# Incidence Geometry

- [Introduction](introduction.md)

- [Construction of Incidence and Coset Geometries](creation.md)

  - [Construction of an Incidence Geometry](creation.md#construction-of-an-incidence-geometry)

    - [`IncidenceGeometry(G): GrphUnd → IncGeom`](creation.md#function-incidencegeometry-grphund)

    - [`Example: Constructors`](creation.md#example-ex-a2453c)

    - [`Example: Constructors`](creation.md#example-ex-2bc35f)

    - [`Example: Constructors`](creation.md#example-ex-65148f)

    - [`Example: Constructors`](creation.md#example-ex-666366)

  - [Construction of a Coset Geometry](creation.md#construction-of-a-coset-geometry)

    - [`CosetGeometry(G, S, I): GrpPerm, Set, Set → CosetGeom`](creation.md#function-cosetgeometry-grpperm-set-set)

    - [`CosetGeometry(G, S): GrpPerm, Set → CosetGeom`](creation.md#function-cosetgeometry-grpperm-set)

    - [`Example: Constructors`](creation.md#example-ex-a1eb9c)

    - [`Example: Constructors`](creation.md#example-ex-4a5ca3)

    - [`Example: Constructors`](creation.md#example-ex-16511c)

- [Elementary Invariants](access.md)

  - [`Points(D): IncGeom → SetIndx`](access.md#function-points-incgeom)

  - [`Elements(D): IncGeom → SetIndx`](access.md#function-elements-incgeom)

  - [`Types(D): IncGeom → SetIndx`](access.md#function-types-incgeom)

  - [`Types(C): CosetGeom → SetIndx`](access.md#function-types-cosetgeom)

  - [`Rank(D): IncGeom → RngIntElt`](access.md#function-rank-incgeom)

  - [`Rank(C): CosetGeom → RngIntElt`](access.md#function-rank-cosetgeom)

  - [`IncidenceGraph(D): IncGeom → GrphUnd, GrphVertSet, GrphEdgeSet`](access.md#function-incidencegraph-incgeom)

  - [`Group(C): CosetGeom → GrpPerm`](access.md#function-group-cosetgeom)

  - [`MaxParabolics(C): CosetGeom → SetIndx`](access.md#function-maxparabolics-cosetgeom)

  - [`MaximalParabolics(C): CosetGeom → SetIndx`](access.md#function-maximalparabolics-cosetgeom)

  - [`MinParabolics(C): CosetGeom → SetIndx`](access.md#function-minparabolics-cosetgeom)

  - [`MinimalParabolics(C): CosetGeom → SetIndx`](access.md#function-minimalparabolics-cosetgeom)

  - [`Borel(C): CosetGeom → GrpPerm`](access.md#function-borel-cosetgeom)

  - [`BorelSubgroup(C): CosetGeom → GrpPerm`](access.md#function-borelsubgroup-cosetgeom)

  - [`Kernel(C): CosetGeom → GrpPerm`](access.md#function-kernel-cosetgeom)

  - [`Kernels(C): CosetGeom → SeqEnum`](access.md#function-kernels-cosetgeom)

  - [`Quotient(C, K): CosetGeom, GrpPerm → CosetGeom`](access.md#function-quotient-cosetgeom-grpperm)

- [Conversion Functions](conversion-functions.md)

  - [`IncidenceGeometry(C): CosetGeom → IncGeom`](conversion-functions.md#function-incidencegeometry-cosetgeom)

  - [`CosetGeometry(D): IncGeom → BoolElt, CosetGeom`](conversion-functions.md#function-cosetgeometry-incgeom)

  - [`Graph(D): IncGeom → GrphUnd`](conversion-functions.md#function-graph-incgeom)

  - [`Graph(C): CosetGeom → GrphUnd`](conversion-functions.md#function-graph-cosetgeom)

  - [`Example: Constructors`](conversion-functions.md#example-ex-cfd076)

- [Residues](residues.md)

  - [`Residue(D, f): IncGeom, Set → IncGeom`](residues.md#function-residue-incgeom-set)

  - [`Residue(C, f): CosetGeom, Set → CosetGeom`](residues.md#function-residue-cosetgeom-set)

- [Truncations](truncations.md)

  - [`Truncation(D, t): IncGeom, Set → IncGeom`](truncations.md#function-truncation-incgeom-set)

  - [`Truncation(C, t): CosetGeom, Set → CosetGeom`](truncations.md#function-truncation-cosetgeom-set)

- [Shadows](shadows.md)

  - [`Shadow(D, I, F): IncGeom, Set, Set → SetIndx`](shadows.md#function-shadow-incgeom-set-set)

- [Shadow Spaces](shadow-space.md)

  - [`ShadowSpace(D,I): IncGeom, Set → Inc`](shadow-space.md#function-shadowspace-incgeom-set)

- [Automorphism Group and Correlation Group](automorphisms.md)

  - [`AutomorphismGroup(D): IncGeom → GrpPerm`](automorphisms.md#function-automorphismgroup-incgeom)

  - [`CorrelationGroup(D): IncGeom → GrpPerm`](automorphisms.md#function-correlationgroup-incgeom)

- [Properties of Incidence Geometries and Coset Geometries](properties.md)

  - [`IsFTGeometry(D): IncGeom → BoolElt`](properties.md#function-isftgeometry-incgeom)

  - [`IsFTGeometry(C): CosetGeom → BoolElt`](properties.md#function-isftgeometry-cosetgeom)

  - [`IsFirm(X): IncGeom → BoolElt`](properties.md#function-isfirm-incgeom)

  - [`IsFirm(X): CosetGeom → BoolElt`](properties.md#function-isfirm-cosetgeom)

  - [`IsThin(X): CosetGeom → BoolElt`](properties.md#function-isthin-cosetgeom)

  - [`IsThin(X): IncGeom → BoolElt`](properties.md#function-isthin-incgeom)

  - [`IsThick(X): CosetGeom → BoolElt`](properties.md#function-isthick-cosetgeom)

  - [`IsThick(X): IncGeom → BoolElt`](properties.md#function-isthick-incgeom)

  - [`IsResiduallyConnected(X): IncGeom → BoolElt`](properties.md#function-isresiduallyconnected-incgeom)

  - [`IsRC(X): IncGeom → BoolElt`](properties.md#function-isrc-incgeom)

  - [`IsResiduallyConnected(X): CosetGeom → BoolElt`](properties.md#function-isresiduallyconnected-cosetgeom)

  - [`IsRC(X): CosetGeom → BoolElt`](properties.md#function-isrc-cosetgeom)

  - [`IsGraph(D): IncGeom → GrphUnd`](properties.md#function-isgraph-incgeom)

  - [`IsGraph(C): CosetGeom → GrphUnd`](properties.md#function-isgraph-cosetgeom)

- [Intersection Properties of Coset Geometries](intersectionproperties.md)

  - [`HasIntersectionPropertyN(C,n): CosetGeom, RngIntElt → BoolElt, BoolElt`](intersectionproperties.md#function-hasintersectionpropertyn-cosetgeom-rngintelt)

  - [`HasIntersectionProperty(C): CosetGeom → BoolElt`](intersectionproperties.md#function-hasintersectionproperty-cosetgeom)

  - [`HasWeakIntersectionProperty(C): CosetGeom → BoolElt`](intersectionproperties.md#function-hasweakintersectionproperty-cosetgeom)

- [Primitivity Properties on Coset Geometries](primitivityproperties.md)

  - [`IsPrimitive(C): CosetGeom → BoolElt`](primitivityproperties.md#function-isprimitive-cosetgeom)

  - [`IsPRI(C): CosetGeom → BoolElt`](primitivityproperties.md#function-ispri-cosetgeom)

  - [`IsPrimitive(C): CosetGeom → BoolElt`](primitivityproperties.md#function-isprimitive-cosetgeom-2)

  - [`IsPRI(C): CosetGeom → BoolElt`](primitivityproperties.md#function-ispri-cosetgeom-2)

  - [`IsWeaklyPrimitive(C): CosetGeom → BoolElt`](primitivityproperties.md#function-isweaklyprimitive-cosetgeom)

  - [`IsWPRI(C): CosetGeom → BoolElt`](primitivityproperties.md#function-iswpri-cosetgeom)

  - [`IsWeaklyPrimitive(C): CosetGeom → BoolElt`](primitivityproperties.md#function-isweaklyprimitive-cosetgeom-2)

  - [`IsWPRI(C): CosetGeom → BoolElt`](primitivityproperties.md#function-iswpri-cosetgeom-2)

  - [`IsResiduallyPrimitive(C): CosetGeom → BoolElt`](primitivityproperties.md#function-isresiduallyprimitive-cosetgeom)

  - [`IsRPRI(C): CosetGeom → BoolElt`](primitivityproperties.md#function-isrpri-cosetgeom)

  - [`IsResiduallyPrimitive(C): CosetGeom → BoolElt`](primitivityproperties.md#function-isresiduallyprimitive-cosetgeom-2)

  - [`IsRPRI(C): CosetGeom → BoolElt`](primitivityproperties.md#function-isrpri-cosetgeom-2)

  - [`IsResiduallyWealyPrimitive(C): CosetGeom → BoolElt`](primitivityproperties.md#function-isresiduallywealyprimitive-cosetgeom)

  - [`IsRWPRI(C): CosetGeom → BoolElt`](primitivityproperties.md#function-isrwpri-cosetgeom)

  - [`IsRWP(C): CosetGeom → BoolElt`](primitivityproperties.md#function-isrwp-cosetgeom)

  - [`IsResiduallyWeaklyPrimitive(C): CosetGeom → BoolElt`](primitivityproperties.md#function-isresiduallyweaklyprimitive-cosetgeom)

  - [`IsRWPRI(C): CosetGeom → BoolElt`](primitivityproperties.md#function-isrwpri-cosetgeom-2)

  - [`IsRWP(C): CosetGeom → BoolElt`](primitivityproperties.md#function-isrwp-cosetgeom-2)

  - [`IsLocallyTwoTransitive(C): CosetGeom → BoolElt`](primitivityproperties.md#function-islocallytwotransitive-cosetgeom)

  - [`Is2T1(C): CosetGeom → BoolElt`](primitivityproperties.md#function-is2t1-cosetgeom)

  - [`LocallySArcTransitive(C): CosetGeom → RngIntElt`](primitivityproperties.md#function-locallysarctransitive-cosetgeom)

- [Diagram of an Incidence Geometry](diagram.md)

  - [`Diagram(D): IncGeom → GrphUnd, GrphVertSet, GrphEdgeSet`](diagram.md#function-diagram-incgeom)

  - [`Diagram(C): CosetGeom → GrphUnd, GrphVertSet, GrphEdgeSet`](diagram.md#function-diagram-cosetgeom)

  - [`Example: diagram`](diagram.md#example-ex-5b021d)

  - [`Example: diagram`](diagram.md#example-ex-ee5e4d)

  - [`Example: diagram`](diagram.md#example-ex-cde392)

  - [`Example: diagram`](diagram.md#example-ex-3dd4a3)

- [C-Groups](cgroups.md)

  - [`HasIntersectionProperty(G): GrpPerm → BoolElt`](cgroups.md#function-hasintersectionproperty-grpperm)

  - [`HasStringProperty(G): GrpPerm → BoolElt`](cgroups.md#function-hasstringproperty-grpperm)

  - [`IsCGroup(G): GrpPerm → BoolElt`](cgroups.md#function-iscgroup-grpperm)

  - [`IsStringCGroup(G): GrpPerm → BoolElt`](cgroups.md#function-isstringcgroup-grpperm)

  - [`CosetGeometryFromCGroup(G): GrpPerm → CosetGeom`](cgroups.md#function-cosetgeometryfromcgroup-grpperm)

  - [`CosetGeometryToCGroup(D): CosetGeom → GrpPerm`](cgroups.md#function-cosetgeometrytocgroup-cosetgeom)

  - [`CoxeterDiagram(G): GrpPerm → GrphUnd`](cgroups.md#function-coxeterdiagram-grpperm)

  - [`Example: cosetgeometrytocgroup`](cgroups.md#example-ex-306b2c)

  - [`Example: cgrouptocosetgeometry`](cgroups.md#example-ex-ffd840)

- [$C^+$-Groups](cplusgroups.md)

  - [`HasIntersectionPropertyPlus(G): GrpPerm → BoolElt`](cplusgroups.md#function-hasintersectionpropertyplus-grpperm)

  - [`IsCPlusGroup(G): GrpPerm → BooElt`](cplusgroups.md#function-iscplusgroup-grpperm)

  - [`CosetGeometryFromCPlusGroup(G): GrpPerm → CosetGeom`](cplusgroups.md#function-cosetgeometryfromcplusgroup-grpperm)

  - [`BDiagram(G): GrpPerm → GrphUnd`](cplusgroups.md#function-bdiagram-grpperm)

  - [`Example: cgrouptocosetgeometry`](cplusgroups.md#example-ex-4866f0)