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(1)On Mikheev's onstrution of enveloping groups Comment.Math.Univ.Carolin

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On Mikheev's onstrution of enveloping groups

Comment.Math.Univ.Carolin. 51,2 (2010)245{252.

Abstrat:Mikheev,startingfromaMoufangloop,onstrutedagroupoidandreported

that this groupoid is infat a group whih, in anappropriate sense, is universal with

respettoenvelopingtheMoufangloop. LaterGrishkovandZavarnitsinegaveaomplete

proofofMikheev'sresults. Herewegiveadiretandself-ontainedproofthatMikheev's

groupoidisagroup,intheproessextendingtheresultfromMoufangloopstoBolloops.

Keywords:Bolloop,Moufangloop,autotopismgroup,groupwithtriality

AMSSubjetClassiation:20N05

Referenes

[1℄ BrukR.H.,ASurveyofBinarySystems,ErgebnissederMathematikundihrerGrenzgebiete,

NeueFolge,Heft20,Springer,Berlin-Gottingen-Heidelberg,1958.

[2℄ DoroS.,SimpleMoufangloops,Math.Pro.CambridgePhilos.So.83(1978),377{392.

[3℄ GrishkovA.N.,ZavarnitsineA.V.,Groupswithtriality,J.AlgebraAppl.5(2006),441{463.

[4℄ HallJ.I.,Moufangloopsandgroupswithtrialityareessentiallythesamething,submitted.

[5℄ MikheevP.O.,EnvelopinggroupsofMoufangloops,UspekhiMat.Nauk48(1993),191{192;

translationinRussianMath.Surveys48(1993),195{196.

[6℄ PugfelderH.O.,QuasigroupsandLoops: Introdution,SigmaSeriesinPureMathematis,

7,Heldermann,Berlin,1990.

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