Incidence Geometry

• Introduction
• Construction of Incidence and Coset Geometries
  • Construction of an Incidence Geometry
    • IncidenceGeometry(G): GrphUnd → IncGeom
    • Example: Petersen Graph as an Incidence Geometry
    • Example: Cube as an Incidence Geometry
    • Example: Constructing the Hoffman-Singleton Graph
    • Example: Rank Four Geometry from Hoffman-Singleton
  • Construction of a Coset Geometry
    • CosetGeometry(G, S, I): GrpPerm, Set, Set → CosetGeom
    • CosetGeometry(G, S): GrpPerm, Set → CosetGeom
    • Example: Petersen Graph as a Coset Geometry
    • Example: Cube as a Coset Geometry
    • Example: Coset Geometry for Sym(6)
• Elementary Invariants
  • Points(D): IncGeom → SetIndx
  • Elements(D): IncGeom → SetIndx
  • Types(D): IncGeom → SetIndx
  • Types(C): CosetGeom → SetIndx
  • Rank(D): IncGeom → RngIntElt
  • Rank(C): CosetGeom → RngIntElt
  • IncidenceGraph(D): IncGeom → GrphUnd, GrphVertSet, GrphEdgeSet
  • Group(C): CosetGeom → GrpPerm
  • MaxParabolics(C): CosetGeom → SetIndx
  • MaximalParabolics(C): CosetGeom → SetIndx
  • MinParabolics(C): CosetGeom → SetIndx
  • MinimalParabolics(C): CosetGeom → SetIndx
  • Borel(C): CosetGeom → GrpPerm
  • BorelSubgroup(C): CosetGeom → GrpPerm
  • Kernel(C): CosetGeom → GrpPerm
  • Kernels(C): CosetGeom → SeqEnum
  • Quotient(C, K): CosetGeom, GrpPerm → CosetGeom
• Conversion Functions
  • IncidenceGeometry(C): CosetGeom → IncGeom
  • CosetGeometry(D): IncGeom → BoolElt, CosetGeom
  • Graph(D): IncGeom → GrphUnd
  • Graph(C): CosetGeom → GrphUnd
  • Example: Neumaier Geometry as a Coset Geometry
• Residues
  • Residue(D, f): IncGeom, Set → IncGeom
  • Residue(C, f): CosetGeom, Set → CosetGeom
• Truncations
  • Truncation(D, t): IncGeom, Set → IncGeom
  • Truncation(C, t): CosetGeom, Set → CosetGeom
• Shadows
  • Shadow(D, I, F): IncGeom, Set, Set → SetIndx
• Shadow Spaces
  • ShadowSpace(D,I): IncGeom, Set → Inc
• Automorphism Group and Correlation Group
  • AutomorphismGroup(D): IncGeom → GrpPerm
  • CorrelationGroup(D): IncGeom → GrpPerm
• Properties of Incidence Geometries and Coset Geometries
  • IsFTGeometry(D): IncGeom → BoolElt
  • IsFTGeometry(C): CosetGeom → BoolElt
  • IsFirm(X): IncGeom → BoolElt
  • IsFirm(X): CosetGeom → BoolElt
  • IsThin(X): CosetGeom → BoolElt
  • IsThin(X): IncGeom → BoolElt
  • IsThick(X): CosetGeom → BoolElt
  • IsThick(X): IncGeom → BoolElt
  • IsResiduallyConnected(X): IncGeom → BoolElt
  • IsRC(X): IncGeom → BoolElt
  • IsResiduallyConnected(X): CosetGeom → BoolElt
  • IsRC(X): CosetGeom → BoolElt
  • IsGraph(D): IncGeom → GrphUnd
  • IsGraph(C): CosetGeom → GrphUnd
• Intersection Properties of Coset Geometries
  • HasIntersectionPropertyN(C,n): CosetGeom, RngIntElt → BoolElt, BoolElt
  • HasIntersectionProperty(C): CosetGeom → BoolElt
  • HasWeakIntersectionProperty(C): CosetGeom → BoolElt
• Primitivity Properties on Coset Geometries
  • IsPrimitive(C): CosetGeom → BoolElt
  • IsPRI(C): CosetGeom → BoolElt
  • IsPrimitive(C): CosetGeom → BoolElt
  • IsPRI(C): CosetGeom → BoolElt
  • IsWeaklyPrimitive(C): CosetGeom → BoolElt
  • IsWPRI(C): CosetGeom → BoolElt
  • IsWeaklyPrimitive(C): CosetGeom → BoolElt
  • IsWPRI(C): CosetGeom → BoolElt
  • IsResiduallyPrimitive(C): CosetGeom → BoolElt
  • IsRPRI(C): CosetGeom → BoolElt
  • IsResiduallyPrimitive(C): CosetGeom → BoolElt
  • IsRPRI(C): CosetGeom → BoolElt
  • IsResiduallyWealyPrimitive(C): CosetGeom → BoolElt
  • IsRWPRI(C): CosetGeom → BoolElt
  • IsRWP(C): CosetGeom → BoolElt
  • IsResiduallyWeaklyPrimitive(C): CosetGeom → BoolElt
  • IsRWPRI(C): CosetGeom → BoolElt
  • IsRWP(C): CosetGeom → BoolElt
  • IsLocallyTwoTransitive(C): CosetGeom → BoolElt
  • Is2T1(C): CosetGeom → BoolElt
  • LocallySArcTransitive(C): CosetGeom → RngIntElt
• Diagram of an Incidence Geometry
  • Diagram(D): IncGeom → GrphUnd, GrphVertSet, GrphEdgeSet
  • Diagram(C): CosetGeom → GrphUnd, GrphVertSet, GrphEdgeSet
  • Example: Diagram of the Petersen Geometry
  • Example: Diagram of the Cube
  • Example: Diagram of the Hoffman-Singleton Geometry
  • Example: Diagram of Neumaier's Geometry
• C-Groups
  • HasIntersectionProperty(G): GrpPerm → BoolElt
  • HasStringProperty(G): GrpPerm → BoolElt
  • IsCGroup(G): GrpPerm → BoolElt
  • IsStringCGroup(G): GrpPerm → BoolElt
  • CosetGeometryFromCGroup(G): GrpPerm → CosetGeom
  • CosetGeometryToCGroup(D): CosetGeom → GrpPerm
  • CoxeterDiagram(G): GrpPerm → GrphUnd
  • Example: Converting the Cube to a C-Group
  • Example: 5-Simplex from a C-Group
• \(C^+\)-Groups
  • HasIntersectionPropertyPlus(G): GrpPerm → BoolElt
  • IsCPlusGroup(G): GrpPerm → BooElt
  • CosetGeometryFromCPlusGroup(G): GrpPerm → CosetGeom
  • BDiagram(G): GrpPerm → GrphUnd
  • Example: 5-Simplex from Its Rotation Group
• Bibliography