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#W  mgmideal.gd                  GAP library                   Andrew Solomon
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#Y  Copyright (C)  1997,  Lehrstuhl D für Mathematik,  RWTH Aachen,  Germany
#Y  (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y  Copyright (C) 2002 The GAP Group
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##  This file contains the declaration of operations for magma ideals.
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##               Left Magma Ideals
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#P  IsLeftMagmaIdeal( <D> )
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##  A *left magma ideal* is a submagma (see~"Magmas") which is closed under
##	left multiplication by elements of its parent magma.
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DeclareSynonym("IsLeftMagmaIdeal", IsMagma and IsLeftActedOnBySuperset);

##  As a sub magma, a left magma ideal has a Parent (the enclosing magma)
##  and as LeftActedOnBySuperset it has a  LeftActingDomain. 
##  We must ensure that these two are the same object when the 
##  left magma ideal is created.
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#F  LeftMagmaIdeal(<D>, <gens> ) 
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##  `LeftMagmaIdeal' returns the magma containing the elements in the 
##  homogeneous list <gens> and closed under left multiplication by elements 
##  of the magma <D> in which it embeds.
##

##  This has to put in the parent and left acting set. Although it is a 
##  submagma, we can't call the generic submagma creation since that
##  requires *magma* generators.
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DeclareGlobalFunction( "LeftMagmaIdeal" );


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#O  AsLeftMagmaIdeal( <D>, <C> )
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##  Let <D> be a domain and <C> a collection.
##  If <C> is a subset of <D> 
##  `AsLeftMagmaIdeal' returns the LeftMagmaIdeal with generators <C>, 
##  and with parent <D>.
##  Otherwise `fail' is returned.
##  Probably more desirable would be to regard <C> as the set of 
##  elements of <D>.
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DeclareOperation( "AsLeftMagmaIdeal", [ IsDomain, IsCollection ] );





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#A  GeneratorsOfLeftMagmaIdeal( <I> )
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##  These are left ideal generators, not magma generators.
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DeclareSynonymAttr( "GeneratorsOfLeftMagmaIdeal", GeneratorsOfExtLSet );


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#O  LeftMagmaIdealByGenerators(<D>, <gens> )
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##  is the underlying operation of `LeftMagmaIdeal' 
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DeclareOperation( "LeftMagmaIdealByGenerators", [IsMagma, IsCollection ] );



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##               Right Magma Ideals
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#P  IsRightMagmaIdeal( <D> )
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##  A *right magma ideal* is a submagma (see~"Magmas") which is closed under
##	right multiplication by elements of its parent magma.
##
DeclareSynonym("IsRightMagmaIdeal", IsMagma and IsRightActedOnBySuperset);

##  As a sub magma, a right magma ideal has a Parent (the enclosing magma)
##  and as RightActedOnBySuperset it has a  RightActingDomain. 
##  We must ensure that these two are the same object when the 
##  right magma ideal is created.
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#F  RightMagmaIdeal(<D>, <gens> ) . . . . . . . . . . 
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##  `RightMagmaIdeal' returns the magma containing the elements in the 
##  homogeneous list <gens> and closed under right multiplication by elements 
##  of the parent magma <D>  in which it embeds.
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DeclareGlobalFunction( "RightMagmaIdeal" );


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#O  AsRightMagmaIdeal( <D>, <C> )
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##  Let <D> be a domain and <C> a collection.
##  If <C> is a subset of <D> that forms a RightMagmaIdeal then 
##  `AsRightMagmaIdeal' returns this RightMagmaIdeal, with parent <D>.
##  Otherwise `fail' is returned.
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DeclareOperation( "AsRightMagmaIdeal", [ IsDomain, IsCollection ] );





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#A  GeneratorsOfRightMagmaIdeal( <I> )
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##  These are right ideal generators, not magma generators.
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DeclareSynonymAttr( "GeneratorsOfRightMagmaIdeal", GeneratorsOfExtRSet );



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#O  RightMagmaIdealByGenerators(<D>, <gens> )
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##  is the underlying operation of `RightMagmaIdeal' 
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DeclareOperation( "RightMagmaIdealByGenerators", [IsMagma, IsCollection ] );



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##               Two Sided Magma Ideals
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#P  IsMagmaIdeal( <D> )
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##  A *magma ideal* is a submagma (see~"Magmas") which is closed under
##	left and right multiplication by elements of its parent magma.
##
DeclareSynonym("IsMagmaIdeal", IsLeftMagmaIdeal and IsRightMagmaIdeal);

##  As a sub magma, a magma ideal has a Parent (the enclosing magma)
##  and as LeftActedOnBySuperset it has a  LeftActingDomain, 
##  and as RightActedOnBySuperset it has a  RightActingDomain.
##  We must ensure that these three are the same object when the 
##  magma ideal is created.
##

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#F  MagmaIdeal(<D>, <gens> ) 
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##  `MagmaIdeal' returns the magma containing the elements in the homogeneous 
##  list <gens> and closed under left  and right multiplication by elements 
##  of the parent magma <D> in which it emeds.
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DeclareGlobalFunction( "MagmaIdeal" );


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#O  AsMagmaIdeal( <D>, <C> )
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##  Let <D> be a domain and <C> a collection.
##  If <C> is a subset of <D> that forms a MagmaIdeal then 
##  `AsMagmaIdeal' returns this MagmaIdeal, with parent <D>.
##  Otherwise `fail' is returned.
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DeclareOperation( "AsMagmaIdeal", [ IsDomain, IsCollection ] );


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#A  GeneratorsOfMagmaIdeal( <I> )
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##  These are ideal generators, not magma generators.
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DeclareAttribute( "GeneratorsOfMagmaIdeal", IsMagmaIdeal );




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#O  MagmaIdealByGenerators( <D>, <gens> )
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##  is the underlying operation of `MagmaIdeal' 
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DeclareOperation( "MagmaIdealByGenerators", [IsMagma, IsCollection ] );




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