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DeclareSynonym( "MapNearRing", TransformationNearRing );
#############################################################################
##
#O PolynomialNearRing( <G> ). . . . . . . . . . compute the nearring
## of all polynomials on the group G
##
## Constructor function for nearrings.
## Returns the nearring P(G).
##
DeclareOperation ("PolynomialNearRing", [IsGroup]);
#############################################################################
##
#P PolynomialNearRingFlag is set to true if the near ring is a
## polynomial nearring
DeclareProperty( "PolynomialNearRingFlag", IsTransformationNearRing );
DeclareSynonym( "IsPolynomialNearRing", HasPolynomialNearRingFlag );
#############################################################################
##
#O EndomorphismNearRing( <G> ). . . . . . . . . . compute the nearring
## generated by all endomorphisms in G.
##
## Constructor function for nearrings.
## Returns the nearring E(G).
##
DeclareOperation ("EndomorphismNearRing", [IsGroup]);
#############################################################################
##
#P EndomorphismNearRingFlag is set to true if the near ring is a
## Endomorphism nearring
DeclareProperty( "EndomorphismNearRingFlag", IsTransformationNearRing );
DeclareSynonym( "IsEndomorphismNearRing", HasEndomorphismNearRingFlag );
#############################################################################
##
#O AutomorphismNearRing( <G> ). . . . . . . . . . compute the nearring
## generated by all Automorphisms in G.
##
## Constructor function for nearrings.
## Returns the nearring A(G).
##
DeclareOperation ("AutomorphismNearRing", [IsGroup]);
#############################################################################
##
#P AutomorphismNearRingFlag is set to true if the near ring is a
## Automorphism nearring
DeclareProperty( "AutomorphismNearRingFlag", IsTransformationNearRing );
DeclareSynonym( "IsAutomorphismNearRing", HasAutomorphismNearRingFlag );
#############################################################################
##
#O InnerAutomorphismNearRing( <G> ). . . . . . . . . . compute the nearring
## generated by all inner automorphisms in G.
##
## Constructor function for nearrings.
## Returns the nearring A(G).
##
DeclareOperation ("InnerAutomorphismNearRing", [IsGroup]);
#############################################################################
##
#P InnerAutomorphismNearRingFlag is set to true if the near ring is a
## InnerAutomorphism nearring
DeclareProperty( "InnerAutomorphismNearRingFlag", IsTransformationNearRing );
DeclareSynonym( "IsInnerAutomorphismNearRing",
HasInnerAutomorphismNearRingFlag );
#############################################################################
##
#O InnerAutomorphismNearRingGeneratedByCommutators( <G> )
## computes the nearring generated by all inner automorphisms
## on G.
##
## Constructor function for nearrings.
## Returns the nearring I(G).
DeclareOperation ("InnerAutomorphismNearRingGeneratedByCommutators",
[IsGroup]);
#############################################################################
##
#P InnerAutomorphismNearRingByCommutatorsFlag is set to true if the
## near ring is a
## polynomial nearring
DeclareProperty( "InnerAutomorphismNearRingByCommutatorsFlag",
IsTransformationNearRing );
DeclareSynonym( "IsInnerAutomorphismNearRingByCommutators",
HasInnerAutomorphismNearRingByCommutatorsFlag );
#############################################################################
##
#O CentralizerNearRing ( <G>, <endos> )
## computes the nearring of all mappings m on G with the property that
## m(e(x)) = e(m(x)) for all e in <endos> and x in <G>.
##
## Constructor function for nearrings.
## Returns the nearring M_{<endos>}(<G>). Note that the resulting nearring
## need not be zerosymmetric if the zero-endomorphism is not included
## in <endos>.
##
DeclareOperation ("CentralizerNearRing", [IsGroup, IsListOrCollection]);
#############################################################################
##
#P CentralizerNearRingFlag is set to true if the near ring is a
## Centralizer nearring
DeclareProperty( "CentralizerNearRingFlag", IsTransformationNearRing );
DeclareSynonym( "IsCentralizerNearRing", HasCentralizerNearRingFlag );
##############################################################################
##
#O RestrictedEndomorphismNearRing ( <G>, <U> )
## Returns the near-ring H(G,U), which is the
## near-ring that is addidtively generated by the set
## of all endomorphisms of G whose range is a subset
## of U.
##
DeclareOperation( "RestrictedEndomorphismNearRing", [IsGroup, IsGroup] );
#############################################################################
##
#P RestrictedEndomorphismNearRingFlag is set to true if the near ring is a
## RestrictedEndomorphism nearring
DeclareProperty( "RestrictedEndomorphismNearRingFlag",
IsTransformationNearRing );
DeclareSynonym( "IsRestrictedEndomorphismNearRing",
HasRestrictedEndomorphismNearRingFlag );
#############################################################################
##
#O LocalInterpolationNearRing ( <NR>, <m> )
## Suppose that NR is a near-ring of functions on G (= NR.Gamma).
## computes the nearring of all mappings on G that can be
## interpolated at any m places by an element in <NR>
##
DeclareOperation( "LocalInterpolationNearRing",
[IsTransformationNearRing, IsInt and IsPosRat] );
#############################################################################
##
#P LocalInterpolationNearRingFlag is set to true if the near ring is a
## Local nearring
DeclareProperty( "LocalInterpolationNearRingFlag", IsTransformationNearRing );
DeclareSynonym( "IsLocalInterpolationNearRing", HasLocalInterpolationNearRingFlag );
#############################################################################
##
#O NoetherianQuotient2 ( <NR>, <Gamma>, <Target>, <Source> )
##
## for transformation nearrings:
## Input conditions: (1) <NR> is a near-ring of functions on <Gamma>,
## (2) <Target> is a subgroup of G,
## (3) <Source> is a subset of G,
## (4) <Target> is a subset of <Source>.
## Then this function returns all mappings in <NR> that
## map <Source> into <Target>.
## Condition (4) guarantees that the resulting functions are closed
## under composition.
##
## for NGroups:
## Input conditions: (1) <NR> is a near-ring acting on the N-group <Gamma>
## (2) <Target> is a subset of <Gamma>
## (3) <Source> is a subset of <Gamma>
## Then this function returns all elements n in <NR> such that
## n <Source> \subseteq <Target>.
DeclareOperation( "NoetherianQuotient2",
[IsNearRing, IsGroup, IsCollection, IsCollection] );
#############################################################################
##
#O CongruenceNoetherianQuotient ( <P>, <A>, <B>, <C> )
##
## for nearrings of all polynomial functions.
## Input conditions: (1) <P> is the near-ring of polynomial
## functions on a group G,
## (2) <A> is a normal subgroup of G,
## (3) <B> is a normal subgroup of G,
## (4) <C> is a normal subgroup of G.
## Then this function returns all mappings in <P> that
## map every element of P into C, and maps two elements that
## are congruent modulo B into elements that are congruent modulo
## A.
DeclareOperation( "CongruenceNoetherianQuotient",
[IsPolynomialNearRing, IsGroup, IsGroup, IsGroup] );
#############################################################################
##
#O CongruenceNoetherianQuotientForInnerAutomorphismNearRings
## ( <I>, <A>, <B>, <C> )
##
## for inner automorphism nearrings.
## Input conditions: (1) <I> is the inner automorphism of G,
## functions on a group G,
## (2) <A> is a normal subgroup of G,
## (3) <B> is a normal subgroup of G,
## (4) <C> is a normal subgroup of G.
## Then this function returns all mappings in <I> that
## map every element of P into C, and maps two elements that
## are congruent modulo B into elements that are congruent modulo
## A.
DeclareOperation( "CongruenceNoetherianQuotientForInnerAutomorphismNearRings",
[IsInnerAutomorphismNearRing, IsGroup, IsGroup, IsGroup] );
##############################################################################
##
#A ZeroSymmetricPart ( <NR> )
## returns the zerosymmetric part of <NR>
##
DeclareAttribute( "ZeroSymmetricPart", IsNearRing );
##############################################################################
##
#P IsClosedUnderComposition ( <L> )
##
## interesting, e.g., for left ideals.
##
DeclareProperty ("IsClosedUnderComposition", IsNearRingElementCollection);
##############################################################################
##
#P NoetherianQuotientFlag ( <N> )
##
## true if N is a left ideal constructed as a noetherian quotient.
##
DeclareProperty ("NoetherianQuotientFlag", IsNearRingElementCollection);
DeclareSynonym( "IsNoetherianQuotient", HasNoetherianQuotientFlag );
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