The following describes functions to analyse further properties of arrangements such as formality, supersolvability, simpliciality, factoredness, inductive factoredness ...
‣ LIsModularPair( L, m1, m2 ) | ( operation ) |
Returns: true or false
Determines if m1 and m2 form a modular pair in L.
gap> A:=AGpql(2,2,3); <HyperplaneArrangement: 6 hyperplanes in 3-space> gap> L:=IntersectionLattice(A); <Geometric lattice: 6 atoms, rank 3> gap> LGroundSet(L); [ [ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ] ], [ [ 1, 2 ], [ 1, 3, 6 ], [ 2, 3, 5 ], [ 1, 4, 5 ], [ 2, 4, 6 ], [ 3, 4 ], [ 5, 6 ] ], [ [ 1, 2, 3, 4, 5, 6 ] ] ] gap> m1:=[1,2]; m2:= [5,6]; LIsModularPair(L,m1,m2); false gap> m1:=[1,2]; m2:= [2,4,6]; LIsModularPair(L,m1,m2); true
‣ LIsModularFlat( L, m ) | ( operation ) |
Returns: true or false
Determines if m is a modular flat in L.
gap> A:=AGpql(2,2,3); L:=IntersectionLattice(A); LGroundSet(L); <HyperplaneArrangement: 6 hyperplanes in 3-space> <Geometric lattice: 6 atoms, rank 3> [ [ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ] ], [ [ 1, 2 ], [ 1, 3, 6 ], [ 2, 3, 5 ], [ 1, 4, 5 ], [ 2, 4, 6 ], [ 3, 4 ], [ 5, 6 ] ], [ [ 1, 2, 3, 4, 5, 6 ] ] ] gap> m1:=[1,2]; m2:= [2,4,6]; LIsModularFlat(L,m1); LIsModularFlat(L,m2); false true
‣ LModularFlatsRk( L, k ) | ( operation ) |
Returns: list
Determines the modular flats in L of rank k.
gap> L:=IntersectionLattice(AGpql(2,1,4)); <Geometric lattice: 16 atoms, rank 4> gap> LModularFlatsRk(L,3); [ [ 1, 2, 3, 5, 6, 7, 8, 11, 12 ], [ 1, 2, 4, 5, 6, 9, 10, 13, 14 ], [ 1, 3, 4, 7, 8, 9, 10, 15, 16 ], [ 2, 3, 4, 11, 12, 13, 14, 15, 16 ] ]
‣ LIsSupersolvable( L ) | ( property ) |
Returns: true or false.
Determines whether the geometric lattice L is supersolvable.
gap> L:=IntersectionLattice(AGpql(2,1,4)); <Geometric lattice: 16 atoms, rank 4> gap> LIsSupersolvable(L); true gap> L:=IntersectionLattice(AGpql(2,2,4)); <Geometric lattice: 12 atoms, rank 4> gap> LIsSupersolvable(L); false
‣ HArrIsSupersolvable( A ) | ( property ) |
Returns: true or false.
Determines whether the arrangement A is supersolvable.
gap> A:=AGpql(2,1,4); <HyperplaneArrangement: 16 hyperplanes in 4-space> gap> HArrIsSupersolvable(A); true gap> A:=AGpql(2,2,4); <HyperplaneArrangement: 12 hyperplanes in 4-space> gap> HArrIsSupersolvable(A); false
‣ OMIsSupersolvable( OM ) | ( property ) |
Returns: true or false.
Determines whether the oriented matroid OM is supersolvable.
gap> O:=OrientedMatroid(3,4,[1,1,1,1]); <OrientedMatroid: 4 elements, rank 3> gap> OMIsSupersolvable(O); false gap> C:=List(Combinations(Roots(AGpql(2,2,3)),3),x->pos(Determinant(x))); [ 1, 1, 1, 1, 0, -1, -1, -1, -1, 0, 1, 0, 1, -1, 0, 1, 1, 1, -1, -1 ] gap> O:=OrientedMatroid(3,6,C); <OrientedMatroid: 6 elements, rank 3> gap> OMIsSupersolvable(O); true
‣ IsFormal( A ) | ( property ) |
Returns: true or false.
Determines whether the arrangement is formal in the sense of Falk-Randell [FR87].
gap> A1 := Arr([ > [1,0,0], > [0,1,0], > [0,0,1], > [1,1,1], > [2,1,1], > [2,3,1], > [2,3,4], > [3,0,5], > [3,4,5] > ]); gap> gap> A2 := Arr([ > [1,0,0], > [0,1,0], > [0,0,1], > [1,1,1], > [2,1,1], > [2,3,1], > [2,3,4], > [1,0,3], > [1,2,3] > ]); gap> IsLEquiv(A1,A2); true gap> IsFormal(A1); true gap> IsFormal(A2); false
‣ HArrIsSimplicial( A ) | ( property ) |
Returns: true or false.
Determines whether the real arrangement A is simplicial, i.e. if all chambers are simplicial.
gap> A:=HyperplaneArrangement([[1,0,0],[0,1,0],[0,0,1],[1,1,0],[0,1,1]]); HArrIsSimplicial(A); <HyperplaneArrangement: 5 hyperplanes in 3-space> false gap> A:=HyperplaneArrangement([[1,0,0],[0,1,0],[0,0,1],[1,1,0],[0,1,1],[1,1,1]]); HArrIsSimplicial(A); <HyperplaneArrangement: 6 hyperplanes in 3-space> true
The following still needs to be tested further...
‣ OMBoundedCpx( OM, g ) | ( operation ) |
Returns: list of list of sign vectors.
Constructs the complex of bounded cells of the affine part of the oriented matroid OM with respect to the element g.
‣ OMSupportsFalkWeights( OM, g ) | ( operation ) |
Returns: list or fail
Only for oriented matroids of rank 3. Determines, whether OM supports a system of weights as described by M. Falk in [Fal95]. This implies, that the Salvetti complex (see SalvettiComplex (4.2-8)) is apsherical. The functionality of the cddinterface package is used, to determine solotions of a system of linear inequalities.
If the algorithm determines a suitable weight system for OM, each list element consists of a weight, and a corner, i.e. a pair of a 2-cell and an adjacent vertex in the bounded complex, given as sign vectors.
gap> O:=OrientedMatroid( > [[1,1,0],[1,-1,0],[1,0,1],[1,0,-1],[0,1,1],[0,1,-1],[0,0,1]] > ); <OrientedMatroid: 7 elements, rank 3> gap> OMIsSupersolvable(O); false gap> OMSupportsFalkWeights(O,3); fail gap> OMSupportsFalkWeights(O,1); [ [ 1/2, [ [ 1, 1, 1, 1, 1, 1, 1 ], [ 1, 0, 1, 0, 1, 0, 1 ] ] ], [ 0, [ [ 1, 1, 1, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 0, 0, 0 ] ] ], [ 1/2, [ [ 1, 1, 1, 1, 1, 1, 1 ], [ 1, 0, 1, 1, 1, 1, 0 ] ] ], [ 0, [ [ 1, 1, 1, 1, 1, 1, -1 ], [ 1, 0, 0, 1, 0, 1, -1 ] ] ], [ 1/2, [ [ 1, 1, 1, 1, 1, 1, -1 ], [ 1, 1, 1, 1, 0, 0, 0 ] ] ], [ 1/2, [ [ 1, 1, 1, 1, 1, 1, -1 ], [ 1, 0, 1, 1, 1, 1, 0 ] ] ], [ 0, [ [ 1, -1, 1, 1, 1, 1, 1 ], [ 1, 0, 1, 0, 1, 0, 1 ] ] ], [ 1/2, [ [ 1, -1, 1, 1, 1, 1, 1 ], [ 1, -1, 0, 0, 1, 1, 0 ] ] ], [ 1/2, [ [ 1, -1, 1, 1, 1, 1, 1 ], [ 1, 0, 1, 1, 1, 1, 0 ] ] ], [ 1/2, [ [ 1, -1, 1, 1, 1, 1, -1 ], [ 1, 0, 0, 1, 0, 1, -1 ] ] ], [ 0, [ [ 1, -1, 1, 1, 1, 1, -1 ], [ 1, -1, 0, 0, 1, 1, 0 ] ] ], [ 1/2, [ [ 1, -1, 1, 1, 1, 1, -1 ], [ 1, 0, 1, 1, 1, 1, 0 ] ] ] ]
‣ HArrIsFactored( A ) | ( attribute ) |
Returns: A list or fail
Determines if A is factored and if so returns some factorization.
gap> A:=AGpql(2,1,3); <HyperplaneArrangement: 9 hyperplanes in 3-space> gap> HArrIsFactored(A); [ [ 1 ], [ 2, 4, 5 ], [ 3, 6, 7, 8, 9 ] ] gap> A:=AGpql(2,2,4); <HyperplaneArrangement: 12 hyperplanes in 4-space> gap> HArrIsFactored(A); fail
‣ HArrFactorizations( A ) | ( operation ) |
Returns: A list
Computes a list of all factorizations of A as partitions of [1..|A|].
gap> A:=AGpql(2,1,3); <HyperplaneArrangement: 9 hyperplanes in 3-space> gap> HArrFactorizations(A); [ [ [ 1 ], [ 2, 4, 5 ], [ 3, 6, 7, 8, 9 ] ], [ [ 1 ], [ 2, 4, 5, 8, 9 ], [ 3, 6, 7 ] ], [ [ 2 ], [ 1, 4, 5 ], [ 3, 6, 7, 8, 9 ] ], [ [ 2 ], [ 1, 4, 5, 6, 7 ], [ 3, 8, 9 ] ], [ [ 3 ], [ 1, 6, 7 ], [ 2, 4, 5, 8, 9 ] ], [ [ 3 ], [ 1, 4, 5, 6, 7 ], [ 2, 8, 9 ] ], [ [ 4 ], [ 1, 2, 5 ], [ 3, 6, 7, 8, 9 ] ], [ [ 5 ], [ 1, 2, 4 ], [ 3, 6, 7, 8, 9 ] ], [ [ 6 ], [ 1, 3, 7 ], [ 2, 4, 5, 8, 9 ] ], [ [ 7 ], [ 1, 3, 6 ], [ 2, 4, 5, 8, 9 ] ], [ [ 8 ], [ 1, 4, 5, 6, 7 ], [ 2, 3, 9 ] ], [ [ 9 ], [ 1, 4, 5, 6, 7 ], [ 2, 3, 8 ] ] ]
‣ HArrIsInductivelyFactored( A ) | ( attribute ) |
Returns: A list or fail
Determines if A is inductively factored and if so returns some inductive factorization.
gap> A:=AGpql(2,1,4); <HyperplaneArrangement: 16 hyperplanes in 4-space> gap> HArrIsInductivelyFactored(A); [ [ [ 1 ], [ 2, 5, 6 ], [ 3, 7, 8, 11, 12 ], [ 4, 9, 10, 13, 14, 15, 16 ] ], true ] gap> A:=AGpql(2,2,4); <HyperplaneArrangement: 12 hyperplanes in 4-space> gap> HArrIsInductivelyFactored(A); false
‣ HArrInductiveFactorizations( A ) | ( operation ) |
Returns: A list
Computes a list of all inductive factorizations of A as partitions of [1..|A|].
gap> A:=AGpql(2,1,3); <HyperplaneArrangement: 9 hyperplanes in 3-space> gap> HArrInductiveFactorizations(A); [ [ [ 1 ], [ 2, 4, 5 ], [ 3, 6, 7, 8, 9 ] ], [ [ 1 ], [ 2, 4, 5, 8, 9 ], [ 3, 6, 7 ] ], [ [ 2 ], [ 1, 4, 5 ], [ 3, 6, 7, 8, 9 ] ], [ [ 2 ], [ 1, 4, 5, 6, 7 ], [ 3, 8, 9 ] ], [ [ 3 ], [ 1, 6, 7 ], [ 2, 4, 5, 8, 9 ] ], [ [ 3 ], [ 1, 4, 5, 6, 7 ], [ 2, 8, 9 ] ], [ [ 4 ], [ 1, 2, 5 ], [ 3, 6, 7, 8, 9 ] ], [ [ 5 ], [ 1, 2, 4 ], [ 3, 6, 7, 8, 9 ] ], [ [ 6 ], [ 1, 3, 7 ], [ 2, 4, 5, 8, 9 ] ], [ [ 7 ], [ 1, 3, 6 ], [ 2, 4, 5, 8, 9 ] ], [ [ 8 ], [ 1, 4, 5, 6, 7 ], [ 2, 3, 9 ] ], [ [ 9 ], [ 1, 4, 5, 6, 7 ], [ 2, 3, 8 ] ] ]
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