vol. lxxiii, 1(2004), pp. 69–73 distributive pairs in biatomic lattices

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1、ActaMath.Univ.Comenianae69Vol.LXXIII,1(2004),pp.69–73DISTRIBUTIVEPAIRSINBIATOMICLATTICESB.N.WAPHAREandV.V.JOSHIAbstract.WeprovethatabiatomiclatticeLisdistributiveifandonlyifeverypairofatomsofLisdistributive.Thisresulthasbeenusedtoobtaincharacterizationsofdistributivepairsint

2、ermsofsemi-distributivepairs,del-relationandperspectivity.Inanatomisticlattice(everynon-zeroelementisthejoinofatomscontainedinit)L,forapairofnon-zeroelementsa,b∈Lwewrite(a,b)P,ifforeveryatomp≤a∨bthereexistatomsq,rsuchthatp≤q∨r,q≤aandr≤b.Liscalledbiatomicif(a,b)Pholdsforallno

3、n-zeroelementsa,b∈L.In[2],Bennettstudiedtheclassofbiatomiclatticesandprovidedmanyimpor-tantexamples.Infact,thesameclasswiththenomenclature“additivelattices”isalsostudiedbyBennett[1].BiatomiclatticesarealsodefinedintermsofP-relation.PropertiesandcharacterizationsofP-relationca

4、nbefoundinMaeda[7](seealsoMaeda[8])forlatticesandinThakare,WasadikarandMaeda[11]forjoin-semilattices.ThefollowingconceptscanbefoundinMaedaandMaeda[6]andMaeda[9].ForalatticeLanda,b∈Lwewrite:(a,b)D(distributivepair)if,(a∨b)∧x=(a∧x)∨(b∧x)foreveryx;(a,b)SD(semi-distributivepair)

5、if,{(a∨b)∧x}∨b=(a∧x)∨bforeveryx;(a,b)M(modularpair)if,c∨(a∧b)=(c∨a)∧bforeveryc≤b;a∇b(del-relation)if,(a∨x)∧b=b∧xforeveryx;a∇˜bif,(a∨x)∧(b∨x)=xforeveryx.Dually,wehavetheconceptsofduallydistributivepair(a,b)D∗,duallysemi-distributivepair(a,b)SD∗andduallymodularpair(a,b)M∗etc.A

6、latticeissaidtobedistributiveif(a,b)Dholdsforalla,b.Itiseasytoprovethat(a,b)D⇒(a,b)SDbutnotconversely;alsoalatticeisdistributiveif(a,b)SDholdsforalla,b∈L;seeMaeda[7].Maeda[9]essentiallyprovedthatforelementsa,binabiatomiclatticeL,(a,b)M∗holdsif(p,q)M∗holdsforatomsp≤aandq≤b.Th

7、ismotivatesustoReceivedJune23,2003.2000MathematicsSubjectClassification.Primary06D10,06D99;Secondary06C05.Keywordsandphrases.Biatomiclattice,distributivepair,modularpair,del-relation,ex-changeproperty.70B.N.WAPHAREandV.V.JOSHIproveanaloguesresultsfordifferentconceptsinlattices

8、.Infact,inthispaper,weprovethefollowingresultinbiatomiclattices.Theorem1.In