,
## respectively. The category `IsRcwaMonoidOverZOrZ_pi' is the union of
## `IsRcwaMonoidOverZ' and `IsRcwaMonoidOverZ_pi'.
##
DeclareSynonym( "IsRcwaMonoidOverZ",
CategoryCollections(IsRcwaMappingOfZ) and IsMonoid );
DeclareSynonym( "IsRcwaMonoidOverZxZ",
CategoryCollections(IsRcwaMappingOfZxZ) and IsMonoid );
DeclareSynonym( "IsRcwaMonoidOverZ_pi",
CategoryCollections(IsRcwaMappingOfZ_pi) and IsMonoid );
DeclareSynonym( "IsRcwaMonoidOverGFqx",
CategoryCollections(IsRcwaMappingOfGFqx) and IsMonoid );
DeclareSynonym( "IsRcwaMonoidOverZOrZ_pi",
CategoryCollections(IsRcwaMappingOfZOrZ_pi) and IsMonoid );
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##
#O RcwaCons( <R> ) . . . . . . . . . . . . . . . . . . Rcwa( R ) for ring R
#F Rcwa( <R> )
##
## The monoid formed by all rcwa mappings of <R>.
##
DeclareConstructor( "RcwaCons", [ IsRcwaMonoid, IsDomain ] );
DeclareGlobalFunction( "Rcwa" );
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#P IsNaturalRcwa( <M> ) . . . . . . . . . . . . . . . . . . . . . Rcwa( R )
##
DeclareCategory( "IsNaturalRcwa", IsRcwaMonoid );
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##
#A ModulusOfRcwaMonoid( <M> ) . . . . . . . . modulus of the rcwa monoid <M>
##
## We define the *modulus* of an rcwa monoid by the lcm of the moduli of its
## elements in case such an lcm exists, and by 0 otherwise.
##
DeclareAttribute( "ModulusOfRcwaMonoid", IsRcwaMonoid );
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##
#O Ball( <M>, <f>, <r> ) ball of radius <r> around the element <f> of <M>
#O Ball( <M>, <p>, <r>, <act> ) " the point <p> under the action of <M>
#O Ball( <M>, <p>, <r> ) . . . as above, where <act> defaults to `OnPoints'
#O RestrictedBall( <M>, <f>, <r>, <modulusbound> ) . . . . "restricted" ball
#O RestrictedBall( <M>, <p>, <r>, <act>, <bound> ) . . . . . . . . . (dito)
#O RestrictedBall( <M>, <p>, <r>, <bound> ) . . . . . . . . . . . . (dito)
#O RestrictedBall( <M>, <f>, <r>, <bounds> ) . . . . . . . . . . . . (dito)
##
## The first variant returns the ball of radius <r> about the element <f>
## of <M>.
##
## The second variant returns the ball of radius <r> about the point <p>
## under the action of <M>.
##
## The third variant is the same as the second --
## <act> defaults to `OnPoints'.
##
## The fourth variant does the same as the first except that it stops
## where extending the ball would yield elements whose moduli exceed the
## bound <modulusbound>.
##
## The fifth variant does the same as the second except that it stops
## where further extending the ball would yield points which exceed the
## bound <bound> in absolute value. If the points are tuples, then the
## bound applies to all coordinates.
##
## The sixth variant is the fifth -- <act> defaults to `OnPoints'.
##
## All balls are understood w.r.t. the stored generators of the monoid <M>,
## respectively w.r.t. the stored generators and their inverses if <M> is
## actually a group.
##
## An option `Spheres' is recognized. If set, the returned ball is split
## into a list of spheres.
##
DeclareOperation( "Ball", [ IsMonoid, IsObject, IsInt ] );
DeclareOperation( "Ball", [ IsMonoid, IsObject, IsInt, IsFunction ] );
DeclareOperation( "RestrictedBall",
[ IsMonoid, IsObject, IsObject, IsObject ] );
DeclareOperation( "RestrictedBall",
[ IsMonoid, IsObject, IsObject, IsObject, IsObject ] );
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##
#O ShortOrbits( <G>, <S>, <maxlng> ) . . . . short orbits of rcwa group <G>
#O ShortOrbits( <G>, <S>, <maxlng>, <maxn> ) . dito, with upper bound <maxn>
#O ShortOrbits( <M>, <S>, <maxlng> ) short forward orbits of rcwa monoid <M>
##
## In the first case, this operation returns a list of all finite orbits of
## the rcwa group <G> of length <= <maxlng>, which intersect nontrivially
## with the set <S>.
## In the second case, it returns a list of all such orbits which do not
## contain a point larger than <maxn>.
## In the third case, it returns a list of all finite forward orbits with
## starting point within the set <S>, of length <= <maxlng>.
##
DeclareOperation( "ShortOrbits", [ IsMonoid, IsListOrCollection, IsInt ] );
DeclareOperation( "ShortOrbits", [ IsMonoid, IsListOrCollection, IsInt,
IsInt ] );
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##
#E rcwamono.gd . . . . . . . . . . . . . . . . . . . . . . . . . . ends here
