GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS

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1 GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS JULIA E. BERGNER AND CHRISTOPHER D. WALKER How do mathematicias cout? For a fiite set, the idea is straightforward, ad may mathematical objects, especially algebraic oes, are just sets with extra structure. Give a fiite group, the first thig we wat to kow is how may elemets it has. However, ofte we eed to move beyod simple coutig. Whe sets are ifiite, we ca still distiguish betwee differet cardialities, but sometimes we wat a alterate way of measurig them. We are more iterested i the dimesio of a vector space tha its cardiality; for topological spaces we use ivariats such as the Euler characteristic. However we assig a umber to a mathematical object, the ext atural questio is oe of realizatio. Ca we fid a example of our mathematical structure which attais a give umber? For ay positive iteger, we kow that there is at least oe group of order, for example, the cyclic group with elemets. The structure we look at i this paper is that of a groupoid, a geeralizatio of a group. Takig the order of a groupoid i the same way that we take the order of a group is ot the most iterestig way to cout it. A alterative defiitio of the cardiality of a groupoid, recetly give by Baez ad Dola [], assigs a positive ratioal umber to ay fiite groupoid. I ice cases, ifiite groupoids also have this kid of cardiality, give by a positive real umber. What about realizatio? Give ay positive real umber, is there a groupoid with that cardiality? We show that a positive aswer to this questio ca be obtaied by reducig it to aother realizatio questio i umber theory, whether ay positive real umber has a Egyptia fractio decompositio. Groups ad groupoids We ca thik of a group via a picture such as the oe i Figure (a), where the elemets of the group are represeted as arrows startig ad edig at a ceter dot. The arrows go i both directios, idicatig that each elemet has a iverse. We ca thik of these arrows as Date: April 8, 203.

2 2 J.E. BERGNER AND C.D. WALKER fuctios with a commo domai ad rage, so they compose i ay order. (a) A group. (b) A groupoid. Figure. Why, however, do the arrows have to start ad ed i the same place? Istead, we ca draw more geeral pictures, as i Figure (b), to visualize groupoids. Arrows compose wheever the rage dot of oe arrow is the domai dot of the other, ad we require this operatio to be associative. Such a structure o the set of arrows is a partially defied operatio, sice ot every arrow ca be composed with ay other arrow. Each dot has a idetity arrow, ad all arrows have iverses with respect to these idetities. (For simplicity, we do ot draw idetity arrows.) A groupoid may have may differet compoets, or collectios of dots which are ot coected to oe aother by ay arrows. For example, a equivalece relatio o a set defies a groupoid. There is a dot for each elemet of the set, ad there is a uique arrow from a dot to ay equivalet dot. The compoets of this groupoid are exactly the equivalece classes. Give a dot i a groupoid, its automorphisms (arrows startig ad edig at that dot) form a group Aut( ). A basic fact about groupoids is that ay two dots i the same compoet have the same automorphism group. Two groupoids are equivalet if they have the same umber of compoets ad if the automorphism group of each compoet of oe groupoid agrees with the automorphism group of the respective compoet of the other. Notice that equivalet groupoids eed ot have the same umber of objects, oly the same umber of compoets. Groupoid cardiality Recall that the order of a fiite group G is the umber of elemets of G as a set, deoted #G. We could defie the order of a groupoid similarly, by coutig the umber of arrows, but the we could have equivalet groupoids with differet order. I particular, a groupoid with a fiite umber of arrows ca be equivalet to oe with ifiitely may arrows, obtaied by addig more isomorphic objects to a give

3 GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS 3 compoet. To remedy this problem, Baez ad Dola [] provide a ew method for coutig a groupoid. They defie the groupoid cardiality of a groupoid G to be G = [ ] G #Aut( ), wheever it is defied, where [ ] deotes a compoet of G. There are may groupoids to which this defiitio does ot apply, for example if at least oe of the automorphism groups is ifiite. However, if a groupoid has ifiitely may compoets ad all the automorphism groups are fiite, the cardiality is still defied if the resultig ifiite series coverges. Let s look at some examples. If G is a group, the G = #G. Two more iterestig examples are depicted i Figure 2. (a)g H. (b) A more complicated example. Figure 2. If G ad H are groups, the takig them together gives a two-compoet groupoid (see Figure 2(a)), called their disjoit uio, ad deoted by G H, ad we ca compute G H = #G + #H. Still more iterestig is the groupoid G i Figure 2(b). Recall that we oly draw o-idetity automorphisms, so the two-dot compoets each have a trivial automorphism group, ad the sigle-dot compoet i the middle has a automorphism group of order 2. Therefore, the groupoid cardiality is = 5 2. The cardiality of a groupoid must always be a positive umber, sice groups have positive order. But how iterestig ca these umbers be? Let E be the groupoid with objects all fiite sets ad arrows all the possible the isomorphisms betwee them. Sice two fiite sets are isomorphic if they have the same umber of elemets, this groupoid has oe compoet for each atural umber. Sice the automorphism group of a set with elemets is the symmetric group S,

4 4 J.E. BERGNER AND C.D. WALKER the cardiality of E is E = [ ] E #Aut( ) = N! = e. So, we see that more iterestig umbers result from groupoid cardiality tha from group order. Hece we ask: is every positive real umber the cardiality of some groupoid? We ca obtai whole umber cardialities by takig dots with o o-idetity arrows. For a ratioal umber of the form we ca use Z/. Ay ratioal umber of the form + is also easy, sice we ca m take a disjoit uio of groups such as (Z/) (Z/m). I fact, for ay ratioal umber m, we ca take (Z/) (Z/), with m copies of Z/. Fially, for a real umber, cosider the coverget series of the fractios i its decimal expasio; take the disjoit uio of all the groupoids givig these fractios. So, the aswer to our questio is yes. We fid, however, the case of m to be usatisfyig, sice we have merely repeated the same group over ad over agai. Ca we istead fid a groupoid with a give cardiality, such that o two compoets of the groupoid have the same cardiality? I other words, ca we write ay ratioal umber betwee 0 ad i the form m = i i, where the positive itegers i are distict? This questio leads directly to the study of Egyptia fractios. Egyptia fractio decompositios The aciet Egyptias had a curious way of workig with ratioal umbers [4]. They had otatio for uit fractios, those with umerator, but they did ot have otatio for more geeral oes. Whe they eeded to represet such a umber, they did so by takig sums of uit fractios with o repeated summads. Such a represetatio of a ratioal umber is called a Egyptia fractio decompositio. For example, 5 8 = 2 +. This otatio might seem cumbersome, but suppose you have 5 muffis to divide amog 8 people. You could 8 measure each muffi carefully ad give each perso exactly 5 of oe, but it is 8 far easier to give each perso half a muffi, the divide the remaiig

5 GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS 5 muffi ito 8 pieces, oe for each perso. Everyoe has ow received 5 8 = 2 + of a muffi. 8 Were the Egyptias limited by this method? Ca ay ratioal umber be writte as the sum of fiitely may distict uit fractios? The aswer is yes. I fact, we ca say somethig eve stroger: every ratioal umber has ifiitely may distict Egyptia fractio decompositios. Let us take existece for grated first, ad show that, give ay Egyptia fractio decompositio, we ca fid aother. For example, 3 4 = but also 3 4 = We get the secod decompositio 20 by applyig the splittig algorithm: More geerally, let = + + ( + ). q = d + + d be a Egyptia fractio decompositio, with d < < d ad apply the splittig algorithm to the last term. For d >, we have d < d + < d (d + ), so the terms of the sum q = d + + d + d + + d (d + ) are all distict ad hece we have obtaied a ew Egyptia fractio decompositio for q. If d =, the o previous terms i the sum are possible, by our assumptio about the deomiators, so q =. Applyig the splittig algorithm gives = 2 +, which is ot a Egyptia 2 fractio decompositio, but applyig it oce more gives = , which is. But do Egyptia fractio decompositios eve exist? Followig [0], suppose that m is a fractio with < m <. The strategy, which goes back to Fiboacci [3], is to fid the largest possible uit fractio smaller tha m, subtract, ad repeat. As we show below, this process evetually termiates. Sice we always take the largest possible uit fractio less tha m, this procedure a example of what is commoly called a greedy algorithm. For example, cosider 4. We first otice that 3 4 = 4 6 < 4 3 < 4 2 = 3,

6 6 J.E. BERGNER AND C.D. WALKER so 4 is the smallest uit fractio less tha 4. We the obtai = Sice 3 is ot a uit fractio, we repeat the procedure ad see that 52 8 = 3 54 < 3 52 < 3 5 = 7, ad subtract to see that 4 3 = To prove that this process must termiate for ay m, write The, we kow that m = d + < m d < d. ( m ) = d + md d. d The iequality m < implies that m < d d, or m(d ) <. We ca multiply to get md m < ad rearrage to see that md < m. I other words, the umerator md is smaller tha our origial umerator m. We ca the fid a positive iteger d 2 such that < md < which gives a remaider whose d 2 d d 2 umerator is strictly smaller tha md. Sice each umerator is a positive iteger strictly smaller tha the last, the process must evetually give a umerator of, termiatig the algorithm. Notice, however, that the greedy decompositio is ot ecessarily the shortest Egyptia fractio decompositio. For example, applyig the greedy algorithm to yields = However, this fractio ca also be decomposed as = A calculator for computig shortest Egyptia fractio decompositios ca be foud at [0]. We also wat to cosider positive irratioal umbers, which do ot have a fiite Egyptia fractio decompositio. We use the fact that,

7 GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS 7 give ay positive real umber x 0, the sequece {a } give by a = has a subsequece a i such that a i = x 0. i= How ca such a subsequece be foud? Give x 0, let a be the uit fractio such that a = < x 0. Let x = x 0 a. Iductively, choose a i so that a i = < x i i i ad x i = x i a i. The usig that a i = x i x i ad takig the sum, we get a i = (x i x i ) = x 0 i= i= which gives the ecessary subsequece. Notice that if x 0 is ratioal, it is preferable to use the origial greedy algorithm to get a Egyptia fractio decompositio of fiite legth. However, if x 0 is irratioal, the we obtai a ifiite series composed of distict uit fractios but which coverges to x 0. I ay case we have aswered our questio: ay positive real umber ca be writte as the sum of distict uit fractios, possibly ifiitely may. Implicatios for groupoid cardiality The results of the previous sectio immediately imply two facts about groupoid cardiality. First, ay positive real umber is the cardiality of a groupoid with o two compoets havig the same cardiality. Secod, ay positive ratioal umber is the cardiality of ifiitely may o-equivalet groupoids, each of which has o two compoets of the same cardiality. Both results have iterestig implicatios. First, whe workig with the cocept of groupoid cardiality, it is typical to have a desired cardiality i mid, the work backward to fid a groupoid with this cardiality. As a example, we already saw that the groupoid E of fiite sets ad bijectios has groupoid cardiality e. We got this cardiality by cosiderig the power series expasio e x = x with x =. What if, istead, we took other values for x?! For example, we might wat a groupoid with cardiality e 2, ad the methods of this paper provide oe. But, as with the case of e, usig the power series e 2 = 2 could suggest a alterative approach; we! could try to idetify groupoids of cardiality 2 which have a useful!

8 8 J.E. BERGNER AND C.D. WALKER descriptio. Sice the summads are o loger uit fractios, such a groupoid would likely be more complicated tha E. Our secod result presets a stark cotrast betwee the theory of groupoids ad that of fiite groups. The classificatio of fiite simple groups is very difficult ad a sigificat mathematical achievemet. Because there are so may groupoids with a give cardiality, we expect that ay classificatio of fiite groupoids would be still more itricate, ad therefore probably itractable. Other approaches I provig the existece of Egyptia fractio decompositios, we used a particular strategy, the greedy algorithm. However, there are may other such algorithms, ad the resultig decompositios ofte look quite differet. Some examples are the greedy odd algorithm [2], the Yokota algorithm [6] ad the Egel series method [5]. The splittig algorithm ca also be used for existece, ot just for fidig ew decompositios [2], ad there is a method makig use of cotiued fractios [3]. There has bee a good deal of research o each of these methods, ad some relevat results iclude fidig decompositios of a give legth [8], [9], [5]. Other topics of iterest iclude fidig Egyptia fractio decompositios where the deomiators fall withi certai itervals [7], or are bouded by some fixed umber [6], or where the decompositios are particularly short or log i legth [4], []. These results facilitate fidig groupoids with further restrictios o the umber of compoets or o the order of their automorphism groups. Ackowledgmets. The authors thak Joh Baez ad the participats i the Groupoid Semiar at UCR for discussios which led to this paper, as well as the referees ad editor for umerous suggestios. The first-amed author was partially supported by NSF grats DMS ad DMS-05766, ad by a UCR Regets Fellowship. summary Two very ew questios about the cardiality of groupoids reduce to very old questios cocerig the aciet Egyptias method for writig fractios. First, the questio of whether ay positive real umber is the groupoid cardiality of some groupoid reduces to the questio of whether ay positive ratioal umber has a Egyptia fractio decompositio. Secod, the questio of how may o-equivalet groupoids have a give cardiality ca be aswered via the umber of distict Egyptia fractio decompositios.

9 GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS 9 Refereces [] J. Baez ad J. Dola, From fiite sets to Feyma diagrams, i Mathematics Ulimited 200 ad Beyod, Spriger, Berli, (200) [2] L. Beeckmas, The splittig algorithm for Egyptia fractios. J. Number Theory 43 (993) [3] M. N. Bleicher, A ew algorithm for the expasio of Egyptia fractios. J. Number Theory 4 (972) [4] M. N. Bleicher ad P. Erdős, Deomiators of Egyptia fractios. J. Number Theory 8 (976) [5] R. Cohe, Egyptia fractio expasios. Math. Mag. 46 (973) [6] E. S. Croot, O some questios of Erdős ad Graham about Egyptia fractios. Mathematika 46 (999) [7], O uit fractios with deomiators i short itervals. Acta Arith. 99 (200) [8] E. S. Croot, D. E. Dobbs, J. B. Friedlader, A. J. Hetzel, ad F. Pappalardi, Biary Egyptia fractios. J. Number Theory 84 (2000) [9] T. R. Hagedor, A proof of a cojecture o Egyptia fractios. Amer. Math. Mothly 07 (2000) [0] R. Kott, Itroductio to Egyptia fractios with iteractive calculators (2008); available at: [] G. Marti, Dese Egyptia fractios. Tras. Amer. Math. Soc. 35 (999) [2] J. Pihko, Remarks o the greedy odd Egyptia fractio algorithm. Fiboacci Quart. 39 (200) [3] L. Pisao, Liber Abaci (tr. Laurece Sigler), Spriger [4] G. Robis ad C. Shute, The Rhid Mathematical Papyrus: A Aciet Egyptia Text. Dover Publicatios, 990. [5] R. C. Vaugha, O a problem of Erdős, Straus ad Schizel. Mathematika 7 (970) [6] H. Yokota, O umber of itegers represetable as a sum of uit fractios. II. J. Number Theory 67 (997) Departmet of Mathematics, Uiversity of Califoria, Riverside, CA address: bergerj@member.ams.org Odessa College, 20 W Uiversity, Odessa,TX address: cwalker@odessa.edu