GeneralizedHibiringsandHibiideals - PDF document

GeneralizedHibiringsandHibiidealsJ¨urge
GeneralizedHibiringsandHibiidealsJ¨urgenHerzogUniversit¨atDuisburg-EssenEllwangen,March2011OutlineHibiringsHibiidealsGeneralizedHibiringsandHibiidealsOutlineHibiringsHibiidealsGeneralizedHibiringsandHibiidealsOutlineHibiringsHibiidealsGeneralizedHibiringsa

ndHibiidealsHibiringsIn1985Hibiintrodu
ndHibiidealsHibiringsIn1985HibiintroducedaclassofalgebraswhichnowadaysarecalledHibirings.Theyaresemigroupringsattachedtoniteposets,andmaybeviewedasnaturalgeneralizationsofpolynomialrings.HibiringsIn1985HibiintroducedaclassofalgebraswhichnowadaysarecalledHibi

rings.Theyaresemigroupringsattachedton
rings.Theyaresemigroupringsattachedtoniteposets,andmaybeviewedasnaturalgeneralizationsofpolynomialrings.Indeed,apolynomialringinnvariablesoveraeldKisjusttheHibiringoftheposet[n1]=f1;2;:::;n1g.HibiringsIn1985Hibiintroducedaclassofalgebraswhichnowadaysarecall

edHibirings.Theyaresemigroupringsattac
edHibirings.Theyaresemigroupringsattachedtoniteposets,andmaybeviewedasnaturalgeneralizationsofpolynomialrings.Indeed,apolynomialringinnvariablesoveraeldKisjusttheHibiringoftheposet[n1]=f1;2;:::;n1g.LetP=fp1;:::;pngbeaniteposet.AposetidealIofPisasubsetof

Pwhichsatisesthefollowingcondition:fore
Pwhichsatisesthefollowingcondition:foreveryp2I;andq2Pwithqp,itfollowsq2I.HibiringsIn1985HibiintroducedaclassofalgebraswhichnowadaysarecalledHibirings.Theyaresemigroupringsattachedtoniteposets,andmaybeviewedasnaturalgeneralizationsofpolynomialrings.Ind

eed,apolynomialringinnvariablesoveraeld
eed,apolynomialringinnvariablesoveraeldKisjusttheHibiringoftheposet[n1]=f1;2;:::;n1g.LetP=fp1;:::;pngbeaniteposet.AposetidealIofPisasubsetofPwhichsatisesthefollowingcondition:foreveryp2I;andq2Pwithqp,itfollowsq2I.LetI(P)bethesetoftheposetideals

ofP.ThenI(P)isasublatticeofthepowersetof
ofP.ThenI(P)isasublatticeofthepowersetofP;andhenceitisadistributivelattice.ByBirkhoff'stheoremanynitedistributivelatticearisesinthisway.ByBirkhoff'stheoremanynitedistributivelatticearisesinthisway.LetKbeaeld.ThentheHibiringoverKattachedtoPisthetoricringK[I(P)]

K[x1;:::;xn;y1;:::;yn]generatedbythese
K[x1;:::;xn;y1;:::;yn]generatedbythesetofmonomialsfuI:I2I(P)gwhereuI=Qpi2IxiQpi62Iyi.ByBirkhoff'stheoremanynitedistributivelatticearisesinthisway.LetKbeaeld.ThentheHibiringoverKattachedtoPisthetoricringK[I(P)]K[x1;:::;xn;y1;:::;yn]generatedbythesetofmono

mialsfuI:I2I(P)gwhereuI=Qpi2IxiQpi62I
mialsfuI:I2I(P)gwhereuI=Qpi2IxiQpi62Iyi.LetT=K[ftI:tI2I(P)g]bethepolynomialringinthevariablestIoverK,and'T!K[I(P)]theK-algebrahomomorphismwithtI7!uI.ByBirkhoff'stheoremanynitedistributivelatticearisesinthisway.LetKbeaeld.ThentheHibiringoverKattachedtoPist

hetoricringK[I(P)]K[x1;:::;xn;y1;:::;y
hetoricringK[I(P)]K[x1;:::;xn;y1;:::;yn]generatedbythesetofmonomialsfuI:I2I(P)gwhereuI=Qpi2IxiQpi62Iyi.LetT=K[ftI:tI2I(P)g]bethepolynomialringinthevariablestIoverK,and'T!K[I(P)]theK-algebrahomomorphismwithtI7!uI.OnefundamentalresultconcerningHibiringsisth

atthetoricidealLP=Ker'hasareducedGr¨obne
atthetoricidealLP=Ker'hasareducedGr¨obnerbasisconsistingoftheso-calledHibirelations:tItJtI\JtI[JwithI6JandJ6I:HibishowedthatanyHibiringisanormalCohen–Macaulaydomainofdimension1+jPj,andthatitisGorensteinifandonlyiftheattachedposetPisgraded,thatis,allmaximalc

hainsofPhavethesamecardinality.Hibishow
hainsofPhavethesamecardinality.HibishowedthatanyHibiringisanormalCohen–Macaulaydomainofdimension1+jPj,andthatitisGorensteinifandonlyiftheattachedposetPisgraded,thatis,allmaximalchainsofPhavethesamecardinality.Moregenerally,foranynitelatticeL,notnecessarilydist

ributive,onemayconsidertheKalgebraK[L]
ributive,onemayconsidertheKalgebraK[L]withgeneratorsy,2L,andrelationsyy=y^y_where^and_denotemeetandjoininL.HibishowedthatK[L]isadomainifandonlyifLisdistributive,inotherwords,ifLisanideallatticeofaposet.LetKbeaeldandX=(xij)i=1;:::;mj=1;:::;namatrixofi

ndeterminates.WedenotebyK[X]thepolynomia
ndeterminates.WedenotebyK[X]thepolynomialringoverKwiththeindeterminatesxij,andbyAtheK-subalgebraofK[X]generatedbyallmaximalminorsofX.LetKbeaeldandX=(xij)i=1;:::;mj=1;:::;namatrixofindeterminates.WedenotebyK[X]thepolynomialringoverKwiththeindeterminatesxij,andbyA

theK-subalgebraofK[X]generatedbyallmaxim
theK-subalgebraofK[X]generatedbyallmaximalminorsofX.TheK-algebraAK[X]isthecoordinateringoftheGrassmannianofthem-dimensionalvectorK-subspacesofKn.LetKbeaeldandX=(xij)i=1;:::;mj=1;:::;namatrixofindeterminates.WedenotebyK[X]thepolynomialringoverKwiththeindeter

minatesxij,andbyAtheK-subalgebraofK[X]ge
minatesxij,andbyAtheK-subalgebraofK[X]generatedbyallmaximalminorsofX.TheK-algebraAK[X]isthecoordinateringoftheGrassmannianofthem-dimensionalvectorK-subspacesofKn.LetbethelexicographicorderonK[X]inducedbyx11�x12�


�x1n�x21�x2

2�


�xm1�xm2�
2�


�xm1�xm2�


�xmn:Wedenoteby=[a1;a2;:::;am]themaximalminorofXwithcolumnsa1a2


am.Thenin()=x1;a1x2;a2


xm;amisthe`diagonal'of.LetS=K[x1;:::;xn]beapolynomialring,amonomialorderonSandASaK-subalgebra.LetS=K[x1;:::

;xn]beapolynomialring,amonomialorderonS
;xn]beapolynomialring,amonomialorderonSandASaK-subalgebra.ThentheK-algebrain(A)generatedbyallmonomialsin(f)withf2AiscalledtheinitialalgebraofAwithrespectto.LetS=K[x1;:::;xn]beapolynomialring,amonomialorderonSandASaK-subalgebra.ThentheK-algebrain(A

)generatedbyallmonomialsin(f)withf2A
)generatedbyallmonomialsin(f)withf2AiscalledtheinitialalgebraofAwithrespectto.Ingeneralin(A)isnotnitelygenerated.AsubsetSAiscalledaSagbibasesofAwithrespectto,iftheelementsf2SgenerateAoverK.ThisconcepthasbeenintroducedbyRobbianoandSweedlerandindependentl

ybyKapurandMadlener.LetS=K[x1;:::;xn]
ybyKapurandMadlener.LetS=K[x1;:::;xn]beapolynomialring,amonomialorderonSandASaK-subalgebra.ThentheK-algebrain(A)generatedbyallmonomialsin(f)withf2AiscalledtheinitialalgebraofAwithrespectto.Ingeneralin(A)isnotnitelygenerated.AsubsetSAiscalledaSagbi

basesofAwithrespectto,iftheelementsf2Sg
basesofAwithrespectto,iftheelementsf2SgenerateAoverK.ThisconcepthasbeenintroducedbyRobbianoandSweedlerandindependentlybyKapurandMadlener.TheoremThemaximalminorsofXformaSagbibasesoftheGrassmannianalgebraA.LetS=K[x1;:::;xn]beapolynomialring,amonomialorderonSandA

SaK-subalgebra.ThentheK-algebrain(A)g
SaK-subalgebra.ThentheK-algebrain(A)generatedbyallmonomialsin(f)withf2AiscalledtheinitialalgebraofAwithrespectto.Ingeneralin(A)isnotnitelygenerated.AsubsetSAiscalledaSagbibasesofAwithrespectto,iftheelementsf2SgenerateAoverK.Thisconcepthasbeenintroduce

dbyRobbianoandSweedlerandindependentlyby
dbyRobbianoandSweedlerandindependentlybyKapurandMadlener.TheoremThemaximalminorsofXformaSagbibasesoftheGrassmannianalgebraA.Whatistheuseofthistheorem?WedeneapartialonthesetLofmaximalminorsofX:[a1;a2;:::;am][b1;b2;:::;bm],aibiforalliThesetLwiththispartialorderi

sadistributivelattice.Wedeneapartialon
sadistributivelattice.WedeneapartialonthesetLofmaximalminorsofX:[a1;a2;:::;am][b1;b2;:::;bm],aibiforalliThesetLwiththispartialorderisadistributivelattice.Theoremin(A)isisomorphictotheHibiringK[L]ofthelatticeL.WedeneapartialonthesetLofmaximalminorsofX:

[a1;a2;:::;am][b1;b2;:::;bm],aibiforal
[a1;a2;:::;am][b1;b2;:::;bm],aibiforalliThesetLwiththispartialorderisadistributivelattice.Theoremin(A)isisomorphictotheHibiringK[L]ofthelatticeL.Indeed,letTbethepolynomialringoverKinthevariablestwith2L,andlet :T!in(A)betheK-algebrahomomorphismwith (t)=in

().OneshowsthattheHibirelationst1t2
().OneshowsthattheHibirelationst1t2t1_2t1^2;1;12LgenerateKer .WedeneapartialonthesetLofmaximalminorsofX:[a1;a2;:::;am][b1;b2;:::;bm],aibiforalliThesetLwiththispartialorderisadistributivelattice.Theoremin(A)isisomorphictotheHibiringK[L]ofthelatt

iceL.Indeed,letTbethepolynomialringove
iceL.Indeed,letTbethepolynomialringoverKinthevariablestwith2L,andlet :T!in(A)betheK-algebrahomomorphismwith (t)=in().OneshowsthattheHibirelationst1t2t1_2t1^2;1;12LgenerateKer .CorollaryThecoordinateringAoftheGrassmannianofm-dimensionalK-subspacesofK

nisaGorensteinringofdimensionm(nm)+1.
nisaGorensteinringofdimensionm(nm)+1.HibiidealsLetPbeaniteposet.TheidealHPK[x1;:::;xn;y1;:::;yn]whichhisgeneratedbythemonomialsuI=Yp2IxpYp62Iyq;II(P)iscalledtheHibiidealofP.HibiidealsLetPbeaniteposet.TheidealHPK[x1;:::;xn;y1;:::;yn]whichhisgenera

tedbythemonomialsuI=Yp2IxpYp62Iyq;II(P
tedbythemonomialsuI=Yp2IxpYp62Iyq;II(P)iscalledtheHibiidealofP.Theorem(a)HPhasalinearresolution.(b)HP=Tpq(xp;yq).HibiidealsLetPbeaniteposet.TheidealHPK[x1;:::;xn;y1;:::;yn]whichhisgeneratedbythemonomialsuI=Yp2IxpYp62Iyq;II(P)iscalledtheHibiideal

ofP.Theorem(a)HPhasalinearresolution.(
ofP.Theorem(a)HPhasalinearresolution.(b)HP=Tpq(xp;yq).Application:LetGbeanitesimplegraphonthevertexset[n].OnedenestheedgeidealIGofGasthemonomialidealinK[x1;:::;xn]withsetofgeneratorsfxixj:fi;jg2E(G)g.HibiidealsLetPbeaniteposet.TheidealHPK[x1;:::;x

n;y1;:::;yn]whichhisgeneratedbythemonom
n;y1;:::;yn]whichhisgeneratedbythemonomialsuI=Yp2IxpYp62Iyq;II(P)iscalledtheHibiidealofP.Theorem(a)HPhasalinearresolution.(b)HP=Tpq(xp;yq).Application:LetGbeanitesimplegraphonthevertexset[n].OnedenestheedgeidealIGofGasthemonomialidealinK[x1;:::;xn]

withsetofgeneratorsfxixj:fi;jg2E(G)g.F
withsetofgeneratorsfxixj:fi;jg2E(G)g.ForwhichgraphsisIGCohen–Macaulay?Theorem(H-Hibi)LetGbeabipartitegraphwithvertexpartitionV[V0.Thenthefollowingconditionsareequivalent:(a)GisaCohen–Macaulaygraph;(b)jVj=jV0jandtheverticesV=fx1;:::;xngandV0=fy1;:::;yngcanbela

belledsuchthat:(i)fxi;yigareedgesfori=
belledsuchthat:(i)fxi;yigareedgesfori=1;:::;n;(ii)iffxi;yjgisanedge,thenij;(iii)iffxi;yjgandfxj;ykgareedges,thenfxi;ykgisanedge.P1234y1y2y3y4x1x2x3x4G(P)TheAlexanderdual:letIbeasquarefreemonomialideal.ThenI=r\j=1PFj;whereforasubsetF[n]weset

PF=(fxi:i2Fg).TheAlexanderdual:letIbea
PF=(fxi:i2Fg).TheAlexanderdual:letIbeasquarefreemonomialideal.ThenI=r\j=1PFj;whereforasubsetF[n]wesetPF=(fxi:i2Fg).TheidealI_=(xF1;:::;xFr)iscalledtheAlexanderdualofI.HereforasubsetF[n]wesetxF=Qi2Fxi.TheAlexanderdual:letIbeasquarefreemonomialideal.The

nI=r\j=1PFj;whereforasubsetF[n]weset
nI=r\j=1PFj;whereforasubsetF[n]wesetPF=(fxi:i2Fg).TheidealI_=(xF1;:::;xFr)iscalledtheAlexanderdualofI.HereforasubsetF[n]wesetxF=Qi2Fxi.Example:I=(x1x4;x1x5;x2x5;x3x5)=(x1;x2;x3)\(x1;x5)\(x4;x5).TheAlexanderdual:letIbeasquarefreemonomialideal.ThenI=r\

j=1PFj;whereforasubsetF[n]wesetPF=(fx
j=1PFj;whereforasubsetF[n]wesetPF=(fxi:i2Fg).TheidealI_=(xF1;:::;xFr)iscalledtheAlexanderdualofI.HereforasubsetF[n]wesetxF=Qi2Fxi.Example:I=(x1x4;x1x5;x2x5;x3x5)=(x1;x2;x3)\(x1;x5)\(x4;x5).ThereforeI_=(x1x2x3;x1x5;x4x5).Theorem(Eagon-Reiner)LetISbea

squarefreemonomialideal.ThenI_isCohen-Ma
squarefreemonomialideal.ThenI_isCohen-Macaulay,ifandonlyifIhasalinearresolution.Theorem(Eagon-Reiner)LetISbeasquarefreemonomialideal.ThenI_isCohen-Macaulay,ifandonlyifIhasalinearresolution.SinceHP=Tpq(xp;yq)andhasalinearresolution,theAlexanderdualH_PisCohen–M

acaulaybytheEagon–ReinerTheorem.Theorem
acaulaybytheEagon–ReinerTheorem.Theorem(Eagon-Reiner)LetISbeasquarefreemonomialideal.ThenI_isCohen-Macaulay,ifandonlyifIhasalinearresolution.SinceHP=Tpq(xp;yq)andhasalinearresolution,theAlexanderdualH_PisCohen–MacaulaybytheEagon–ReinerTheorem.ButH_P=(fxpyq:p

qg)istheedgeidealofabipartitegraphsati
qg)istheedgeidealofabipartitegraphsatisfyingtheconditions(i),(ii)and(ii).ThisprovesonedirectionoftheclassicationtheoremofCohen-Macaulaybipartitegraphs.GeneralizedHibiidealsandHibiringsLetPbeaniteposetandI(P)thesetofposetidealsofP.Anr-multichainofI(P)isachain

ofposetidealsoflengthr,I:I1I2


Ir=P
ofposetidealsoflengthr,I:I1I2


Ir=P:GeneralizedHibiidealsandHibiringsLetPbeaniteposetandI(P)thesetofposetidealsofP.Anr-multichainofI(P)isachainofposetidealsoflengthr,I:I1I2


Ir=P:WedeneapartialorderonthesetIr(P)ofallr-multichainsofI(P)bysettingI

I0ifIkI0kfork=1;:::;r.GeneralizedHi
I0ifIkI0kfork=1;:::;r.GeneralizedHibiidealsandHibiringsLetPbeaniteposetandI(P)thesetofposetidealsofP.Anr-multichainofI(P)isachainofposetidealsoflengthr,I:I1I2


Ir=P:WedeneapartialorderonthesetIr(P)ofallr-multichainsofI(P)bysettingII0ifIkI0kfo

rk=1;:::;r.ThepartiallyorderedsetIr(P)is
rk=1;:::;r.ThepartiallyorderedsetIr(P)isadistributivelattice,ifwedenethemeetofI:I1


IrandI0:I01


I0rasI\I0where(I\I0)k=Ik\I0kfork=1;:::;r,andthejoinasI[I0where(I[I0)k=Ik[I0kfork=1;:::;r.Witheachr-multichainIofIr(P)weassociateamonomialuIinthepol

ynomialringS=K[fxij:1ir;1jng]inrni
ynomialringS=K[fxij:1ir;1jng]inrnindeterminateswhichisdenedasuI=x1J1x2J2


xrJr;wherexkJk=Qp`2Jkxk`andJk=IknIk1fork=1;:::;r.Witheachr-multichainIofIr(P)weassociateamonomialuIinthepolynomialringS=K[fxij:1ir;1jng]inrnindeterminateswhichisdeneda

suI=x1J1x2J2


xrJr;wherexkJk=Qp`2Jkx
suI=x1J1x2J2


xrJr;wherexkJk=Qp`2Jkxk`andJk=IknIk1fork=1;:::;r.WedenotebyHr;PthemonomialidealinSgeneratedbythesemonomialsandbyRr(P)theK-subalgebrageneratedbythemonomialgeneratorsofHr;P.Witheachr-multichainIofIr(P)weassociateamonomialuIinthepolynomialr

ingS=K[fxij:1ir;1jng]inrnindetermi
ingS=K[fxij:1ir;1jng]inrnindeterminateswhichisdenedasuI=x1J1x2J2


xrJr;wherexkJk=Qp`2Jkxk`andJk=IknIk1fork=1;:::;r.WedenotebyHr;PthemonomialidealinSgeneratedbythesemonomialsandbyRr(P)theK-subalgebrageneratedbythemonomialgeneratorsofHr;P.Forr=2thei

dealHr;PisjusttheclassicalHibiideal,andR
dealHr;PisjusttheclassicalHibiideal,andRr(P)theHibiringoftheideallatticeI(P)ofP.LetTbethepolynomialringoverKinthesetofindeterminatesftI:I2Ir(P)g.LetTbethepolynomialringoverKinthesetofindeterminatesftI:I2Ir(P)g.Furthermorelet':T!Rr(P)bethesurjectiveK-alge

brahomomorphismwith'(tI)=uIforallI2Ir
brahomomorphismwith'(tI)=uIforallI2Ir(P).LetTbethepolynomialringoverKinthesetofindeterminatesftI:I2Ir(P)g.Furthermorelet':T!Rr(P)bethesurjectiveK-algebrahomomorphismwith'(tI)=uIforallI2Ir(P).TheoremTheset=ftItI0tI[I0tI\I02T:I;I02Ir(P)incomparableg

isareducedGr¨obnerbasisoftheidealLr=Ker
isareducedGr¨obnerbasisoftheidealLr=Ker'withrespecttothereverselexicographicorder.LetTbethepolynomialringoverKinthesetofindeterminatesftI:I2Ir(P)g.Furthermorelet':T!Rr(P)bethesurjectiveK-algebrahomomorphismwith'(tI)=uIforallI2Ir(P).TheoremTheset=ftItI

0tI[I0tI\I02T:I;I02Ir(P)incomparablegi
0tI[I0tI\I02T:I;I02Ir(P)incomparablegisareducedGr¨obnerbasisoftheidealLr=Ker'withrespecttothereverselexicographicorder.CorollaryRr(P)isanormalCohen–Macaulaydomainofdimensionn(r1)+1CorollaryLetPbeaniteposet.Thefollowingconditionsareequivalent:IRr(P)isGorens

tein.IR2(P)isGorenstein.IPisgraded.
tein.IR2(P)isGorenstein.IPisgraded.CorollaryLetPbeaniteposet.Thefollowingconditionsareequivalent:IRr(P)isGorenstein.IR2(P)isGorenstein.IPisgraded.Proof:OneshowsthatRr(P)=R2(P[r1]).CorollaryLetPbeaniteposet.Thefollowingconditionsareequivalent:IRr

(P)isGorenstein.IR2(P)isGorenstein.I
(P)isGorenstein.IR2(P)isGorenstein.IPisgraded.Proof:OneshowsthatRr(P)=R2(P[r1]).FinallyweconsiderthegeneralizedHibiidealHr;PanditsAlexanderdual.LetCPamultichainoflengthr,i.e.,C=fp1;p2;:::;prgwithp1p2


pr.LetCbethesetofallmultichainsoflengthrof

P.LetCPamultichainoflengthr,i.e.,C=
P.LetCPamultichainoflengthr,i.e.,C=fp1;p2;:::;prgwithp1p2


pr.LetCbethesetofallmultichainsoflengthrofP.WedenethemonomialuC=Qri=1xi;piandletIr;P=(fuC:C2Cg).LetCPamultichainoflengthr,i.e.,C=fp1;p2;:::;prgwithp1p2


pr.LetCbethesetofallm

ultichainsoflengthrofP.Wedenethemonomia
ultichainsoflengthrofP.WedenethemonomialuC=Qri=1xi;piandletIr;P=(fuC:C2Cg).TheidealsIr;Pmaybeinterpretedasfacetidealsofacompletelybalancedsimplicialcomplexes,asintroducedbyStanley.LetCPamultichainoflengthr,i.e.,C=fp1;p2;:::;prgwithp1p2


pr.LetCbet

hesetofallmultichainsoflengthrofP.Weden
hesetofallmultichainsoflengthrofP.WedenethemonomialuC=Qri=1xi;piandletIr;P=(fuC:C2Cg).TheidealsIr;Pmaybeinterpretedasfacetidealsofacompletelybalancedsimplicialcomplexes,asintroducedbyStanley.Theorem(a)Hr;Phasalinearresolution.(b)H_r;P=Ir;P.LetCPamultich

ainoflengthr,i.e.,C=fp1;p2;:::;prgwith
ainoflengthr,i.e.,C=fp1;p2;:::;prgwithp1p2


pr.LetCbethesetofallmultichainsoflengthrofP.WedenethemonomialuC=Qri=1xi;piandletIr;P=(fuC:C2Cg).TheidealsIr;Pmaybeinterpretedasfacetidealsofacompletelybalancedsimplicialcomplexes,asintroducedbyStanley.Theore

m(a)Hr;Phasalinearresolution.(b)H_r;P
m(a)Hr;Phasalinearresolution.(b)H_r;P=Ir;P.CorollaryThefacetidealofacompletelybalancedsimplicialcomplexarisingfromaposetisCohen–Macaulay.J.A.EagonandV.Reiner,ResolutionsofStanley-ReisnerringsandAlexanderduality.J.ofPureandAppl.Algebra,130,265–275(1998).V.Ene,J

.Herzog,F.Mohammadi,Monomialidealsandtor
.Herzog,F.Mohammadi,MonomialidealsandtoricringsofHibitypearisingformaniteposet,Europ.J.Comb.32,404–421,(2011)J.HerzogandT.Hibi,Distributivelattices,bipartitegraphsandAlexanderduality.J.AlgebraicCombin.22,289–302(2005).J.HerzogandT.Hibi,MonomialIdeals.Springer(201

0).T.Hibi,Distributivelattices,afnesem
0).T.Hibi,Distributivelattices,afnesemigroupringsandalgebraswithstraighteninglaws,in“CommutativeAlgebraandCombinatorics”(M.NagataandH.Matsumura,eds.)Adv.Stud.PureMath.11,North-Holland,Amsterdam,93–109(1987).R.Stanley,BalancedCohen-Macaulaycomplexes.Trans.Amer.Mat