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Posets

Posets and finite spaces.

1.0.2

25 May 2020

Ximena Fernández
Email: xfernand@dm.uba.ar
Homepage: http://mate.dm.uba.ar/~xfernand
Address:
Pabellón I - Ciudad Universitaria (1428)
Buenos Aires
Argentina

Kevin Piterman
Email: kpiterman@dm.uba.ar
Homepage: http://mate.dm.uba.ar/~kpiterman
Address:
Pabellón I - Ciudad Universitaria (1428)
Buenos Aires
Argentina

Iván Sadofschi Costa
Email: isadofschi@dm.uba.ar
Homepage: http://mate.dm.uba.ar/~isadofschi
Address:
Pabellón I - Ciudad Universitaria (1428)
Buenos Aires
Argentina

Contents

2.1 The Poset type

  2.1-1 Ordering
  2.1-2 Set
  2.1-3 \=
  2.1-4 Size
  2.1-5 Iterator
  2.1-6 \in

2.2 Examples of posets

  2.2-1 EmptyPoset
  2.2-2 MinimalFiniteModelSphere
  2.2-3 TheWallet
  2.2-4 TheTriangle
  2.2-5 FiniteModelProjectivePlane

2.3 Plotting posets

  2.3-1 DotCode
  2.3-2 DotCodeToFile
  2.3-3 PlotHasseDiagram
  2.3-4 PlotMorseMatching

2.4 Homomorphisms

  2.4-1 IdentityMap
  2.4-2 CompositionPosetHomomorphisms
  2.4-3 SourceMap
  2.4-4 TargetMap
  2.4-5 UnderlyingFunction
  2.4-6 Inverse
  2.4-7 ImageMap
  2.4-8 ImageMap
  2.4-9 \*
  2.4-10 \*
  2.4-11 \^

2.5 Constructing poset homomorphisms in different ways

  2.5-1 PosetHomomorphismByFunction
  2.5-2 PosetHomomorphismByFunctionNC
  2.5-3 PosetHomomorphismByImages
  2.5-4 PosetHomomorphismByMapping
  2.5-5 IsomorphismPosets

2.6 Construction of posets

  2.6-1 PosetByFunction
  2.6-2 PosetByFunctionNC
  2.6-3 PosetByOrderMatrix
  2.6-4 PosetByOrderMatrix
  2.6-5 PosetByOrderMatrixNC
  2.6-6 PosetByOrderMatrixNC
  2.6-7 PosetByOrderRelation
  2.6-8 PosetByHasseDiagram
  2.6-9 PosetByCoveringRelations

2.7 Representations of the ordering

  2.7-1 RelationByPoset
  2.7-2 OrderMatrix
  2.7-3 OrderingByIndex
  2.7-4 CoveringRelations
  2.7-5 UpperCovers
  2.7-6 UpperCovers
  2.7-7 LowerCovers
  2.7-8 LowerCovers
  2.7-9 MaximalElements
  2.7-10 MinimalElements
  2.7-11 MaximumPoset
  2.7-12 MinimumPoset
  2.7-13 IndicesElementsAbove
  2.7-14 IndicesElementsBelow

2.8 Saving posets

2.8-1 SavePosetAsListsOfElementsBelow
2.8-2 SavePosetAsOrderMatrix

3 Classical stuff

3.1 Elements above and below

  3.1-1 ElementsAbove
  3.1-2 ElementsStrictlyAbove
  3.1-3 ElementsBelow
  3.1-4 ElementsStrictlyBelow
  3.1-5 ElementsComparable
  3.1-6 ElementsStrictlyComparable
  3.1-7 RemovePoint
  3.1-8 RemoveCoveringRelation

3.2 Grading and height

  3.2-1 Grading
  3.2-2 Height
  3.2-3 Height
  3.2-4 ChainsPoset
  3.2-5 MobiusMatrix
  3.2-6 MobiusFunction

4 Construction of new posets

4.1 Categorical constructions

  4.1-1 SubPoset
  4.1-2 CoproductOp
  4.1-3 Coproduct
  4.1-4 Join
  4.1-5 ConePoset
  4.1-6 SuspensionPoset
  4.1-7 OppositePoset
  4.1-8 MappingCylinderPosetHomomorphism
  4.1-9 QuotientPoset
  4.1-10 WedgePosets
  4.1-11 HomPosets
  4.1-12 Endomorphisms
  4.1-13 AutomorphismGroup
  4.1-14 Automorphisms

4.2 Natural maps

4.2-1 NaturalMaps

5.1 Face poset and order complex

  5.1-1 FacePoset
  5.1-2 OrderComplex
  5.1-3 FacePoset
  5.1-4 OrderComplex
  5.1-5 BarycentricSubdivision
  5.1-6 BarycentricSubdivision
  5.1-7 BarycentricSubdivision
  5.1-8 BarycentricSubdivision

5.2 Topological invariants

  5.2-1 PosetHomology
  5.2-2 PosetHomology
  5.2-3 PosetHomology
  5.2-4 EulerCharacteristic
  5.2-5 FundamentalGroup

5.3 Fixed Point Property

5.3-1 FixedPointsPosetHomomorphism
5.3-2 HasFixedPointProperty

6 Homotopy theory

6.1 Beat points and homotopy type

  6.1-1 IsBeatPoint
  6.1-2 IsUpBeatPoint
  6.1-3 IsDownBeatPoint
  6.1-4 UpBeatPoints
  6.1-5 DownBeatPoints
  6.1-6 BeatPoints
  6.1-7 Core
  6.1-8 CoreParallel
  6.1-9 IsContractible
  6.1-10 HomotopyEquivalence

6.2 Weak points and simple homotopy type

  6.2-1 IsWeakPoint
  6.2-2 IsUpWeakPoint
  6.2-3 IsDownWeakPoint
  6.2-4 WeakPoints
  6.2-5 UpWeakPoints
  6.2-6 DownWeakPoints
  6.2-7 WeakCore
  6.2-8 MinWeakCore
  6.2-9 IsCollapsible

  6.3-1 FundamentalGroupByColoring
  6.3-2 SpanningForestHasseDiagram
  6.3-3 RandomSpanningForestHasseDiagram
  6.3-4 SpanningTreeHasseDiagram
  6.3-5 RandomSpanningTreeHasseDiagram
  6.3-6 SpanningCollapsibleSubdiagram
  6.3-7 RandomSpanningCollapsibleSubdiagram

6.4 Connectivity

6.4-1 PathInPoset
6.4-2 ConnectedComponents

6.5 Fences and homotopies

6.5-1 Fence
6.5-2 AreHomotopic

7 More reductions

7.1 QC Reductions

  7.1-1 IsDownQCReduction
  7.1-2 IsUpQCReduction
  7.1-3 IsQCReduction
  7.1-4 QCCore
  7.1-5 QCCores

7.2 Osaki reductions

  7.2-1 IsDownOsakiReduction
  7.2-2 IsUpOsakiReduction
  7.2-3 DownOsakiReduction
  7.2-4 UpOsakiReduction
  7.2-5 DownOsakiCore
  7.2-6 UpOsakiCore

7.3 Middle reductions

  7.3-1 IsDownMiddleReduction
  7.3-2 IsUpMiddleReduction
  7.3-3 IsMiddleReduction
  7.3-4 MiddleReduction
  7.3-5 MiddleReductionCore

7.4 Edge reductions

  7.4-1 IsDownEdgeReduction
  7.4-2 IsUpEdgeReduction
  7.4-3 IsEdgeReduction
  7.4-4 EdgeReduction
  7.4-5 EdgeReductionCore

7.5 Random reductions

  7.5-1 RandomWeakPointReduction
  7.5-2 RandomQCReduction
  7.5-3 RandomMiddleReduction
  7.5-4 RandomEdgeReduction
  7.5-5 RandomOsakiReduction
  7.5-6 RandomReductionCore
  7.5-7 RandomReduction

8 Posets and presentations

8.1 The poset associated to a group presentation

8.1-1 PosetFpGroup

8.2 Discrete Morse theory for presentations

8.2-1 MorsePresentation
8.2-2 GreedySpanningMatching

8.3 Colorings and presentations

8.3-1 ColoringPresentation

9 Posets of subgroups of a finite group

9.1 Posets of subgroups

  9.1-1 PosetOfSubgroups
  9.1-2 PosetOfpSubgroups
  9.1-3 ElementaryAbelianpSubgroups
  9.1-4 PosetOfElementaryAbelianpSubgroups
  9.1-5 RadicalpSubgroups
  9.1-6 PosetOfRadicalpSubgroups
  9.1-7 RobinsonPoset
  9.1-8 OrbitSubdivisionBoucPoset
  9.1-9 OrbitSubdivisionQuillenPoset
  9.1-10 OrbitSubdivisionBrownPoset
  9.1-11 EulerCharacteristicQuillenPoset

9.2 Poset of subspaces of a vector space

9.2-1 PosetOfSubspaces
9.2-2 PosetOfProperSubspaces

10 The Andrews-Curtis conjecture

10.1 Potential Counterexamples

  10.1-1 AkbulutKirbyGroup
  10.1-2 RapaportGroup
  10.1-3 GordonGroup

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