GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS

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1 GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS JULIA E. BERGNER AND CHRISTOPHER D. WALKER Abstract. I this paper we show that two very ew questios about the cardiality of groupoids reduce to very old questios cocerig the ethods of aciet Egyptias for writig fractios. First, the questio of whether ay positive real uber occurs as the groupoid cardiality of soe groupoid reduces to the questio of whether ay positive ratioal uber has a Egyptia fractio decopositio. Secod, the questio of the uber of o-equivalet groupoids with a give cardiality ca be aswered via the uber of uique Egyptia fractio decopositios.. Groups ad groupoids The goal of this paper is to coect the very old ethod of Egyptia fractios with the oder idea of classifyig algebraic objects. We describe a algebraic object called a groupoid, which has a well-defied cardiality give by positive ratioal uber. Groupoids have bee aroud for soe tie, but oly recetly has their cardiality bee defied ad ivestigated []. The cardiality of a groupoid has bee used to defie a ew categorificatio process for describig liear algebraic objects such as Hall algebras ad Hecke algebras [2]. What we are iterested i here are iplicatios for groupoid cardiality which follow fro old results about Egyptia fractios decopositios. We will work up to the defiitio of a groupoid by first givig a visual iterpretatio of a group. First, recall the defiitio of a group. Defiitio.. A group is a set equipped with a operatio which is associative, has a idetity eleet, ad has iverses. Exaple.2. The itegers Z for a group uder additio. Exaple.. The group Z/ of itegers odulo, also fors a group uder additio. Exaple.4. The syetric group o letters, S, has eleets give by the uber of ways of perutig the eleets of the set {, 2,..., }. The group operatio is copositio of perutatios. We will thik of a group usig the followig kid of picture. Date: July 9, 20. The first-aed author was partially supported by NSF grat DMS

2 2 J.E. BERGNER AND C.D. WALKER There is oe dot i the iddle, ad the eleets of the group are draw as the arrows. The arrows are draw to go i both directios to idicate that each arrow has a iverse arrow that cacels it out. We ca thik of these arrows as fuctios fro the ceter dot to itself. All these fuctios start ad ed at the sae place, so we ca copose the i ay order. Usig this fuctio aalogy, oe ight ask why they all have to start ad ed i the sae place. Istead, we ca draw ore geeral pictures, where we have ore tha oe dot. For exaple oe ca use a diagra like the followig to visualize a groupoid. We ca thik of the set of arrows as havig a operatio defied wheever the arrows atch up (the rage dot of oe arrow is the doai dot of the other), ad we require this operatio to be associative. Each dot has a idetity arrow, ad all arrows have iverses with respect to these idetities. Defiitio.5. A set with a partially defied operatio satisfyig the above coditios is called a groupoid. A groupoid ay have ay differet copoets, or collectios of dots which are ot coected to oe aother by ay arrows. Exaple.6. A equivalece relatio o a set defies a groupoid. The eleets of the set for the dots, ad there is a uique arrow fro a dot to ay other dot which is equivalet to it. The copoets of this groupoid are exactly give by the equivalece classes. Give ay dot i a groupoid, its autoorphiss (arrows startig ad edig at that dot) for a group Aut( ). A basic fact about groupoids is that ay two dots i the sae copoet have the sae autoorphis group. Therefore, we get a group associated to each copoet of a groupoid. Two groupoids are equivalet if they have the sae uber of copoets ad if the autoorphis group of each copoet of oe groupoid agrees with the autoorphis group of the respective copoet of the other. Reark.7. We would be reiss if we did t ake soe referece to categories here. I the defiitio of a group, if we were to drop the requireet that each eleet have a iverse, we would have the defiitio of a ooid. Siilarly, i our defiitio of groupoid, we could drop the requireet that each arrow have a iverse; such a object is kow as a category. We will ot eed ore geeral categories i this paper, but ay of the cocepts we preset here ca be foralized i the cotext of category theory. 2. Groupoid cardiality I this sectio, we show how to defie the cardiality of a groupoid, i a siilar spirit to defiig the order of a group.

3 GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS Defiitio 2.. The order of a fiite group G is the uber of eleets of G as a set, deoted #G. Exaple 2.2. The group Z/ has eleets, so #Z/ =. Exaple 2.. The syetric group S has! eleets. Exaple 2.4. The group of itegers Z is ot a fiite group, so we say it has ifiite order. We could defie the order of a groupoid siilarly, just coutig the uber of arrows, but we would like a ore refied versio for two reasos. First, we would like a defiitio which ca be used for at least soe ifiite groupoids, ot just fiite oes. Secod, we would like a defiitio which gives us soe iforatio about the differet copoets of the groupoid. The followig defiitio was developed by Baez ad Dola []. Defiitio 2.5. The groupoid cardiality of a groupoid G is whe the su is defied. G = G #Aut( ), Of course, there are ay groupoids for which this defiitio does ot apply, for exaple if at least oe of the autoorphis groups is ifiite. If a groupoid has ifiitely ay copoets ad all the autoorphis groups are fiite, the cardiality is still defied if the resultig ifiite series coverges. Exaple 2.6. If G is a group, the G = #G. Reark 2.7. With regard to groups, the ters order ad cardiality are ofte used iterchageably, ad our otatio for groupoid cardiality is cooly used for the order of a group. Here we distiguish the two sice, as the previous exaple shows, this defiitio of groupoid cardiality does ot restrict to the defiitio of order i the special case of a group. However, it should be oticed that, for a group, the groupoid cardiality retais the sae basic iforatio, as the order has just bee iverted. Exaple 2.8. If G ad H are groups, the their disjoit uio G H is a groupoid. I other words, we thik of takig the groups G ad H, settig the side by side, ad regardig the two of the together as a two-copoet groupoid. The cardiality of G H is G H = #G + #H. Exaple 2.9. Cosider the groupoid G give by the followig picture:

4 4 J.E. BERGNER AND C.D. WALKER Recall that, i such a picture, we oly draw o-trivial autoorphiss at each dot. This eas the two copoets with two dots each have a trivial autoorphis group (oly eleet), ad the sigle dot copoet i the iddle has a autoorphis group of order 2. Therefore, the groupoid cardiality is G = = 5 2. Lookig at the defiitio, takig the cardiality of a groupoid ust always be a positive uber, sice groups have positive order. But how iterestig ca these ubers get? Exaple 2.0. Let E be the groupoid with dots the fiite sets ad the arrows the isoorphiss betwee the. This groupoid has oe copoet for each atural uber >. The cardiality of this groupoid is E = #Aut( ) [ ] E = #S N =! N = e. This last exaple shows that groupoid cardialities do ot have to be ratioal ubers! This groupoid had ot oly a ifiite uber of objects, but eve a ifiite uber of copoets, yet the forula for its cardiality gave a coverget series. I particular, this exaple illustrates that we ca get ore iterestig thigs happeig with groupoid cardiality tha we do with group order. Oe ight, however, ask the followig realizatio questio. Questio 2.. Ca we get ay positive real uber as the cardiality of soe groupoid? Sice ay real uber ca be writte as a coverget series of ratioal ubers (give by its decial expasio), it suffices to deterie whether we ca obtai ay positive ratioal uber as a groupoid with fiitely ay copoets. We ca get whole ubers by takig dots with o o-idetity arrows. Therefore, the iterestig part of the questio is whether we ca fid a groupoid whose cardiality is q for ay ratioal uber 0 < q <. Ay ratioal uber of the for is easy, sice we ca just take the groupoid cardiality of ay group of order, for exaple Z/. Furtherore, ay ratioal uber of the for + is also easy, sice we ca use a disjoit uio of groups such as Z/ Z/. I fact, for ay ratioal uber, we ca take Z/ Z/, where there are copies of Z/. So the aswer to our questio is yes. However, this last step is ot etirely satisfyig, sice we have just repeated the sae group over ad over agai. We could istead ask if there are ore iterestig ways to obtai such a groupoid. For exaple, could we istead fid a groupoid

5 GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS 5 with a give cardiality, such that o two copoets of the groupoid have the sae cardiality? I other words, ca we write ay ratioal uber betwee 0 ad i the for = i where all the positive itegers i are distict? This questio leads us to the aciet study of Egyptia fractios.. Egyptia fractio decopositios Much of what we kow about aciet Egyptia atheatics cae fro the discovery of the Rhid Papyrus i the 850s [5]. Our iterest here is cocered with their curious way of workig with fractios. The aciet Egyptias oly had a otatio for fractios of the for, ot for ore geeral oes such as 7. Furtherore, whe they eeded to write ore 2 coplicated ratioal ubers, they did so by takig sus of these uit fractios where o suads were repeated. Fro our perspective, the questio arises whether the Egyptias were liited by this ethod. Give ay ratioal uber q with 0 < q <, ca it be writte as the su of fiitely ay distict uit fractios? We shall refer to a such a su as a Egyptia fractio decopositio. Exaple.. This idea ight soud cubersoe, but it ca actually be useful i practice. As a eleetary exaple, suppose you have 5 uffis that you wat to divide aog 8 people. You could easure the all carefully ad give each perso 5 8 of a uffi, but it would be far easier to give each perso of a uffi ad 2 the divide the reaiig uffi ito 8 pieces, so each perso gets 5 8 = of a uffi. Notice that Egyptia fractio decopositios are ot uique. For exaple 4 = but also 4 = We get the secod equatio by applyig the splittig algorith which says = + + ( + ). The followig theore aswers our questio. Theore.2. Every ratioal uber has a Egyptia fractio decopositio. I fact, we have a eve stroger result. Theore.. Every ratioal uber has ifiitely ay distict Egyptia fractio decopositios. Let us first look at the ai idea behid the proof of Theore.. The iportat fact that we are goig to use is the Egyptia fractio decopositio for give by i =

6 6 J.E. BERGNER AND C.D. WALKER This su is give by applyig the splittig algorith twice to the uber. To see how this equatio ca be applied, we retur to our exaple of 4 = We ca divide the first equatio by 4 to get 4 = ( ) = Hece, we ca replace 4 i our origial decopositio with its expasio give here to obtai 4 = But the we ca go back to our expressio for ad divide it by 24, obtaiig 24 = leadig to a still loger decopositio 4 = Further decoposig each fial ter leads to a ifiite uber of Egyptia fractio decopositios for a give positive ratioal uber, oce we establish via Theore.2 that at least oe exists. We ow foralize the arguet fro this exaple ito a proof. Proof of Theore.. Let q be a ratioal uber ad let q = + + d d be a Egyptia fractio decopositio, so that each d i is distict. Further assue that d,, d is a icreasig sequece of itegers. The use the decopositio for used above to observe that = + +. d 2d d 6d The q = d d 2d d 6d is aother decopositio. Because d < 2d < d < 6d, the sequece d,..., d, 2d, d, 6d is still a icreasig sequece; i particular, we have guarateed that we have ot repeated ay deoiators. Therefore, give ay Egyptia fractio decopositio, we ca fid aother distict oe. We ow look at a exaple to illustrate oe approach to provig Theore.2, as give i []. Suppose that is a fractio with < <. The strategy, which goes back to Fiboacci [4], is to fid the largest possible uit fractio saller tha ad show that the process of doig so evetually teriates. Sice we are always takig the largest possible Egyptia fractio less tha our give ratioal uber, this procedure is called the greedy algorith.

7 GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS 7 For exaple, cosider 4. We first otice that 4 = 4 6 < 4 < 4 2 = so is the sallest uit fractio less tha 4. We the take the differece ad 4 obtai that 4 = Sice the secod fractio i this su is ot of Egyptia type, we repeat the procedure ad see that 8 = 54 < 52 < 5 = 7 ad subtract to see that 4 = To show that such a process always works, we eed to prove that it ust teriate, givig a fiite uber of Egyptia fractios. Agai, our arguet of proof is take fro []. Proof of Theore.2. For ay, write I other words, we kow that The iequality iplies that or We ca ultiply to get ad rearrage to see that = d + < d < d. ( d ) < d < d, (d ) <. d < d <. = d + d d. I other words, the uerator d is saller tha our origial uerator. We ca the fid a positive iteger d 2 such that < d < d 2 d d 2 which gives a reaider whose uerator is strictly saller tha d. Sice each uerator is a positive iteger strictly saller tha the last, the process ust evetually give a uerator of, teriatig the algorith.

8 8 J.E. BERGNER AND C.D. WALKER Notice, however, that the greedy decopositio ay ot ecessarily be the shortest Egyptia fractio decopositio. For exaple, applyig the greedy algorith to 8 40 yields 8 40 = However, this fractio ca also be decoposed as 8 40 = A calculator for coputig shortest Egyptia fractio decopositios ca be foud at []. 4. Iplicatios for groupoid cardiality The results of the previous sectio iediately iply two iterestig facts about groupoid cardiality. First, the existece of Egyptia fractio decopositios as give by Theore.2 gives us a iterestig groupoid with this uber as the cardiality. Theore 4.. Ay positive ratioal uber is the cardiality of a groupoid with o two copoets havig the sae cardiality. The secod result follows fro Theore.. Theore 4.2. Ay positive ratioal uber is the cardiality of ifiitely ay distict such groupoids. Not oly ca we fid ifiitely ay groupoids with the sae cardiality, but we ca choose the so that o two are equivalet to oe aother. Both of these results have iterestig iplicatios to research. For istace, whe workig with the cocept of groupoid cardiality, it is typical to have a desired cardiality i id, the work backward to fid a groupoid which has this value as its cardiality. As a exaple, we already saw that the groupoid E of fiite sets ad bijectios has a groupoid cardiality of e. We got this cardiality by cosiderig the power series expasio for e x : e x = x! with x =. What if, istead, we took other values for x? For exaple, we ight wat a groupoid with cardiality e 2, ad so we could start with the power series expasio: e 2 = 2!. The first thig to ote is that this su o loger uses just uit fractios. We could the try to write differet decopositios for each fractio 2 util we fid! soethig that looks failiar. The evetual goal is to the fid a groupoid that is ore iterestig tha siply disjoit uios of groups. Also, the secod result is i stark cotrast to the theory of fiite groups, where oe of the great recet achieveets was the classificatio of fiite siple groups. Thaks to Theore 4.2, it would be expected that ay kid of classificatio for fiite groupoids would be still ore itricate.

9 GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS 9 Refereces [] J. Baez ad J. Dola, Fro fiite sets to Feya diagras, i Matheatics Uliited 200 ad Beyod, eds. B. Egquist ad W. Schid, Spriger, Berli, 200, pp Also available as arxiv:ath/0004. [2] J. Baez, A. Hoffug, ad C. Walker, Higher Diesioal Algebra VII: Groupoidificatio. Theory Appl. Categ. vol. 24, 200, No. 8, [] Ro Kott, [4] Leoardo Pisao, Liber Abaci (tr. Laurece Sigler), Spriger [5] Gay Robis ad Charles Shute, The Rhid Matheatical Papyrus: A Aciet Egyptia Text. Dover Publicatios, 990. Departet of Matheatics, Uiversity of Califoria, Riverside, CA 9252 E-ail address: bergerj@eber.as.org, cwalker66@ath.ucr.edu

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