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INCIDENCE GEOMETRY

 
Acknowledgements
 
Introduction
 
Construction of Incidence and Coset Geometries
      Construction of an Incidence Geometry
      Construction of a Coset Geometry
 
Elementary Invariants
 
Conversion Functions
 
Residues
 
Truncations
 
Shadows
 
Shadow Spaces
 
Automorphism Group and Correlation Group
 
Properties of Incidence Geometries and Coset Geometries
 
Intersection Properties of Coset Geometries
 
Primitivity Properties on Coset Geometries
 
Diagram of an Incidence Geometry
 
Bibliography







DETAILS

 
Introduction

 
Construction of Incidence and Coset Geometries

      Construction of an Incidence Geometry
            IncidenceGeometry(G) : GrphUnd -> IncGeom
            Example IncidenceGeometry_Constructors (H142E1)
            Example IncidenceGeometry_Constructors (H142E2)
            Example IncidenceGeometry_Constructors (H142E3)
            Example IncidenceGeometry_Constructors (H142E4)

      Construction of a Coset Geometry
            CosetGeometry(G, S, I) : GrpPerm, Set, Set -> CosetGeom
            CosetGeometry(G, S) : GrpPerm, Set -> CosetGeom
            Example IncidenceGeometry_Constructors (H142E5)
            Example IncidenceGeometry_Constructors (H142E6)
            Example IncidenceGeometry_Constructors (H142E7)

 
Elementary Invariants
      Points(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
      MinParabolics(C) : CosetGeom -> SetIndx
      Borel(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 IncidenceGeometry_Constructors (H142E8)

 
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
      IsThin(X) : CosetGeom -> BoolElt
      IsThick(X) : CosetGeom -> BoolElt
      IsResiduallyConnected(X) : IncGeom -> 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
      IsWeaklyPrimitive(C) : CosetGeom -> BoolElt
      IsResiduallyPrimitive(C) : CosetGeom -> BoolElt
      IsResiduallyWealyPrimitive(C) : CosetGeom -> BoolElt
      IsLocallyTwoTransitive(C) : CosetGeom -> BoolElt

 
Diagram of an Incidence Geometry
      Diagram(D) : IncGeom -> GrphUnd, GrphVertSet, GrphEdgeSet
      Example IncidenceGeometry_diagram (H142E9)
      Example IncidenceGeometry_diagram (H142E10)
      Example IncidenceGeometry_diagram (H142E11)
      Example IncidenceGeometry_diagram (H142E12)

 
Bibliography

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