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FINITE PLANES

 
Acknowledgements
 
Introduction
      Planes in Magma
 
Construction of a Plane
 
The Point-Set and Line-Set of a Plane
      Introduction
      Creating Point-Sets and Line-Sets
      Using the Point-Set and Line-Set to Create Points and Lines
      Retrieving the Plane from Points, Lines, Point-Sets and Line-Sets
 
The Set of Points and Set of Lines
 
The Defining Points of a Plane
 
Subplanes
 
Structures Associated with a Plane
 
Numerical Invariants of a Plane
 
Properties of Planes
 
Identity and Isomorphism
 
The Connection between Projective and Affine Planes
 
Operations on Points and Lines
      Elementary Operations
      Deconstruction Functions
      Other Point and Line Functions
 
Arcs
 
Unitals
 
The Collineation Group of a Plane
      The Collineation Group Function
      General Action of Collineations
      Central Collineations
      Transitivity Properties
 
Translation Planes
 
Planes and Designs
 
Planes, Graphs and Codes







DETAILS

 
Introduction

      Planes in Magma

 
Construction of a Plane
      FiniteProjectivePlane< v | X : parameters > : RngIntElt, List -> PlaneProj
      FiniteProjectivePlane(W) : ModTupFld -> PlaneProj
      FiniteAffinePlane< v | X : parameters > : RngIntElt, List -> PlaneAff
      FiniteAffinePlane(W) : ModFld -> PlaneAff
      Example Plane_Constructors (H141E1)

 
The Point-Set and Line-Set of a Plane

      Introduction

      Creating Point-Sets and Line-Sets
            PointSet(P) : Plane -> PlanePtSet
            LineSet(P) : Plane -> PlaneLnSet

      Using the Point-Set and Line-Set to Create Points and Lines
            V . i : PlanePtSet, RngIntElt -> PlanePt
            V ! [a, b, c] : PlanePtSet, SeqEnum -> PlanePt
            V ! [a, b] : PlanePtSet, SeqEnum -> PlanePt
            V ! x : PlanePtSet, Elt -> PlanePt
            Representative(V) : PlanePtSet -> PlanePt
            Random(V) : PlanePtSet -> PlanePt
            L . i : PlanePtSet, RngIntElt -> PlanePt
            L ! [a, b, c] : PlaneLnSet, SeqEnum -> PlaneLn
            L ! [m, b] : PlaneLnSet, SeqEnum -> PlaneLn
            L ! S : PlaneLnSet, SetEnum -> PlaneLn
            L ! l : PlaneLnSet, PlaneLn -> PlaneLn
            Representative(L) : PlaneLnSet -> PlaneLn
            Random(L) : PlaneLnSet -> PlaneLn
            Example Plane_points-lines (H141E2)

      Retrieving the Plane from Points, Lines, Point-Sets and Line-Sets
            ParentPlane(V) : PlanePtSet -> Plane, PlanePtSet, PlaneLnSet
            ParentPlane(L) : PlaneLnSet -> Plane, PlanePtSet, PlaneLnSet
            ParentPlane(p) : PlanePt -> Plane, PlanePtSet, PlaneLnSet
            ParentPlane(l) : PlaneLn -> Plane, PlanePtSet, PlaneLnSet

 
The Set of Points and Set of Lines
      Points(P) : Plane -> { PlanePt }
      Lines(P) : PlaneLnSet -> { PlaneLn }

 
The Defining Points of a Plane
      Support(P) : Plane -> { Elt }
      Support(l) : PlaneLn -> SetEnum
      Support(P, p) : Plane, PlanePt -> .
      Example Plane_supp (H141E3)

 
Subplanes
      sub<P | L> : Plane, List -> Plane
      SubfieldSubplane(P, F) : Plane, FldFin -> Plane, PlanePtSet, PlaneLnSet
      Example Plane_sub (H141E4)

 
Structures Associated with a Plane
      VectorSpace(P) : Plane -> ModTupFld
      Field(P) : Plane -> FldFin
      IncidenceMatrix(P) : Plane -> AlgMatElt
      Dual(P) : Plane -> Plane, PlanePtSet, PlaneLnSet
      Example Plane_sub (H141E5)

 
Numerical Invariants of a Plane
      Order(P) : Plane -> RngIntElt
      NumberOfPoints(P) : Plane -> RngIntElt
      NumberOfLines(P) : Plane -> RngIntElt
      pRank(P) : Plane -> RngIntElt
      pRank(P, p) : Plane -> RngIntElt
      Example Plane_invar (H141E6)

 
Properties of Planes
      IsDesarguesian(P) : Plane -> BoolElt
      IsSelfDual(P) : PlaneProj -> BoolElt

 
Identity and Isomorphism
      P eq Q : Plane, Plane -> BoolElt
      P ne Q : Plane, Plane -> BoolElt
      IsIsomorphic(P, Q: parameters) : Plane, Plane -> BoolElt, Map
      P subset Q : Plane, Plane -> BoolElt

 
The Connection between Projective and Affine Planes
      FiniteAffinePlane(P, l) : PlaneProj, PlaneLn -> PlaneAff, PlanePtSet, PlaneLnSet, Map
      ProjectiveEmbedding(P) : PlaneAff -> PlaneProj, PlanePtSet, PlaneLnSet, Map
      Example Plane_embedding (H141E7)

 
Operations on Points and Lines

      Elementary Operations
            p eq q : PlanePt, PlanePt -> BoolElt
            p ne q : PlanePt, PlanePt -> BoolElt
            l eq m : PlaneLn, PlaneLn -> BoolElt
            l ne m : PlaneLn, PlaneLn -> BoolElt
            p in l : PlanePt, PlaneLn -> BoolElt
            p notin l : PlanePt, PlaneLn -> BoolElt
            S subset l : { PlanePt }, PlaneLn -> BoolElt
            S notsubset l : { PlanePt }, PlaneLn -> BoolElt
            l meet m : PlaneLn, PlaneLn -> PlanePt
            Representative(l) : PlaneLn -> PlanePt
            Random(l) : PlaneLn -> PlanePt

      Deconstruction Functions
            Index(P, p) : PlanePt -> RngIntElt
            Index(P, l) : PlaneLn -> RngIntElt
            p[i] : PlanePt, RngIntElt -> FldFinElt
            l[i] : PlaneLn, RngIntElt -> FldFinElt
            Coordinates(P, p) : Plane, PlanePt -> [ FldFinElt ]
            Coordinates(P, l) : Plane, PlaneLn -> [ FldFinElt ]
            ElementToSequence(p) : PlanePt -> [ FldFinElt ]
            ElementToSequence(l) : PlaneLn -> [ FldFinElt ]
            Set(l) : PlaneLn -> { PlanePt }
            Example Plane_decon (H141E8)

      Other Point and Line Functions
            IsCollinear(P, S) : Plane, { PlanePt } -> BoolElt, PlaneLn
            IsConcurrent(P, R) : Plane, { PlaneLn } -> BoolElt, PlanePt
            ContainsQuadrangle(P, S) : Plane, { PlanePt } -> BoolElt
            Pencil(P, p) : Plane, PlanePt -> { PlaneLn }
            Slope(l) : PlaneLn -> FldFinElt
            IsParallel(P, l, m) : Plane, PlaneLn, PlaneLn -> BoolElt
            ParallelClass(P, l) : Plane, PlaneLn -> { PlaneLn }
            ParallelClasses(P) : PlaneAff -> { { PlaneLn } }
            Example Plane_elt-other (H141E9)

 
Arcs
      kArc(P, k) : Plane, RngIntElt -> SetEnum
      CompleteKArc(P, k) : Plane, RngIntElt -> SetEnum
      IsArc(P, A) : Plane, { PlanePt } -> BoolElt
      IsComplete(P, A) : Plane, { PlanePt } -> BoolElt
      Conic(P, S) : Plane, { PlanePt } -> SetEnum
      QuadraticForm(S) : { PlanePt } -> RngMPolElt
      Tangent(P, A, p) : Plane, { PlanePt }, PlanePt -> PlaneLn
      AllTangents(P, A) : Plane, { PlanePt } -> { PlaneLn }
      AllSecants(P, A) : Plane, { PlanePt } -> { PlaneLn }
      ExternalLines(P, A) : Plane, { PlanePt } -> { PlaneLn }
      Knot(P, C) : Plane, { PlanePt } -> PlanePt
      Exterior(P, C) : Plane, { PlanePt } -> { PlanePt }
      Interior(P, C) : Plane, { PlanePt } -> { PlanePt }
      Example Plane_arcs (H141E10)

 
Unitals
      IsUnital(P, U) : Plane, { PlanePt } -> BoolElt
      AllTangents(P, U) : Plane, { PlanePt } -> { PlaneLn }
      UnitalFeet(P, U, p) : Plane, { PlanePt }, PlanePt -> { PlanePt }
      Example Plane_unital (H141E11)

 
The Collineation Group of a Plane

      The Collineation Group Function
            CollineationGroup(P) : Plane -> GrpPerm, GSet, GSet, PowMap, Map
            LineGroup(P) : Plane -> GrpPerm, PowMap, Map
            CollineationGroupStabilizer(P, k) : Plane, RngIntElt -> GrpPerm, GSet, GSet, PowMap, Map
            CollineationSubgroup(P) : Plane -> GrpPerm, GSet, GSet, PowMap, Map

      General Action of Collineations
            y ^ g : Elt, GrpPermElt -> Elt
            y ^ G : Elt, GrpPerm -> GSet
            Image(g, Y, y) : GrpPermElt, GSet, Elt -> Elt
            Orbit(G, Y, y) : GrpPerm, GSet, Elt -> GSet
            Orbits(G, Y) : GrpPerm, GSet -> [ GSet ]
            Stabilizer(G, Y, y) : GrpPerm, Elt -> GrpPerm
            Action(G, Y) : GrpPerm, GSet -> Hom(Grp), GrpPerm, GrpPerm
            ActionImage(G, Y) : GrpPerm, GSet -> GrpPerm
            ActionKernel(G, Y) : GrpPerm, GSet -> GrpPerm
            Example Plane_CollineationGSet (H141E12)
            Example Plane_Collineation (H141E13)
            Example Plane_baer (H141E14)

      Central Collineations
            CentralCollineationGroup(P, p, l) : Plane, PlanePt, PlaneLn -> GrpPerm, PowMap, Map
            CentralCollineationGroup(P, p) : Plane, PlanePt -> GrpPerm, PowMap, Map
            CentralCollineationGroup(P, l) : Plane, PlaneLn -> GrpPerm, PowMap, Map
            IsCentralCollineation(P, g) : Plane, GrpPermElt -> BoolElt, PlanePt, PlaneLn
            Example Plane_cent-coll (H141E15)

      Transitivity Properties
            IsPointTransitive(P) : Plane -> BoolElt
            IsLineTransitive(P) : Plane -> BoolElt
            Example Plane_trans (H141E16)

 
Translation Planes
      BaerDerivation(q2) : RngIntElt -> PlaneAff, PlanePtSet, PlaneLnSet
      BaerSubplane(P) : PlaneProj -> PlaneProj, PlanePtSet, PlaneLnSet
      OvalDerivation(q: parameters) : RngIntElt -> PlaneAff, PlanePtSet, PlaneLnSet

 
Planes and Designs
      Design(P) : Plane -> Dsgn, SetIncPt, SetIncBlk
      FiniteAffinePlane(D) : Inc -> Plane, PlanePtSet, PlaneLnSet
      FiniteProjectivePlane(D) : Inc -> Plane, PlanePtSet, PlaneLnSet
      Example Plane_designs (H141E17)

 
Planes, Graphs and Codes
      LineGraph(P) : Plane -> Grph
      IncidenceGraph(P) : Plane -> Grph
      LinearCode(P, K) : Plane, FldFin -> Code
      Example Plane_codes (H141E18)

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013