International Mathematical Forum, Vol. 6, 2011, no. 62, 3079-3086EnumeratingAG-GroupswithaStudyofSmaradacheAG-GroupsMuhammadShah1DepartmentofMathematicsQuaid-i-AzamUniversity,Pakistanshahmathsproblem@hotmail.comCharlesGretton2andVolkerSorgeDepartmentofComputerScienceUniversityofBirmingham,UKcharles.gretton@gmail.com,V.Sorge@cs.bham.ac.ukAbstractAG-groups are a generalisation of Abelian groups. They correspond to groupoidswithaleft identity, uniqueinverses, andsatisfytheidentity(xy)z =(zy)x. WepresenttherstenumerationresultforAG-groupsuptoorder11andgivealowerboundfororder12. ThecountingisperformedwiththenitedomainenumeratorFINDERusingbespokesymmetrybreakingtechniques. WehavealsodevelopedafunctionintheGAPcomputer algebrasystemtocheckthe generatedCayleytables. Thisnotediscussesafewobservationsobtainedfromourresults, someofwhichinspiredustoexamineanddiscussSmaradacheAG-groupstructures.MathematicsSubjectClassication: 20N99Keywords: AG-groupoid,AG-group,Smaradache,enumeration,modelgeneration1 IntroductionInthestudyof small algebraicstructures moregeneral thangroups, manyinterestingquestions, such as open existence, classication, and counting problems, have been solvedbysoftware tools that enable ecient enumerationof structures. Typicallythis taskinvolves identifying and exploiting symmetries in the problem at hand. Loops with inverseproperty(IP-loops)uptoorder13havebeencountedwithmodelgeneratorsusinghand1MuhammadShahthankstheSchool of ComputerScience, Universityof Birmingham, for hostinghim,andtheRamsayResearchFundfortheirkindnancialsupport.2Charles Gretton is funded by the European Communitys Seventh Framework Programme [FP7/2007-2013]undergrantagreementNo. 215181,CogX.3080 M.Shah,C.Gretton,V.Sorgecraftedsymmetrybreakingconstraintsandpost-hocprocessing[1]. Monoidsuptoorder10andsemigroupsuptoorder9havebeenenumerated[3] witho-the-shelf constraintsatisfaction software by employing lexicographic symmetry breaking constraints computedusingaGAPimplementationof themethodsdescribedin[5]. Asimilarapproachwasmorerecentlyusedfor countingAG-groupoids groupoids that areleft invertive, inthesense(ab)c=(cb)auptoorder6[2]. Finally, alsorelatedistheenumerationofquasigroupsandloopsuptosize11usingamixtureofcombinatorialconsiderationsandbespokeexhaustivegenerationsoftware[7].Inthispaperwepresenttheenumerationof AG-groups. AnAG-group, alsocalledanLA-group, isanAG-groupoid(G, )withleftidentity, anduniqueinverses. Withastructureadmittingnon-associativity, AG-groups liebetweenquasigroups andAbeliangroups. AG-groups were conceived by M.S. Kamran for his PhD thesis [6], rst appearedin[8] andthenmorerecentlyhavebeenstudiedin[12]. AG-groupsgeneraliseAbeliangroups. AnAG-groupsatises the Lagrangetheorem[8]. Moreover, it caneasilybeveried that AG-groups satisfy both the medial, i.e. (ab)(cd) = (ac)(bd), and paramedial,i.e. (ab)(cd) = (db)(ca),laws(see[12,Lemma1]).InSec. 2werstcountthenumberof non-isomorphicAG-groupsof orderupto11andgivealowerboundfororder12. Wethendiscusssomeoftheobservationswehavemade when examining the results (Sec. 3) and in particular develop and study a new typeofAG-groupstructure,SmaradacheAG-groups(Sec.4).2 CountingAG-groupsuptoIsomorphismWe counted AG-groups by exhaustive enumeration using the FINDER system. Our start-ingpointistheapproachdevelopedforcountingIP-loopsuptoisomorphismin[11, 1].Here, asinthatwork, FINDERgeneratesasetof candidatetableswhichcontainsonetableforeachminimal elementgivenbyalex(icographic)orderovereachisomorphismclass. Apost-processingsteppost-hocprocessingisusedtorejecttablesthatarenotminimal intheirisomorphismclass.3InordertopreventFINDERfromgeneratinganimpractical number of candidatetables, further symmetrybreakingconstraints areposed. Moreover, thevalidityof post-hocprocessingisdependentontheseconstraintsbeingsatised. Inour workwehavehadtomodifythoseconstraints. Asummaryoftheconstraintsfrom[11] intheirreducedformforAG-groupsisgiveninthefollowingdenition.Denition2.1. (Symmetry Breaking Constraints) Let Nbe the order of the AG-group,withelementsx, y {0 . . ., N 1}andleftidentitye. Letf(x)abbreviate(e +1)(e +x)andlet FLAGbeabooleanvariablethat isset if therst sixelementsof theAG-groupareself-inverseand(e + 1)(e + 2) isnot self-inverse. Wethendenethefollowing10constraints:3Weconsideritanimportantitemforfutureworktodeveloplex-leaderconstraintsthatcapturethesymmetrybreakingthatiscarriedoutduringthispost-hocstep.numeratingAG-groups 3081(i) e x, (ii) x1< (x + 2),(iii) (x1= x x < y) y1= y.ForoddvaluesofN:(iv) f(1) < (e + 4), (v) (x> 1 2x < N) f(x) < (e + 2x).ForevenvaluesofN(vi) f(1) = e,(vii) (FLAG 0 < x 1 (e + x)1= (e + x)) (f(x)1) = f(x),(x) (FLAG 1 < x < y (e + y)1= (e + y)) f(x) < f(y).The constraints implythat the Cayleytable of the AG-groupwill be lledinanascending order, where e is always the lexicographical minimal element (i.e., 0). They alsohave that elements which are self-inverse are ordered rst, and otherwise that an elementis adjacent to its inverse in the ordering. We have omitted the constraint x1= x x = efrom[1],becausethisisaninvalidsymmetrybreakerforAG-groupcounting.Our enumerationyields all Cayleytables explicitly. WecanthereforevalidatethegeneratedAG-groupsusingthe GAP[4]computer algebrasystem. Becausethere wasnofunctionalitypresentintheGAPloops package[9] totestwhetheraCayleytableisanAG-groupornot, wehaveimplementedourownfunctionthatperformsthistest. TheresultsofourenumerationwerevalidatedusingourGAPfunction.4Thealgorithmisastraightforward implementation, testing that (1) the Cayley table is a Latin square; i.e., allelements occur exactly once in every row and every column, (2) the identity (xy)z= (zy)xholds,and(3)thereexistsaleftidentity.OurvalidatedresultsaregiveninTable1. Wereportthenumberofnon-isomorphicAG-groups having order up to 11, and give a lower bound for order 12. In Table 1, for eachorder we give the total number of AG-groups up to isomorphism,which is further brokendownintoassociativeandnon-associativeAG-groups. Note, associativeAG-groupsareAbelian groups. For each order we also give the total number of CPU-seconds required toenumerateall groupsandthenumberof tablesgeneratedbyFINDERthatweretestedforlex-minimalityinpost-hocprocessing. AllcountingwascarriedoutonanIntelquadcoreCPUQ9650with4GBofmemory. Itshouldbenotedthatourcountingprocedureusesnegligiblecomputermemoryresources.Inthe remainderofthe paper wediscuss someoftheobservationswemadeusingourenumerationresultsand,inparticular,proposeanewinterestingclassofAG-groups.3 AG-groupofSmallestOrderThesmallestAG-groupwhichisnotagroupisoforder3. TheCayleytableforthatisgiveninExample3.1.Example3.1. SmallestAG-groupoforder3 :4Pleasecontacttheauthorsbyemail foreithertheGAPsourcecode, oracopyof theenumeratedtables.3082 M.Shah,C.Gretton,V.SorgeOrder AG-groups Assoc Non-Assoc CPU-Time post-hoctests1 1 1 0 < .01 02 1 1 0 < .01 13 2 1 1 < .01 24 4 2 2 < .01 65 2 1 1 < .01 76 2 1 1 < .01 467 2 1 1 .47 978 10 3 7 8.44 7969 5 2 3 102.37 359910 2 1 1 1, 735.25 1614411 2 1 1 15, 206.26 8640612 7 2 5 NA NATable1: ResultsofAG-groupenumeration. 0 1 20 0 1 21 2 0 12 1 2 0ClearlyeveryAbeliangroupisanAG-group, howevertheconverseiscertainlynotalways true. Wenownoteanumber of contrasts betweengroups andAG-groups. Inparticular, weestablishtheexistenceof non-associativeAG-groupsi.e. non-Abeliangroupsoforderpandp2wherepisaprime. Indetail,ourobservationsare:1. Every group of order p is Abelian. We nd that non-associative AG-groups of orderpexist. TheAG-groupoforder3inExample3.1isnotanAbeliangroup.2. Everygroupoforderp2isAbelian. Wealsohavethatnon-associativeAG-groupsoforderp2exist. TheAG-groupinExample4.3isnotanAbeliangroup.3. Everygroupwhichsatises the squaringproperty(ab)2=a2b2is Abelian. Al-thougheveryAG-groupclearlysatisesthesquaringproperty,anAG-groupisnotnecessarilyAbelian.4 SmaradacheAG-groupsIn[10] PadillaRaul introducedthenotionof aSmarandachesemigroup, herewrittenS-semigroup. AnS-semigroup is a semigroupA such that a proper subset of A is a groupwithrespecttothesameinducedoperation[15]. SimilarlyaSmarandachering, writtenS-ring,isdenedtobearingA,suchthatapropersubsetofA isaeldwithrespecttotheoperationsinduced. ManyotherSmarandachestructureshavealsoappearedinthenumeratingAG-groups 3083literature. ThegeneralconceptofSmarandachestructuresisthat, ifaspecialstructurehappenstobeasubstructureofageneralstructure,thenthatgeneralstructureiscalledSmarandache. InthatspiritweproposeSmarandacheAG-groupshereandstudythemwiththehelpofexamplesgeneratedduringtheenumerationgiveninSec.2.Denition4.1. Let GbeanAG-group. GissaidtobeaSmaradacheAG-group(S-AG-group) if Ghas apropersubset P suchthat P is anAbeliangroupunder thetheoperationofG.TheAG-groupsGinExamples4.3and4.4areS-AG-groups, whereastheAG-groupGinExamples3.1and4.6arenot.ThefollowingtheoremguaranteesthatanAG-grouphavingauniquenontrivial ele-mentoforder2isalwaysanS-AG-group.Theorem4.2. Ifthereisauniquenontrivialelementaoforder2inanAG-groupGthen{e, a}isanAbeliansubgroupofG.Proof. Takea Gsatisfyinga2= e. Nowwehavetoidentifyanelementforthe ? cellinthefollowingtable: e ae e aa ? eTakingy=aeandusingtheparamediallawwehavey2= (ae)2=e2a2=a2=e. Thus,yhasorder2. BecauseG hasasingleelementoforder 2,wehavey= ae = a. Thus,aistherequiredelement,andourtablecannowbecompleted: e ae e aa a eHere{e, a}isanAG-subgroupofGoforder2,andisthereforeanAbeliangroup,henceGisanS-AG-group.WeillustrateTheorem4.2,byconsideringthefollowingexample.Example4.3. AnAG-groupoforder8: 0 1 2 3 4 5 6 70 0 1 2 3 4 5 6 71 1 0 3 2 6 7 4 52 2 3 1 0 5 6 7 43 3 2 0 1 7 4 5 64 6 4 7 5 3 0 2 15 7 5 4 6 0 2 1 36 4 6 5 7 2 1 3 07 5 7 6 4 1 3 0 23084 M.Shah,C.Gretton,V.SorgeHere1 has order 2,and 1 is the unique such elementofG. Hence {0, 1} is an AbeliansubgroupofG.TheconverseofTheorem4.2isnottrue. Thatis, ifanAG-groupGhasanAbeliansubgroup,thenitisnotnecessarythatGwillhaveauniquenon-trivialelementoforder2. Thiscanbeobservedwiththefollowingexample:Example4.4. AnAG-groupoforder9: 0 1 2 3 4 5 6 7 80 0 1 2 3 4 5 6 7 81 2 0 1 5 6 7 8 3 42 1 2 0 7 8 3 4 5 63 5 3 7 8 0 4 2 6 14 6 4 8 0 7 2 3 1 55 3 7 5 6 1 8 0 4 26 4 8 6 1 5 0 7 2 37 7 5 3 4 2 6 1 8 08 8 6 4 2 3 1 5 0 7The AG-group in Example 4.4 has {0, 7, 8} as Abelian subgroup and hence is a Smaran-dacheAG-group. However, theelementof order2isnotunique. Infact, therearetwonontrivialelementsoforder2,namely{1, 2}.Remark4.5. BecauseAG-groupssatisfyLagrangesTheorem,theuniqueelementoforder2canexistinAG-groupsofevenorderonly.However, if Ghas morethanoneelement of order2, thenitis notnecessarythatGwill haveanAbeliansubgroup. InExample4.6all non-trivial elementsof Gareoforder 2, however wealsoseethat Ghas noAbeliansubgroup. Notethat Ghas fourproper AG-subgroups,namely {0, 1, 2} , {0, 3, 7} , {0, 4, 6} , and {0, 5, 8}. None of thoseiscommutative.Example4.6. AnAG-grouporderof9: 0 1 2 3 4 5 6 7 80 0 1 2 3 4 5 6 7 81 2 0 1 4 5 3 7 8 62 1 2 0 5 3 4 8 6 73 7 6 8 0 2 1 5 3 44 6 8 7 1 0 2 4 5 35 8 7 6 2 1 0 3 4 56 4 3 5 8 6 7 0 2 17 3 5 4 7 8 6 1 0 28 5 4 3 6 7 8 2 1 0AG-groupssatisfyLagrangesTheorem, soAG-groupsof primeordercannothaveaproperAG-subgroup,hencecannot haveaproper Abeliansubgroup. Werecordthisfactasthefollowingtheorem.numeratingAG-groups 3085Theorem4.7. AnAG-groupGofprimeordercannotbeanS-AG-group.ThenotionofS-AG-groupcanbegeneralisedtoS-AG-groupoidasfollows.Denition4.8. Let Sbe an AG-groupoid. Sis said to be an Smaradache AG-groupoid(S-AG-groupoid) if Shas a proper subset Psuch that Pis a commutative semigroup undertheoperationofS.The examples given in the case of AG-groups can also be considered for S-AG-groupoids. We now provide two further examples to show that this notion holds generally.Example4.9. AnAG-groupoidorder4. 0 1 2 30 0 0 2 31 0 1 2 32 3 3 0 23 2 2 3 0Example4.10. AnAG-groupoidorder4. 0 1 2 30 1 2 0 31 3 0 2 12 2 1 3 03 0 3 1 2The AG-groupoid Sin Example 4.9 has a proper subset {0, 1} which is a commutativesemigroup, andthereforeSisanS-AG-groupoid. Althoughsomewhattedious, onecancheck manually that the AG-groupoid Sin Example 4.10 has no proper subset having thedesiredproperty,andthereforewehaveSisnotaS-AG-groupoid.5 FutureWorkDuetotheobvious limitationsofenumerativecounting,wehaveonlybeen abletostudyAG-groups of small order here. Animportant future directionis topursue countingalgebraically, especiallywheresuchworkcaninformaconstructiveprocedureforlargerAG-groups. WehavealsobeguncharacterisingS-AG-groupshere, howeverthatnotionrequiressignicantfurtherstudy.References[1] A. Ali andJ. Slaney. CountingLoopswiththeInverseProperty. QuasigroupsandrelatedStructures,Vol.16(2008):1316.3086 M.Shah,C.Gretton,V.Sorge[2] A. Distler, M. Shah and V. Sorge. Enumeration of AG-groupoids. In Proc. of CICM,2011,p.114.[3] A. Distlerand TomKelsey.Themonoidsoforderseight,nine&ten.InAnn.Math.Artif.Intell.,Vol.56(2009),No.1,321.[4] GAPGroups, Algorithms, andProgramming, Version4.4.12, 2008. http://www.gap-system.org[5] S.Linton.Findingthesmallestimageofaset.InProcofISSAC,2004,p.229234.[6] M.S. Kamran. Conditions for LA-semigroups to resemble associative structures,Ph.D. Thesis, Quaid-i-AzamUniversity, Islamabad, 1993. Available at http://eprints.hec.gov.pk/2370/1/2225.htm[7] B.D. McKay, A. Meynert, andW. Myrvold. 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In Collected Papers, Abaddaba,Oradea,Vol.III(2000),p.7881.Received: June, 2011