Boolean Representations of Simplicial Complexes and Matroids

179 Pages·2015·2.314 MB·English

by  John Rhodes, Pedro V. Silva (auth.)

Springer Monographs in Mathematics
John Rhodes
Pedro V. Silva
Boolean
Representations
of Simplicial
Complexes and
Matroids
Springer Monographs in Mathematics
Moreinformationaboutthisseriesathttp://www.springer.com/series/3733

John Rhodes • Pedro V. Silva
Boolean Representations
of Simplicial Complexes
and Matroids
123
JohnRhodes PedroV.Silva
DepartmentofMathematics DepartmentofMathematics
UniversityofCaliforniaatBerkeley UniversityofPorto
Berkeley,CA,USA Porto,Portugal
ISSN1439-7382 ISSN2196-9922 (electronic)
SpringerMonographsinMathematics
ISBN978-3-319-15113-7 ISBN978-3-319-15114-4 (eBook)
DOI10.1007/978-3-319-15114-4
LibraryofCongressControlNumber:2015931927
Mathematics Subject Classification (2010): 03E05, 03G10, 05-00, 05B25, 05B30, 05B35, 05D15,
05E45,06-00,06A11,06A12,06B15,15B34,16Y60,51E14,55P15,55U10
SpringerChamHeidelbergNewYorkDordrechtLondon
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InmemoryofGian-CarloRota

Acknowledgments
Both the authors thank Anne Schilling, Benjamin Steinberg, James Oxley, and
StuartMargolisfortheirvaluablecomments.
JohnRhodeswouldliketothankhisremarkablewife,LauraMorland,forallher
loveandsupport.
PedroV.SilvaisgratefultohiswifeMargaridaforherenduringsupportandfor
everythingelse!Healsoacknowledgessupportfrom
(cid:129) TheEuropeanRegionalDevelopmentFundthroughtheprogrammeCOMPETE
and the Portuguese Government through FCT (Fundação para a Ciência e a
Tecnologia)undertheprojectPEst-C/MAT/UI0144/2013
(cid:129) CNPq(Brazil)throughaBJT-Agrant(process313768/2013-7)
vii

Contents
1 Introduction .................................................................. 1
2 BooleanandSuperbooleanMatrices ...................................... 9
2.1 TheBooleanandtheSuperbooleanSemirings........................ 9
2.2 SuperbooleanMatrices................................................. 11
3 PosetsandLattices........................................................... 17
3.1 BasicNotions........................................................... 17
3.2 RepresentationofPosets............................................... 21
3.3 _-GeneratingSubsetsofLattices...................................... 22
3.4 TheLatticeofFlatsofaMatrix........................................ 23
3.5 MatricesVersusLattices ............................................... 24
3.6 c-Independenceandc-Rank............................................ 28
4 SimplicialComplexes........................................................ 31
4.1 TheCombinatorialPerspective........................................ 31
4.1.1 Matroids ........................................................ 32
4.1.2 Graphs .......................................................... 33
4.2 Flats..................................................................... 34
5 BooleanRepresentations.................................................... 39
5.1 SuperbooleanandBooleanRepresentations .......................... 39
5.2 TheCanonicalBooleanRepresentation............................... 41
5.3 LowDimensions........................................................ 48
5.4 LatticeRepresentations ................................................ 49
5.5 TheLatticeofLatticeRepresentations................................ 50
5.6 MinimumDegree....................................................... 58
5.7 Examples................................................................ 62
5.7.1 TheTetrahedronComplexesT3andT2 ....................... 62
5.7.2 TheFanoMatroid.............................................. 68
5.7.3 TheUniformMatroidU3;n..................................... 73
5.7.4 SteinerSystems ................................................ 80
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