\Chapter{General lattices}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Constructing general lattices}
\>GeneralLattice( <coll>, <geq>, <string> )
The function `GeneralLattice
' constructs a lattice from a list of elements of
the lattice <coll>, an operation <geq>, which decides whether an element is
greater or equal than another and a string which is used when the lattice is
printed. It is assumed that whenever an element $i$ occurs after an element
$j$ in <coll> then $j$ is not smaller than $i$ whith respect to <geq>.
As an example
`GeneralLattice( Subgroups( GTW6_2 ), IsSubgroup,
"subgroup" )
'
will return the lattice of subgroups of the group $6/2$.
`GeneralLattice( NormalSubgroups( GTW6_2 ), IsSubgroup,
"normal subgroup" )
'
will return the lattice of normal subgroups of the group $6/2$.
`GeneralLattice( NearRingIdeals( LibraryNearRing( GTW6_2, 3 ) ), IsSubset,
"ideal" )
'
will return the ideal lattice of the library nearring
number 3 on the group
$6/2$.
\beginexample
gap> GeneralLattice( Subgroups( GTW6_2 ), IsSubgroup,
"subgroup" );
GeneralLattice( 6 subgroups )
gap> GeneralLattice( NormalSubgroups( GTW6_2 ), IsSubgroup,
>
"normal subgroup" );
GeneralLattice( 3 normal subgroups )
gap> GeneralLattice( NearRingIdeals( LibraryNearRing( GTW6_2, 3 ) ),
> IsSubset,
"ideal" );
GeneralLattice( 2 ideals )
\endexample
Once a general lattice is generated its elements are represented by their
position in the list <coll>. This list can be accessed via `AsList
' or
`AsSortedList
'.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Comparing lattice elements}
\>Less( <L>, <i>, <j> )
The function `Less
' checks whether the -th element of the lattice
is smaller than the <j>-th element and returns the according boolean value
`true
' or `false'.
\>SubCoverOfJI( <L>, <i> )
\>Join( <L>, <i>, <j> )
\>Meet( <L>, <i>, <j> )
\>IsJoinIrreducible( <L>, <i> )
\>JoinIrreducibles( <L> )
\beginexample
gap> n := LibraryNearRing( GTW8_4, 3 );
LibraryNearRing(8/4, 3)
gap> i := NearRingIdeals( n );
[ NearRingIdeal(...), NearRingIdeal(...), NearRingIdeal(...),
NearRingIdeal(...), NearRingIdeal(...), NearRingIdeal(...) ]
gap> l := GeneralLattice( i, IsSubset,
"ideal" );
GeneralLattice( 6 ideals )
gap> JoinIrreducibles( l );
[ 2, 3, 4, 5 ]
gap> i{last};
[ NearRingIdeal(...), NearRingIdeal(...), NearRingIdeal(...),
NearRingIdeal(...) ]
gap> List(last, Size);
[ 2, 4, 4, 4 ]
\endexample
\>IsCoveringPair( <L>, <pair> )
\>IsSC1Group( <G> )
\>IsProjectivePairOfPairs( <L>, <pair1>, <pair2> )
\>AlphaBar( <G> )
.
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