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University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 6-1959 Topological Groupoids Ronson J. Warne University of Tennessee - Knoxville Recommended Citation Warne, Ronson J., "Topological Groupoids. " PhD diss., University of Tennessee, 1959. http://trace.tennessee.edu/utk_graddiss/2968 This Dissertation is brought to you for free and open access by the Graduate School at Trace: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Ronson J. Warne entitled "Topological Groupoids." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Mathematics. Orville G. Harold, Major Professor We have read this dissertation and recommend its acceptance: E. Cohen, Edward D. Harris, T. A. Fisher, D. D. Lillian, W. H. Flecther Accepted for the Council: Dixie L. Thompson Vice Provost and Dean of the Graduate School (Original signatures are on file with official student records.) May 27, 1959 To the Graduate Councila I am submitting herewi th a thesis written by Ronson J. Warne entitled •Topological Groupoidsn. I re comme nd that it be ac cepted in part ial fulfillment of the requirements for the degree of Doctor of Philosophy, wi th a major in Mathematics. We have read this thesis and rec ommend its acceptance: c� b, b.L-t-.P� Y,')./.3� Ac cep ted for the Council: 'roPOLOOTCAL GROUPOIDS A THESIS Submitted to 'lhe Graduate Council of The.University of Tennessee . Pa rtial in Fulfillment of the Requirements for the des:ree of _Doctor of Phil<!>s�hy Ronson - J. Warne June . 19S9 ACKNOWLEDGEMI!N T The author wishe s to acknowledge his indebtedness to Professor Orville G. Harrold, Jr., for his dtrection and assis tance in the pre paration of this thesis. 45'7778 TABLE OF CCNTENTS PAGE CHAPTER I. II. TII. IN'IRODUC '!'ION • • • • • • • . • • • • Gl!NERALI'J!ES CN 'tOPOLOGICAL GROUPOIDS • • • 0 • 1 • • • • • $ GROUPOIDS IRREDUCIBLY COONEXj'I'ED BE'!WEEN 'IWO IDEMPOT!N 'IS IV. • APPLICA TIOO S . • . . . . . . . . • Monothetic groupoids BIBLIOGRAPHY • • • • • • • . . . . . . • • • • • • • • . . . . . . . . • . . • Topological groupoids with QUasi groups . . a zero . . . . • • • • • • • • • • • • • • • • • • . . . . . . . . 18 33 33 38 43 48 "· ·CRAP'mt I IN 'IRODUC TION groupoid A defined for and a t the se t is a every pair of pertiee many Ore 1 to associativity. [�]*, [T]1 Frink (h.], a is an An element a (1) onl7 In Chapter· 2, etc. • we a group oid if (hencefo r th, t opological in groupoids first this their for pro- motif, that s tr uctures , i.e. Hausmann and we wi l l • we will use the follov- a G ••• , , is use the a �n • common , e 0 and has r/ 0 all powers gener aliz a ti ons brackets refer to I!! V(all). positive integers or a are UDl-que. will prove theorems r el ating to ideapotenta, 1'trUJ!lbers in square end of this p aper . proper of ·neighborhoods for property (L , a , • abbreviati,on V(�) [V(�)V(a)] n (V(�)v(ar) ]V(a) e ) every triplet ·fr all V(a) of a for a �n-1 said to haw f} ]) Many of the se theorems are at the G, topological • a or It groupoid of � r m oreover, the Our aim [5], Etherington (2?], is a groupoid of certain topological �3 • ·� a a of V(�) the study topological a , "' • a wi l l denote the empty s t and space, and, We note, in relation to the (0 n to element of_ a iff introduced in_ [ , G ia If ab (22]. a • a , ·a� t,' (L i:f' and • G is continuous in �arri so n " i.ng. :not ation1 etc. e G dealt with non-associatiw algebraic and Stein If b to inv estigate the relati on au thors have Al�ert a, called a topological groupoid. dissertation is two-fold: ( 2) elements groupoid , �then G is own sake J which a single valued product ti me a Hausdorff top ol ogical same multi.plic at ion .in the ��ace G . G in of theorems numbers in ide als, in mob the bibliogr aphy 2 theor:r. wise we establish the .following ·theoremss For example, bic o��pae t groupoid tains at G __ a . and �- 2 • .3, ·tiona gt'Ov:poicl c<��pact G A is _th en is bicompac�, G is topological group if groupoid, each llinimal left pro�ded each element of cloeed. algebraic subgroup bicompact o '!'he id eal and ide al , G If (Defin!: is a bi which is a bic011pa ct each m1. nimal right ideal is cl�sed K , the minimal c losed set or idempotents or ideal of' G 1 is center · a t opolo gi cal groupoid is · ·�hapter a G is ideal K denote the n closes bico��pact, or 0 is set of � G so that . cormected.. groupoid Ir is vi th a a G suc h �at unit, the minimal closed groupoid vi th a x e. i�.f' I unit, groupoid such each element of _ I is center assoeiat.i ve cuts I · • I in o If G is a llnit, no element of' eonditi on is satisfied. groupoid vi th a zero, eTery left has a left G a connected (right} ideal of a I let exists a there is a -certain neighborhood G G or connected groupoids. G [17] groupoid with .. disc ussio n It . non-aimple a connected xy • yx • u then no e lement of ·- with e l� en ts . of' is bicompact and Tided to pologi cal topological of a . ... - a right ideal, and a mirrl:mal 2. 2}. Every algebraic subgroup is contained in a maximal associatiTe '£5]. a a unique minimal c lo s ed has - -· I is an algebraic subgroup ot . -rhese c oncepts ar e definea in Chap te r II 2.12, and G A property � con element with an minimal closed left ideal, a subgroupoid. closed 7 If' Every bicompact groupoid �ni1181 � losed If ha• that least one idempotento ��p��oglc�. gro�po�_d gro up� _ G An element in that G , b i eompac t, connected, I It G cuts G is is G provided a connected conne c ted pro (right) unit. In Chapter ni, we consider bico.pact gr oupo:l de G with a zero n u it and a o�� �e with c tf � , 0 cancellation law. e t/ 0 , then ���rges to terval xe < n � ., 0 y, which is homeomorphism. We note and e G € y, c x, for 1 -<-. x E. y , a nd where I represen t s the positive inte gers , x If o states that an t/ u x a,nd n that the unit a as bers if the multiplication from f i G . or has '-n onto the unit usual the above theorem may be replaced In particular, any continuous Dllllti- the cancellation law, obeys of the results 0 G. of acts has property [0, 1] •associati�N-like" properties of ties, we establish some G if every element of space is the usual multiplication acts �s a unit, �d every element the prop erv has is omorph ism as well as an order preserving . [0, 1] plication on G ey ... tud7 cy < center associative id empo tent . next s . - . • there exists a .f'tmc tion multlpl1eat1on, 1 :x:, ex and which of real number s under the usual topol o gy and the [OJI 1] by-- any- !J ar u and ex and .r} yc 1!1 If y. • ye xc Our main result tL , t� en property x implies {r't � � e� 0 which are irreducibly c�nnected between r e al num- a zero, as 0 . :We Usin g these proper- theory for this type ot of mob gr-oupoid. In Chapter �oal is � grou po d s monothetie IV, with we are concerned with a zero, and . quui gt-o ups . We consider the a a t is dense in 0 net (bn�n �ere a� each idemp ot en t is a c lose d ideal of ply AC � (biti eo_ · A topological groupoid • iff there exists finite "word s11 in monothetic groupoids, topolo- G of G ?-. a unit and suppose that E., G • .'! P such that We call • the c ol le ction a G . G Let U Q • A , • i , and AA .. • A • clue• n} � is A ·c G 0 gener a to r of a o o of all � (setr definition --4ct3-) This n�t ·Ifbicompact, '"nC\.' {'bt,ai such that a} 0 1m- be a monothetie groupoid with where P and Q are G disjoiut, closed, and non-emptyo . - "words" nite - in �ny - . and a of generator - a mono the tic conditions for P Then - groupoid - Q each contai,n infinitely many G. We g1. ve necessary and sufficien t to be a monothetic selld.-groupct next consider the general theory ot topological groupoids . If a E, 0 pr �ded . . and a8 ��0 • is q uni ue A nil lett (right, two-sided) ideal of ments. It J. G or for all n (right, then a called is two-aided ) ideal G or We . a zero • nilpotent a a left is G which consists entirely of nilpotent ele G is elementwise bic011pact, every right (lett, two-aided) ideal ia either a nil ideal or contains non-zero idempotents provided eT�ry �lement or - A has property � o '!he groupoid N-gr�poid it� its nilpotent elements form � -�·� prop ert y A , then G neighborhood radical 1 with G ti- (Def. 0 V0 or 4.9) of a is an an is said G open se t. to be an If eTery element N-groupoid provided there exists a which consisis entirely or :nilpotent elements. bico 11pa ct or N - groupoid 1he is open provided each nilpotent ele�n� ·is right associative f 5 J. We show that any bicompact subset or G _ is b�ded. Finally, we consider the special groupoids vi th a zero men _ tioned in Chapter III . We discuss the concepts of nil ideal, radical• and "arbitraril,- We close this �eOJ.'T or two chapter - vi th q�-gr-oupa� that ax • b known tl_lat a • . � eleaenta �. xa small• 'bicompact ideal neighborhoods :ra a and and b A. an exactly one finite implies 0 in these groupoid&. application or topological techniques to gronpoid ill or G G is called a quasi-group , there exists exactly one :r E groupoid that G such that ya • sa�isfies - x • y for all a, x, b € x • It itt tor € G) is arrr G such is well the cancellation law (ax • 7 the a7 a quasi-grou.p. We give conditions for a bieompact cancellation groupoid to be a quasi-group. CHAPTER II O!NERAIJ:'l'IES ell TOPOLOGICAL GROUPOIDS - - · J·- - - _De �Di �ion . -- �·��i� on ain e(i in o ent. 2.2. A 0 ot is a subset of 0 for all UX • X • XU An eleDtent e in . e. x Clearly1 i.f G .. G 0 E € G • is a non-void set of G ia a non-void set (1:1) a two-sided· ideal G in it is unique G an iden tity exists in 0 X an idempo te nt itf • An element all is called is termed a .!.!!! itt Q € tor G (ro C T) J ! u Definition 2.6. b � '1' C Definition 2, S. An e lemen t • xO . elem en ts a, which is both a lett ideal and a right ideal. Clearl71 it a zero exists . . iff C err that G Detini tion 2.4. Ox • in which a sin gle 'Valu ed (1) A� (rig!tt) ideal eich o contained in .or G aubgroupoid in a groupoid _ such that TT G Definition 2.). ! a set ab is de fin ed tw eTery pair procluc\ 1' A £OUpoid is 2.1. E 0 and is termed is an an ide��potent. iden titz ot 0 • 1 it is unique and is an idem- potent� �rinitioa 2. 7. A groupoid b1 and c ··in G G in which (ab)c • a('bc) tor· all a, is. called. a semi-group. Defini tioa 2�8. If. G is Jt groU.poid and at the ,same time a Haus- · dortt topological space and) moreover 1 the multiplication. in the groupoid G Q is continuous in , then G � th var iab�ee w1 th re�t to the topological space is called a topological groupoi d ., Definition 2 .. 9. A topological groupoid in which the multiplication is. �.. associati'Ye.i� ��1� � topological semi-group It !1T 8 and . tor bicoapact are bicoapact groupoid then in a 0 of a topological groupoid subsets T1 x of Cia 1 aG 1 0� 1'2 • ie. bieompact It thus follows that !o and Ga� a , 1 G if G are bicowapact • We also obeerw that it topological gr011poid Q '1'1 01, gical gr�pold ��er �· of and Ta are connec ted subsets of a T1Ta is ccnmected. , then A subgroapoid The �· a Br. Tyen onotf' e theorem T1_x T2 c onti nuous illage the - is - a !a and 1'1Ta ie also lticOllpact. then ia 1'1 ar a topological groupoid G is itselt a tepolo- r elati "fa topelo17. f'olloving theorem gi"Yea a s ta t elll ent of' the asseciati "f8 1• i11 \opolo gio al sroupoida in teras of' neigbborhoode. . .. �•or• 2.1. !. necessary � suffici ent condition �! �opol ogical ... �· groupoi d G � !?_! .n ,! �(a) [T(b)V(e_) � t! -and �(c) a of a , b � � , � !!! triplets � c Suppon the Proof a mente 1 , and of o neighborhoods "T( al o:r a c o ndi ti on G , gioal. !?.! l'leighborhooda V(a) , _T(b) �' respecti"Vell is such that V(b) [V(a)V(b) ly(�) n V (a) [V(b)V(c)] On {V(a)V(b) ]V(c) topological semi-gr:oup !!. � �.!!! a satisfied · G : there exist ele- '!hen one • can tiDd V(c) of' c such that of' b , and .' But this • 0 and (ab)o r/ a(bc) !! b_, c , the other h81'1Cl1 the condition i s clearlz is a ooatr �ictioa. satisfied a tor topolo- aemi�gr-oup. We will Le .. 2.1. ftbaeid.-voup!! Proota b denote the topolocf.cal closure of a eu s e t S � G !!_.!.topological groupoid �!!,! G ·• � S11ppose a !! ! E. S ! oloaed subaemi-Eottp!! and b € S o Let V(ab) of S 0 'be G !.! 'by ! ! • an arbi- • 7 trar)" neighborho od o£ V(b) o£ a e. 1hen there exists neighborhoods • and b respectiTely such that V(a)V(b) the:re exists sa ab V(b) Thus, • and sa S f, 81 Next, suppo se there exists - -1 a(be) (ab)c b , and -· Y(c) can nten one • of c such that e e , sa T(a) n 'T{a) [Y(b)V(c)] all a , b , - - and , L� 2. �· "'a2 • . in . [V(a)V(b) ]V(o) , •3 E. ·V(c . Y(a) S and • S such that e. V(a) of a , n V(a) [V(b)T(c)] of ot V(b) • D • such that S � Thus a1sas3 e. [V(a)V(b) ]V(c) {ab)c • a(bc) and frYr � a ! topolopcal groupoid • ;heii i .!!!_!! � A !!. rig11t is ! closed riiht (left, two-sided) • Definition 2.10. We will make use of the f ollowing notationt " " " - a , "n a • � a al a -1 a a2 a • t� haw • • , • ••• 2.11. An Property (L_ iff element a " tor all positive integers n Examp�e � �1. a An of [V(;-n)v(�) )V(a) eTe�t or neighborhoods V(aD) not � " €- But, • ha-.a ve � Definition a •1 and V(ab) !' ; � G !! (lett, _two-aided) 0 and c C e Hen�e, we ha'\'8 a contradiction • c Analogoual7, ideal of Y(b) , . and neighborhoods fi nd s1 and ab S a , b , �t, there exi s t elements s1 , �· 11 such that S Y(ab) n C. e1aa f, and V(a) a a /\ topological. - ,... {1, cr ete topolG£7 a 2, 1} with G or " &r and is said , �� . T(a) ot • finite t opolo gi cal gr011pcnd which is topological semi-group although eftrJ' eleaent has b7 the set groupoid n V(�� [V(�)V{a) J � 0 o£ an , Y(aJ-) and r e:xaple of • propert7 the following Dntltiplic�tion table CL ia liftn and the dis 8 Leiau 2. 3. G · po��ive n �d . �ntegera _ 2 X 2 1 ) X X � Suppose this r r and 11 __ may x �!!! .!!!!, powers such !!_ a (a8ar)a were not 1, 2, or ) It !!! eleMnt be � � aD(ara) • the case. " " all tor 1hen there !!!· positive exists J " (allar)a r/ a11(ara) that a !!:!. lllligue. A. A It is first shown that Proott �n�'-��· 3 3 X 2 G !.! ! topological groupoid. ·Let (2_�, iiaa properiz 1 1 1 " Thus, there • V(�) V(� , and V(a) or � , ;r �d a respec But this ti��- such that ('V(�'V(�) JV(a). Q V(�n) [v(a!}v(a)] • 0 : contradicts property � . �us, (aDal1a • aD(�a) tor all positi-ve inte-· _ It is next shown that an+w. aD � for ·all posi t1 gars n and r �s� neighborhoods , , . • _ � · A A A ·. A A • • integ��s n and m tor .m sume it is .. true :• � �+l �(;ra) - � • .... �s is true for • • and -r (;-�)a • • m • pr ove �t is �+ra • 1 ve • by definition. true for �+r+l �. m • 'thus, r + all We 1 • wi�l as Now, the powers of a unique. are De�ni ti�n i!_f for every {a•an nth _ � . . ±} power , of a 2 . 1�. � G where a ___ �P� 2�2: An ex�le is �ot bic0111pac't.. ' Let X1' • x � '• 1/2 tor all {aDtn e I} C is t�e r �d [a,i/2:] l • of a.;· are an- po�itive integers and f!_f an a bicompact ele•ntvise (o, 1) with G• - elementvise bicCIIlpact is G �ch that 8.ll the powers , is �o�tained in , I topological groupoid A. in the .G.·· Let subset of unique,· is -� the set �·unique �:- · bi�OI!'P�t groupoid which ia usual topoloa. -Define ' a 6 G , �a7 ·. a:· < lf2 • . . 1hen , . 'Jheorem 2. 2. _ � elementvise bicompact groupoid -�!!.!!. G element !!!2, Fopertz a, eontaine �least �idempotent • M . Proof. Suppo se Lemma 2.3, �he po�rs of A ":( _ • [an�_n � I} is {air� � v} and . and G n�te �nterse�tion is a positive integer. n U {_A'( • Y" t semi -group . {anan z �ho�se Va(z) of 7 C n and z an � } I , �u e Ti / u � • V(Xa ) 1 i 1 td n • > such • Since Since z Ti • v 1 - u ::::.. 1 • 1 1 y Q [anan Q • is and (} an ( } A to • , and ) {!.tj 1 , D obeys n To prove this it • 1here.fore, there exist D s uch that Va(z) or and z such that is bicompact D of the n) 1, 2, • •• V(z) be neighborhoods . Va (y) i•1 i V(z) . and open set containing ·n an inte ger e, D , there exist integers ••• I in • there is , } f.. Suppose there exists • ::> D (1 V(y) It will be shown z , and V(y) C E. D (i • 1, 2, and x4 D V(xa ) 1 that ld V(z41} Let • -v(y) Ti+1 or Let • in Xa.ljDV(xa) Since :J D respectively e D x Xa Let has the fi U • forma a group . for every y, V(xal .o V(y} Q • 0 tha t � of n [.lVCii (z) V(z) (1 Put {an is a co��BU.tati ve cl ose d 1hen there is a • 0 C D D · for all D • -/ fl A --<•1 Since D finite subcollection a so that _ ":�xa) D '7Xa z Va(z). n VaJy)V(xa) XD � yD �ueh t�at neighborhoods we. can • Since • �- D It remains to show that , t!lat is, �- _'fD Then is bicompact by virtue or I} C., G , ••• Hence it is cle ar that is s ufficien t to show that D } 1, 2, • is a commutati-ve closed group. D in '!hen , by • a topolo gic al semi-group by Lemma 2.1. property and the a ss oc ia tive law. 7 CL As is eustolllal7 denote these are unique. the elemen trise bicompactness or that has propert y a represents the positive integers, is a topological semi-group. I Hence a an , etc., where powers by where E a !!!. Vi avi €. V(z ) j • c , c + l , ••• � 1 u and so such that for every } • Then Vi • �- n·,cte> , c•1 element n ACtc) E d e•1 exists a neighborhood - !Js.;tt;} , GD d � "lhus, ... "• aTk - Then since Q is 0 V(d) € and D . � is a group. D . LeBIIla 2�_k. .. � g (2) �. some then y • g (.x) __..__, f. ...... € ��.t. U. ag(a) c • .... G • x g(B) x vi th :x !!!:, .. I! . I _ !!_ r is a Proof't It will ix • i for all x Then there exists be E, z A Hen c e , we obtain • is the idmtity or This • xD 1 D • D e2. • � � !!! powers 2f a !!:!!! !!1 A . � {3) A (! continuous function)� g(x) x , C A B C G '\c • c and sane e a subset compact. If � xy e 1 �, we have � x ! be � (1) i\XT !! £ .:: H:�noe we may obtain �n _ open • U g(B) C G\c If [18]. a B n U XCI in I �!E. algebraic subgroup_2!, � compact submob � by Lemma A � then , I o e, I 2.,1. We will next so that Suppose there is a such that s, Va(z) z , Ya(z) or n topolo�ical is ! topological subgroup:· sufficient to show that one may fin d n eighborhoods for all !!! is !. map g !'hen • Theorem 2.3 • subgroup. V(d) o and we have � con�adictiono . . groupoid e !:. function I.r not, then tor some G If 0 !E element �� and G '!hen • t a k E such that G � !. topological groupoid A� G t \. g(B{. includin � _ Q , and since ��idempoten t element. Let c Proot t g(x) "ttc D , there ! necessary �sufficient condition �!!:.element- . � G .!!:! unigue is • contained in V(z) n V( y)Q • � bi�ompact gcoupoid � � - Choose � • V(,}V(d) C V(y) Q Corollarr 2 1 or G d V(z) e .x or C D . open set containing an !Jlere exists an integer contradicts the fact that tor all r\ A(tc ) �1 and it is easily shown that 0 · 10 .. 7Xe xi • i y S, for all i Va(1) or Va(y)V(:J:a) • Xa in y, aDd a S :x sueh that for all 0 i show that i . is a A yi and � I. Thus V(Xa) or Since I xa is com- • {vCxa;t): paet, there exists a :rlnite s ubcolle etio n k i�V{s:Gi) so that v<xJ •s the may find neighborhoods V(s) that n V(y)Q such -t:Jlat E; a V(z) 0 • • and V(z) we have a contradiction. Ax • i for It where is a D X x all € :V(y) Now, there exist elements V(y) f, b %A 'lhus, . of i topologl.cal and and if m > n m €. D , then m > m • (iii) p e D , p m p -,.. n A __ ,... and is a function � v if far any open set 0 d1 E. such that v v{d•) €. 0 d e a and A V(y)Q E. i . of b and Similarly the f\mc tion A onto itself. x � x-l 1hus, I set ;>) (D , is a non-empty set T E' if then m , D � p n € and n , and m, D (ii) • p if then for some • on a directed set a E X and a , and any D If • ·I d E. D , there is a QJ v(d•) E. d I E: X is a cluster point of the net if given any open set s uc h that if to a set D and (11) dt 0 :::> a €. x is a point containing a 1 d , then • Definition lett D m if containing d1 > d Of convergence or the �et there is a 2.4 tha t n � p �s a topological space, we s�:r (1) D €. a binary relation > satia.tying (i) together w1 th D x respectively so b(b -la) • one subgroup. Definition 2.13, [9]. A directed belong to all is a continuous function O:f A a Now • is a group. follows immediately :t'r<lll Le:nona e But, o i :for • and y z } of k . .• i��V{xa.i) - Q Let • 2, 1, • k and 'lhus I A ::J i 11 2.1ho (ri ght, two-aided) sided) ideal and A C: A Dd.nimal � (right, two-sided) ideal such that if L , then A • L A ideal L is a is any left (right, two • We note that a minimal two-sided ideal or a groupoid G is unique. 1� A For let and AB C A since be fl B 2.4� Theorem �property B Q.. • mtnimal n A Thus, • 0 two sided ideals or an is B {anln g Then, e. } I A ideal and � G �! topological groupoid _ A Then • B • £l T{n) • {antn ot 8 r} . ot T n. is that D Since show � Y ho� w�-- may �hal!, • • • • • and . x yx E D �oup, a .. • since D Thus o:t y • ba-1 {aDu� e, I} gr:oupoid. 7J.. n e. x 2. 2, {a8t � 0(ll:1') D D 2, an is The r e also exists • • • • o xy E a • 0 defined v 2 .3. � a is group. Next, we . Let arbitrary neighbor or x and O(y} aY+iy E an E. {ansn I} e . But, there exists • Thus . two-sided ideal of a that find neighborhoods O(x) Y • 1, is Suppose • B € D 6 • } I O(x) O(y} or for Hence • Similarly • Since D is a �� cannot contain pr operly any ·two-sided i,dealo For suppose A is Theorem Let _ n or all cluster points is the set topological group by 1heorem C O(xy} for two-sided ideal or But , a E one can '!hen &"+iy+n e, O(:xr) � ... t�-sided ideal or the mob · is ·a _rr... 1, 2, {l, {•1•1 Y} • as in the proof or 'lheorem y .;.s��h that O(�)O(y) Y _ A group � !_! � minimal � sided ideal tJO . C?mpaet, it is {ant!l f. I} or D 'Ihe set Proof. -/ Q !!!, a ar e (��the powers !! compact, � � � all cluster points � � .!!,! f�rme ! topological B and suppose unique) an • n o y Let E D and e: A. . By the remark aboTe, such. 2.!). Hence A • D Every compact • Thus groupoid a €. A o Then ya • b · E ·tet C2 be the {Aa1 e."} a is a minimal two-sided ideal an ideal is uniqueo 0 hae !. minimal closed family of all closed subgroupoids of be any nest [9] in � A D !!2_ _ P.roo.t'. • D o Since 0 ·• "'/... has the finite • 13 intersection property, Q A4 mtnimal C Aa for all pri nci ple 0 a e 1\ and hence is a clo sed groupoid. � '!bus, • r 9]. 2.6. '.lb.eorem n Aa r/ - a: 2../\ has a minimal member by the G Every com pact groupoid contains the unique minimal closed ideal. ---- Proofs ?"l_ {BY"' • : e, 1\ Y } is a closed ideal of 'l'hus, 2. 7. Theorem ideal and ! . in i s contained Prooft 0 � x �a • B c Let. Aa A� Let • us A be any ot' x • Aa � B 1' Aa , x{yz) · _. nten A� S xy y Aa for all • Y for every G � 1\ • preT.i.ously. �.!minimal closed lett ! top ological group oid G algebraic subgroup of the topological - E. C Let a -· C2 be the family of all subgroups of consider 1hus • We may assume (:xy) z 0 and hence !. maximal algebraic subgroup. Let and E. Then x Q BY'; uniqueness was established alS!braie subgroup or partially ordered by inclusion. {2; By C Every compact groupo� �eorem 2.8. �oupoid Q BY" But, • 1hen, o Let The proof of this theorem is similar to th at given for 2.6. - G Q, • minimal closed right ideal • . . Prooft 1heorem � nest i n be minimal_ member_. �� a has a G (2 be the fami l:r of all closed ideals of Let But, e B • (j A� A� A� C e c and , __ . {Aai • {Aata e 1\} tor s ome e '7 Let A-..r a B €. z • Supp ose • A-..r be the identity of 1\ and hence e for some a x f: Au , y y, . and A • a: z Then is an identity t, I\ B e., . z are in � , and , e e B • 1\ and i s the identity or B Therefore v �n A., are Q, Then • We may assume • A l;>e any chain in X , y ' and for some x, Hence x /\ <c., � /\} e Let . and a: Cfntaining G • Let x-1 e Aa C B • �d xx-1 f�� al� e = _ � I\ a. Now!.. suppose memb$r x-.1% = 0 A o� 1hen, B• ::> A and B' • t:Z • C .::> !4 a maximal has algebraic subgroup o f G containing the an is [9), lennna H�_nce, by Zorn's • B' B is a group containing A and B Thus, • member. maximal • Theorem 2.9. !! G !!_!. compact topological groupoid, eaoh !.!!! !!:! algebraic subgroup � G !!, .! compact topological group. Proofa If A is an algebraic sub group, I is compact and hence _ is a topological group - - D$f1ni tion • .M Theor• in G if - x(cy) an - c • f x 1 •• y €. G • G of a groupoid for all - A is maximal, element (xc)y a ... A Since 2 • .3. - We shall call 2.1S. center associative .. by _ � [ 5) borem 2 .10. If G_ is !. bipompact groupoid, � minimal !!!.!_ ·- ,. � ' I � ' ._ideal and each minimal right ideal � G !! closed provided � element . of K !!!!_ minimal closed ideal � 0 , !! center associative '!.!! 0 . - . , • G Prooft Let L be a minimal left ideal of - - a E. C Ka • L for L since L is a lett ideal. b (Ka) C (bK) a C Ka since every element or and Ka • - Si��S.:ly, K is center associative in G Th.us, Ka is a le:rt ideal • L because of the minimality of L . . ·- . - . .. i.t R is a a right ideal contained in R for a € we show that minimal Hence, L is closed. • G right ideal of R • aK Hence • R , aX is and R is closed. 'l'heorem 2.11. !!!! � 2.! idempotent& E !! .! topological gronpoi'! is closed. ---- Proofa p � i . rvs_p) J�_n It If' ! • 0 1 the result is trivial. Suppose E .f 0 and p ; p2 , there exists a neighborhood V(p) v(p) ··- o . stnce idempotent f exists in V(p) P • c, i , -v(p) _n _ !his contradicts E ; o.r 0 p *uch that , i.e., rvCp) 12 . n v{p) an .; a·; lS p2 • p , 'lhus, e p E_ , and We next discus s topol ogic al groupcids E is c losed. This will � be co nne c tivi ty as related or briefi:r the concept discussed more :f'ully in Chapter �· � � minimal closed ideal K � G is connected. 1hen, C • uC C GC Proof: Let C be a componen t or K • is a connected ·ac is c lo se d, K Def�nit�on 2.1 �· I denote the set or "1' e, G A and B If G is Let � x that I be to due i ... !. conne c ted � which� Theorem 2�13� pact and_ !.!£!!_ element � - -· Prooft � sume element { __ [18• s is and of K 0 • • • Theorem A n G in Q n v.; � � po nt s. B Q cuts and I !_ if Cl • 1] o 2 let C 2!_ -�!!; � C ! !.!. center associative � CO!ft G , ��element • • u , center g(s) • associati-ve. x·la for eiatiw property, it is easily seen Let X s €, all that r e I G is a • we 'Ihus the result follows immediatel7. · or I • P. Hausdorf:f' space � X C i n means non:degenerate. W� r��st note t�at I is is :f' s) � x_ If I Ddnimality G !!, !. connected groupoid � � I !!. !!. o :f' I cut s I in G Sine• I iff there exists a A\B Wallace _ _ ne the r ·=c.·� �oapoid with_� unit, let Let. us say that ! compact �ubset � I with !! � � �points, � , Qn u.;o utv 2.4. topological a C Simi.lar;J.y ··CG " • K fro m the agree that us We next state a theorem Lemma GC • c C G!C • are separat ed. G-P•JJUV, C • thus, xy • yx • u so �hat K -!.- or elements such Definition 2.18. i.e., subset is cl osed o C II:r • !!!.!, G � !. compact, connected voupoid !!_!!! ! 1heorem 2.12. Since to a as- m y a �oup since e�eh and 1Jsing X , the u � · n��ne center ·�-�o homeomorphism o:f' ·· G onto K .• 16 itselt. I Since ciative , • I f(I) groupoid and each element of a is Now assume • f(u) • x , each poi nt ot I 2.1J. leliiD& 0 1 Hence, .u g(x) • u then Definition contains no u cuts I in cuts does not cut cuts I 0 in I in 2�14. It "'V(y) tripiei ,2! neighborhoods 7�' reepectivelz!!?!.!!! � I K We note that B are each then since B B �- � • u reasoning C xK u �a� __ as t , is non K [6]. . 1 € xA. x g 1 _G € 1hus, G , G , r x • s • by tha t � G t�9(x) has cuts 1 mple , G • G r/ may assume has 1 • • in G in • ·itt '"'G [17] �simple hence and B 'u gr:oupoid xA r/ 0 E: u G\.K cuts 1 and A and • It :x: € . has a point in both e yt a right inTe ns .• u If K C A\ u. that , A G 1 B , and i.e., Utilising the h:ypothesis 1 a lett inverse. and applying elaeRtary, algebraic techniques sees that s) I to be simple K A '-.•. r/ 0 t{s) • x where I . 9(x) has a uniqu inverse x cuts I virttte of 'lheorem 2.12. x and • s r(t(s)) • G 1herefore, s onto 1 tself. eaeh • Let 9{z) g(s ) • and Clearly the maps i� a ���om.orphie� of r(u) si u • We 1� Theoram 2.1, ziDg the hypothesis , og.e e e, cuts � V(s) !!!, y , x , and s G � � yx • u, � � , the connected set Siud.l�l:r, easily o� sees G continua· lherefore, • x f., I n [V(y)V(:x:)]V{s)' f .q !.2!:_ every s € and o A n B , V(:x) , x , is connected U A G • G · Since Proofs then x If • G is said �!�such�_ Y(y) [V(:x:�V(s)] element or G !!, !. compact, connected, G asso center 1hen si�e • proper closed ideal.s. 'lheOra is But, this co ntradicts • Thus , no • A groupoid 2.19. G 0 in I I • s and g B utili and it are c ont inuous. Hence, since point of • g(t(s}) • s and f � consider the u cuts is a cut p�int or ' 17 G But • has at least two non cut points of itself. B Then it is easily shown that one of them. contradiction and hence g {x) then G cuts • u Ex�le 2.3. u Hence • no x • Now, if I cuts in G cannot cut e X G • · E B' u This is a G cuts I . .. Define - U8llal--.,; topology. (O,. l ] aob • ·a2o where the . asserted. as multiplication on the right is the. usual multi- . of' re�l �mbers. [0,1] with this nmltiplication is clearl,- a· topolog;!.cal. groupoid but not a It is non-simple, topological semi-group.. connected, satisfies the nei ghborhood condition of the above theorem, I cute · and I I the following multiplication on a is a left unit, be [0,1] be the mit interval of real numbers with Let the p!f cation G cannot cut p p Let (with respect to, 1) • 1 1 Obviously, no element of • [ 0,1]. 2.1$. Let G be Theorem ! co�nected groupoid � !. (1) If G � !. � �� eve:ey left ideal � (2) If' G !!!:.2.• !! -connectedo G !!!:! ! right unit, every right ideal 2!_ G is �- neeted. {3) !! G has !. right unit _ is connected. - (1) Let Proota x E Hence x E. Hence Ox L <; L . � Hence, R � R Hence R • 'Y.LGx x · and � unit, every ideal � G 0 E. be a right ideal o f' • YR:xa X G It • x Gx _for all € L , x €.. x €. R , tor all x € L • is connected. Ox is connected since each Remark. 2!.! be a lett ideal of' L • (2) Let and 0 e. G xG If • � R • '. ' xG is connected. (1) . and·. (2) .. We_ note that 0 could have been replaced by lett zero in .(1) (3) __ L is connected since each xG C R . The proof of and by right zero in (2). {j). is 'similar to those g:t ven in • CHAPTER CROUPOIDS IRREDUCIBLY p��nte Definition 3.1. �. and b IIT CCNNEC TED BE'IWEJ!N A space S b it and only it it is connected and is the only con and b • a and is a cut point, separating the space into exactly two components, one and the other containing order relation in .S b S - in e every x . y by defining < S _ _ } x X i.f We.. can introduce a linear . t. c , c the component "1 x E S , the set A(x) is pr�cisely the component of - b � S , by-replacing - x and the set in S - x vi �h e. � a S , B (x) '• . ..[21]. ... tor y } :> x {ycy € s, It is clear such that a S: x � b a tor b in the order relation. ·- ibis order relation induces top�logy, x {7ay • that we can define a reverse order relation in S tor every x S b • �� pr e cise l ! the component or a in S £. b [21] o x [21]. Under this order relation, Moreover, .for any y a such a space is Hausdorff every point different frart containing a or two is irreducibly connected between nected subset of itself containing both It 'IWO IDEMPOTENTS a topology in basis elements _or the form W • S {y:s , the so-called order < y< t and s, t E s} ,[21]. Since we wish to use the order top�logy in this section, w� prove that i.f S topology or _. is compact, the order topology is equivalent to the original s Consider or s Since , • ta (S, U) __... (S, y) where U is the original topology . ..., is the order topology, Y C t1 [2ll · and since S and f is the identity function. is compact, f r is a homeomorphism ex- tabliahing our assertion that th e topologies are equivalent. 19 Now, S and S suppose the two relations order We shall It S . ponent � c S - ; 0 r/ x , x if s and c It � S C) , - s < - y • <= <. A onnec ted �s, t] __ is poin ts Let the S � y s so that !:. t } � 0 x � u • 'lhe set • b t < a y <. t � a , <.. s U B whe re A €. X T ¢_ T }t. T T B A u B - X - € T x. , then • T � 1 [s, t] be the two non cut and y B p C and T hus , we and [s,t] such t hat €. are B To oin ts of y is By ·Y t/ • '!hen, a cut point of S • Hence separated. • S • A Since A nor B contains both s _ not irreducibly connected between s and A S is a connected set contained in _[6], neit� er C T s if Further, • � A which implies both S � that is ! connected subset of n Let • � C a com- ! compact Hausdorff space irreducibly .2.2!! If and a connected set meeting [s, t] . T t ��points in Proof. have b S , E W e shall choose • .! connected Hausdorff space � S be nected between � S €. n (S T Lemma 3.2. b S be Let !!!2. B !!! separated, and g A "1' u in defined have { • and a unit zero and is compact and is irreducibly connected between follows easily that Lemma 3.1. n o- with a [21]. t and we [a, t] denote J s, t) is co mpact if � topological groupoid is irreducibly cormeoted between one of s is a L emma .3.1, s,· t and t t Thus, • Y . B and e. ar e Otherwise, • T T • is a Hence • Lemma 3• .3. Let S be ! compact Hausdorff topological is irreducibly connected between ,.!!!2. points be subsets of 5 uch s that a and b • pa c e !� s Let A and B I 20 (2') A n B • 0 (.3) A ; 0 I B ; {4 ) ..!:!!!!. there a � A y and __ �..,. � for all --s Since ��ppose -- � n B r/ 0 exi sts B ment U ( '(' ) � €. A n B a that "> such that a "< Y3 € V' B u{-y-) n B "'Y"3 <: • n B ; Y a • £., e a 0 Y1 and for all A • Y E. .. A whereas Corollary -ti, 3.1. 11!'-. Y'a • • Supp o se Then there is a basis ele 0 . This • n A < "Y for all such that E,. � �3 • for all B contra � a ftr all et Suppose there exi sts > Y � Yi. .::: J) sue� that � A n B; 0 or A and B <:. • a � � B � Y"1 1 containing Y • V'2 Vi C:::: Y12 • Y3 implies Thus V) E. B • B • Hence Suppose there Suppose "2 A such E y e. contradicting the fac t that A and < as sume A n ! r/ 0 and let Next 0 a .S. Y a Then • U(Y) He nc e , Y e. � • A such that i . V € � � Y"1 -! . ys tJxiste � such that 1 such � I nB "< e B E, � ntus there exi sts a basis element U( v) • are two elements _ _ Let Hence , • � Y for all and. . � e; � � V tor all B E. � is connec ted, containing dicta the fac t that Y €. !!!2. £ A a if 0 is � {and only one) Y E. S Proof. there A U B S • {1) a for all A Then there • < Y'2 E. implies Thi s contradicts 2). Under � hypothesis 2!_ Lemma 3 . 3, either A eon taine ! large st element � B contains a smalle st element . 'Proof ... Y It Y' � A , Y is the largest element in A is the sma�lest _!lement in B. By (1) - - _ If y·. E one of t hes e cases mus t occur, ' while (2) • implies both c annot occur toge ther . B 21 Let S Definition 3 . 2 : two � onnected be ?'e�n a be a compac t Hausdorff space �rreducibly points . Let E be a subset or S E. y --� �-- such that x � y .for all x . 7 bounded below, . and call Definition 3 . 3 . . a Let E E lower bound or T. (b) • If is o� > x is • Let be a subset of E Suppose y has the following properties E E , we say that b e a c ompac t Hausdorff space irreducibly S connected be � en two points . of If there is • (a) S • y is a lower bound E y , then x is not a lower bound of the gre atest lower bound E of Then • [clearly from (b) , there is a� �o �t one such · y . ] We shall use the abbrevi ation •int. • or "g .l.b . " � . .t:� "gre ate� t lower bound"• LeDml& _3.4. Let S be !' compact Hausdorff topological space � is irreducibly connected between � points void subset may • S - B E take i� it auf' .rices to A and a iff there is an of lower bound prove that A E x � e. B , a s x • E is a lower ·bound of • Let , and every To prove the existence of the inf . , • E Since We now verify that � E there is 'lhus, 'lb.us • an a <. y e. A x E. E such that p for all , and a � A B is non-de generate, is bounded below, there exists a y tor all �-· > x such that sati sfy the hypothesis of Lemma 3 • .3 . B E E � x has a greatest member . Evidently (1) and (2) hold . Since Let B be the following set t o be non-degenerate . Clearly, no member of B A Then !!!!:l � c6nsists of a single element the result is immediate . C. B � • �mber of E If or elements : A • or S !:!! _.! greatest lower bOllD.d . E Proof. 'lhu�� _w� a � b e. S such that y r/ � 0 x r/ 0 . Hence (3) holds . If � :;:> x A and • If � � B a e, A , and (h) is • 22 satisfied. Hence, by Corollary 3 . 1, eith er contains a smallest element . Le� p e., B • _ > �' • X Definition 3.4. cancellation law iff y G � o , ·with c -/ 0 such that > �I X � > �' e , G vi th a zero is A groupoid :xc • x Choose o s o that B ye or ex said cy implies that x • to satisfy the ,. 7 tar all x 1 • !. c ompac t groupoid � !. !!!:2. . s atisfies .:!:!!.! c anc ellati on !!!• < ex A \. • . '!hen x <. y implies cy for _all x , y , c € G � c r/ 0 _ Proof . Let :x: , y £ G such that x <. y ., � !. � 0 J,et k. � inf' , • pose k :> . 0 yc {xa \ • C, for all e } C. A Xa [xa1u ] } Clearly, • We next show that xk l.� espeetively such that There is a n eighb orh oo d C V{yk) • a E. � ne i ghborhoods V(xk) V(xk) hood 'V(k) 'f and • k Then A r/ 0 and of k Thus xk V(yk) <. Suppose xk 'f €.. V'(k) yk such that Xf(k) V(yk) > a A • a XXa > and Then yk > Y .• and l'Xa and y by the cancellation law Hence there exist diaj oint • <. V C V(xk) e:. yk • xk of imply Hence , o x • or xk and imply u exi sts by Le111111& 3 . 4 . We sup yk • yk , then • since such that xV· (k) C V (xk) Xa: e an If xk and V(yk) € and of k V(k) and we have a contradiction. e V(xk) But there is we have a c ontradic ti on . a yc Le t there _ exist di s j oint open neighborhoods V(xk) and V(yk) yV: {k) < xo • {xatxe - �• � is not � � 0 i s irreducibly c onnected between 0 � u • Suppose . B or This completes the proof. � G be 'lheorem 3 .1. � E x Since the smallest member of B • cont ains a largest element We prove the sec ond alternative cannot hold . '!hen there is > p such that A respe c t� vely such that yk • But, there is a neighbor and yT-(k) C T()'k) • We 23 may take V (k) Xa e V(k) , c e, V(k) • xc <. ot k an Xa for yc < ye XC for all [0, t r/ 0 for · all - There exi s ts a Henc e , • c to be an open int erval about k c) e. c 0 • and Let xc .3 . 5. � G satisfies the c ancellation <. t € G • A yc t .<: such that A. e. <:. XC k ye for all Henee· • This c ontradic ts the definiti on and suppose -/ c 0 '!hen there is • simi.lar argument yields � cy ex be ! groupoid � !!!1 G � .! � !. �· 0 and � idempoten ts except c ontains G If • Proo.:f' . t c . But, o an • Lemma u an d [ t, u-) • v · ( k) €- t Since there is o (xa, U ) £, C k • .:f'f • _, .. Let .:f'u !. and . hence Corrollary which !! be an idempotent of f !f. 3.2. irreducibly t u • .G G t r/ and suppose 0 'lhen • ...by t�e cancellation law . ! is !. � compact groupoid with 0 connec ted between and �!� and whic h satisfies the u c ancellation law, then (1) ry < (2) it x .s (.3) i:t X � property (L min (x, y _ idempotent . to Proof. xy � x by yw implies pr operty { :z:ll t n Lemma x Theorem � � , � .3 .4 · (1) • r} yv u y) � !! w X :::. v , then -j U , { n x sn 1 .!d:!!!!. xw 2., {7fl& · I} yv: s. .!2_ !_!! c onverges n e., y s. t} c onverge s 0 . � u 3 .l o (2) by The orem � x -;;,. xf implies x 3 . 1. � xy S. !: y impli es Henc e r'J . . • and y xw xw 5 yv by 1he orem must clus ter at an idempotent t that the monotone decreasing sequence • .:::: u yw (3) • 3.1. impli es and Since :n e r} v has x By �e orem But, it follows � S.. w 2 . 4, from has a lind t . .� 7fl • _Ir x r/ { , I} then . . :z:ng n 5 is bounded away from Thus by Lemna 3o 5, lim x!l • 0 and lim xD r/ u n n -� 3 . 6 o � G b e .! compa� groupoid wi� ! � � !. � lim Henc e u , n t o u o o which !!.. irreducibly c onne c ted between cancellation !,!! and E O . o · and 3 . 7• . Lemma Gx 'Ihus • • [0, x ] � which satisfies .!!:!! • s € He nc e , • n time � . _ each ous , . E fn ( l:l'tJ.) C_ G y into G G by 6n --+- fn s define (We • by or Lemma 2. 3) vir tue c an cellati on law. . Let rem 3.1 � i f x r/ 0 , assert that .3 . 1. 1heorem :x, y - • Suppose • Henc e ab • aa a 0 • It follows 7.'!1 Suppose < x S. nten a, . x has 1 0 - 0 and fn ( 6� , E G h • .; G an '!hen o a � u a y <. x unique o The n x "P for all positi ve �th 2 y x where are G Since o This i mpli e s nt h root. �..: ab· s i nce a r/ 0 , 1 is continu fn S ince o root. has a unique square b 'lhus, • By induction, E satisfies that all powers of x note is a cormected set c ontaining x r/ 0 G G by fn (y) • xD __ • • [0, x] semi -gr:oup. fn is onto �d� in particular, that any x pose a2 • b 2 Suppos e o G !!,!! property is irred.uibly connected between 0 and u We now xG be the diagonal or the cartesian product of G n is the proje c ti on of unique for all x and [0, x ] e t !!! abeli an topological �n Let For 0 � G � !. c o!pac t groupoid � ! � and !. � G .. !!. Proof . c z By' a similar ar gum n , • � cancellation !!!, � every element 2f � o • ax for some Z' Gx 1 which !.! irreducibl.z 4&0nn.e cted between 0 � u is � � G · Gx t� a c onnected se t containing Since X by C oroll ary 3 . 2 . = Z u xG !2!_ all x ·in • � C: [0, x] Gx 0 e ( 01 X] . • 0 � Gx ·:J [01 x] by Lemma 3 . 2o If , x Gx Then • . Let · x Proof . a 24 b2 � a • b by Sup- ' • a 2> the root . < x '� integers by' Theom o We c all this result (1) • Let us define f'or x € G , x r1 0 p and q are positive integers . It is inmedi ate from the fact that the ..t�th roots are tmique, tha t r is well d efined for any positive dyadic rational r , regardless of the representation of r By the same gument , straight forward verification establishes th a t we where �· x • ar e an { 2) We claim further that i.f r and s � , then r � • 'lhe�e exists a rationals such that dyadi c rational t such that r < rt xr xt By Corollary 3 . 2, r t • s By (2) , xrxt � xr . Therefore, if {rn} is' a monotone_ increasing sequence or dyadic rationals, {rn} is a monotone decreasing sequence in G For E. G x r/ O, u let D �r :r is a p�sitive dyadic r ational} • By (2} , D is an abelian submob of G We claim D G Let V (O) be. any neighborhood or zero. Since r , xn� 0 b7 ���llary 3 . 2 . 'rhus , V(O) n D rf 0 Let V (u) be any basis ele ment containing u Let y e V(u) with y < u Clearly 7n� 0 �.D 1}hus there exists a positive integer n such that r < x He nc , by {1) and the linearity of the ordering y < x1/fl and V(u) n D rf 0 Le t t be an arbitrary element of G , t rf 0 and t rf u and. let B {s : a -� s <. b} be an �bitr� basis ele�nt in the der topology containin g t W'i thout any loss of gen erali ty may assume are dyadic 8 x + .. 8 x • • • • • 8 • x • X , � m • • X • u . • • e • • • I or • • , we • • a 0 < < b < u R' ' l . u.b . a maeta r a positive dyadic ration are non-empty o Since B r' ' and sequence as r r' incre ase s , �erefore1 we as sume • a is n, By .Le111111a to c . . 3 . 4, ; Let w � u E or r' rv ' • .f+P)c � for s ome r Le t o {Pn} e, V • But, · • There exists a + Pk ::::>- �+Pk a mob by Lemma 2 . 1. • i s a mono convergin g to r' {?n} ;?! b n Henc e {:l } countable dense set and since G • w s·o that C W Pk r1 � r• .f •: .z!!k 1 • Since C learly G < r' • , be a monotone n � onverges [0, a ] and n D '!hen • for all U U • 0 • V r/ 0 , {Pn} such in the sequence This implies • e uv C w w and From thi s c ontradiction it follows that __ tween two points, b c UV such that �'!s � . i� • ? c be the r' If' • - [0, a] numbers S. r ' ' r' {r} Si nc e • R' open set c ontaining V and r [o, � ] has a g l,b , � joint .from _ Let· x8, r � implies multiplic ation, there exi st open s ets e U z'Pt e. that an be and C., U {� _ - w1 th al} monotone decreasin g se�uence such that By the continuity . , <. s r R" be the g.l . b . of increasin g sequen�e of dy�die rationals of {zl'n} rational} a positive dyadic have upper and lower bounds respec tively. R' of 1 R' and decreasing tone D • R' ' and r . R' Clearly, R' {r_s_r � b , {r t� � , • R' Let • G is abeli an . is dis :iJ • G Since D • . is a is c ompac t and irreducibly connee.ted be- is hOl'lle omorphic to I 1 the unit interval of real [ 21 ] . hample 3. 1 . An example o r a finite groupoid wi th a zero and a unit whi'ch �beY£! the c anc ellatiC?n law and in which every element }.las � operty a , . but which is non-as sociative and non-abeli an is given by G • {1, 2,3,4,5} following with table o multiplic ation 0 1 2 3 4 , 0 0 0 0 0 0 0 1 0 1 2 3 h , 2 0 2 1 , 3 h 3 0 3 4 1 5 2' 4 0 4 , 2' 1 3 0 , 3 4 2 1 S' 1 We s ee �{ 23) 4 . Definition is an �. eleme nt .J 2 {34). . and 2 -· • 3"3 .. � • 2 An algebraic nilpotent 3. 5. x - .. -· � is such that • �f � groupoid unique for s ome n and with tor a zero n this 1 0 . De.rini tio n 3 .6. x ment such � that We next . state ��a � � � � 3 .8 . a Let An • 0 to� S � Suppo se func ti o n n S F a semi -group is an ele • to Fauc e t t !. compact Assume f'tlrther � � there exi sts !. some r es ult due � idempotents . • algebrai c nilpotent of [3 ] . � �-group � !� � !. � !!_ irreducibly connected between 0 �S c ontain s � �� algebraic nilpoten t& . · � S onto I , .!:!!!. unit interval (01 1] � !:!.!!, numbers !!.!:!! the usual topology � � � multipll:ation, � , !! !i! isomarphism !!. !!!!. !! !!!. order Lemma of G 3.9. Let G � .! has property Cb and - --- G p re groupoid s ervin g home omorphi sm. � ! .!!!:2. � � every element satisfies contains � !!2,!!-.!!!:2, •lgebrait:. nilpotents . the c a ncellatio n law. 'lhen G Suppo se x!l Proof. or we rt-1 • 0 x • se e th at 0 !hen xn-l o 0 or xn- 2 X • Similarly, 0 0 • o • :x: 0 0 • � e X ntus o X • 1 0 Proceeding in this way, • The orem 3 . 2 . G £!. !. compact groupoid Let which is irreducibly connected between 0 2!, G � property and u with o !. � � !. unit A s sume � element � � G satisfies the ·cancellation law� (L there exi sts a func tion r [ o, 1] G onto I , the uni t interval fr om 1hen !!.!_ real numbers under the uaual topology and the usu al multipl�oation, which !:! � ii OJbo�hin ..!! !!,!! !! 3 Thi s result follows immediately from Lemmas Proor . and 3.9. Corollary J . J . idempotent .!!! order preserving h ome o morphism � r � G be .! compac t groupoid with which is center associative . nec ted between 0 and f G 1 • .$, 3 . 1, .3.8., !. � and another Assume G i s irreducibly � satisfies the cancellation law, and every element of G !!.!! pr ope rty (L • Then there exi sts fun c ti on a f from G � I , the � interval [0, 1 ] � real numbers � the usual topo5 and � usu al mul tiplic at�on, which is � isomorphism � well .!!. � order preserving homeomorphism. Pr� . G Since ro Siml.larly, Of i s a connected set c ont aini n g • fG • t act e :is a right unit on G.f is a • two eided unit on G G Since • and • t and 0 f , · is a center as soci ative idempotent G and a left tmit on fG • G • Hence f the result follows from Theorem 3 . 2 . Remark. We note that any continuous multiplicati on on [ 01 1 ] with the usual topology i s the usual multiplication .for real numbers if the multiplic ation obeya the cancellation law, unit, an d every element of 0 ae ts as a z er o , [0, 1 ] has property � • 1 ac ts as a 29 EXample 3 . 2 with the usual has topology c ellation law. of real [0, 1] which is non-as soci ative and non-abelian, which as a zero , which has 0 that An example of a continuous multiplication on We define s as a ri ght unit, and which obeys the can 1 ab 2 wh ere the last multiplic ation is • aob numbers o We next consider the as sociative-like properties pos se ssed by compae t groupoid s with a zero and a unit which twe en and 0 � G be .3 . 10 . � is irreducibly law., cancellation and in b G [0, a ]b 0 � Now, x C < ( Ga)b Similarly, [0, ab ] a • • Let z 0 '- C [0, a ]b a (b G) ( ab) G nr . Thus (ab) G for all a € (0, a ]b z '5. ab Gx is a • !,! irred'ticibly xG a [O, x ] Lemma 3 . 6 � [01 a]b • We will next show that 'lhen z • xb by Corollary .3 . 2 by (Ga)b Lemma 3 . 6 where and z E. (Ga) b and be !. compac t groupoid wi th !. � � Gx • ., 'lhus, • [0, ab ] . G (ab) • = • 0 [0, x ] two sided ideal . and u and which the .--- sati sfies .. - -is ! � s i d e d ideal of G • (Gx) a • Also, [0, x]a xG ! compact groupoid connec ted between 2 !. unit C: [0, x ] � Gx by Corollary 3 . 2 , [O, ax ] Similarly � .3.12 " . If G !.!, which • by Lemma 3 ., 2 . [0, ab ] • G(ab) Le t G a( Ox) �mm� 3 . 2, and a (b G) and � irreduciblz connec ted between Proof . Gx . G (ab) C [0, a )b Henc e, cancellation law. • � and 0 by virtue of Lemma 3 ., 6 . G (ab) Lemma .3 . 11 . which (Ga) b [ 0, ab ] • [0, ab ] connec ted between 'lhen !!!2, ! � u �� which s ati s fie s � !. compac t groupoid with ! � • Proof. :J irreducibly connected be- and which obey the cancellation law. u T.Mma are 0 and is � u C [ 0, x ] • [0, xa ] a two sided ide al . !. � 2 !. � and which obeys the .2!!!,- o 30 (Gx) G cellation law � [0, x ) [0, u ) E:: z 0 � t z e [0, x] (Gx) G � u 3 .13 Proof . = , • � !!!' za e � z a � 0 X for all y � x z and - G (xG) Thus , Henc e, • J3y the above method , - • • and [0, x ] [0, u ] � G and x . by Corollary 3 . 2 zu e ( Gx) G • ... [0, x ] • • G i s ! compact groupoi d wi th ! � and ! 'lmi t � b] a [ O, • !.! 3 .14. ( GV) G then V !! G a all 0 and u � which ob e ys the � a and b in G • [0, ab ] • [0, a )b • (aG)b by Corollary 3 . 2, !. compact groupoid � ! .� and .! unit which G z € G(v G) C G(VG) (GV) G (GvG) a az .f; GVG ideal of . . C. . G � for • G (VG) a of If Proof. G (VG) xt co nn ec ted between a ( Gb) all sub s e ts e where (0, x ] 0 is _irreducibly c onne cted between z two-sided � � 3 . 2, and Lemma 3 . 6 . Lemma tor yt - a (Gb) • (aG)b law, = [O, x ] n by virtue of Lenuna 3 . 6 . G (xG) If � irreducibly o ellation [0, x ] [0, x] • shown that .. Lemma Lemma � yt � 0 z If • it i s easily which a is G(xG) w [0, x] [O, u ] Then z • Thus, • [0, x ) [ 0, u ] • (Gx) G We first note that Proor . Let and . x e o all G (x G) • If C GvG all and u (GV) G a and which satisfies the --cancella- - G (VG) is a two ideal of sided 0 • (Gv) G for so me z E ( GV) G , by Lennna 3 . 12 and (GV) G C G (VG) z GvG E GVG , C GVG a and - in G z € for . all • Thus a GVG for some € v V Si mi l arly • v V i n G by Lemma 3 . 12 . a ( ClV) G • G (VG) Hence , • is Henc e , • S�milal'ly1 a two sided • Lemma 3 . 15 . If G is ! c ompac t groupoid with ! � � ! � whi ch is irreduoibly c o nnec ted between 0 � u � which s ati sfie s � . c ancellation !!!' the n GV and VG � � sided ideals of G for all sub- 31 se ts V G of Proof . az E. a ( Gv za € GV • If C.. GV z C Qv C GV ) for all v� s ame method, € a € z , Gv for all G a GV and for some G in V o Henc e 3 .. ll o by Lemma S imi larly, is a two-sided ideal of � G i s a two-s ded ideal of .. € v G By o the • We next apply the above "as soci ative-like " propertie s to extend s ome of the -results given by and Numa.kara.. [12 ] Koch and Wallace [16 ] for mobs to compac t groupoids with a zero and a uni t which are irreducibly c o n nected b e ·ween 0 and De.fini ti o n 3. 7. G i de al of J (A) CG A If J (A) , let and which ooey the c ancellation law. , a groupoid, 0 • 'Ihe orem x E.. - � G 3.3. J(A) Sinee A Lemma 1) V !!!• �oof . • is Then , if If J (A) 'lhen x be A U YG U GV U E. V G C J (A) Theorem A' If • !. c ompac t groupoid A 'lhus • includes no i de al of J (A) C J (A) lhen 0 3 ol4 i� c ontained in be ! J (A) V c· A about J (A) • r/ 0 . Le t by Lemma Wallace x and Lemma c ompact groupoid with which i s irreducibly connected between c anc ellation law. Suppose 3 . t2 . [18� such that 3 .1$, thi s se t Therefore is open. G Let By Lennna o unit i s ope�. i s c ompact , we apply a lemma of C .A !. and which s atis fies the G U GXG � !_ !!.!:2. xG and hence 3.4. we are done . U Ox u J (A) i s open , , and � U GVG and 0 t o produee an open s e t 3 .14 a nd Lemma A 0 • open and is an ide al of x include s an , the null set . which is irreducibly connec ted between cancellation .A and be the union of all ideals c on tained in i s the large s t ide al contained in J (A) G , u \u • [0, u) 0 and !! u !. zer9 - � ! unit and which s ati sfies the maximal proper � �al £!:_ G • 32 ll c ontains :very proper i de al 2!_ Proor . · If a , 0 , th� � e sult is trivial o a two sided id eal of Thus, A C G\u G o a [ O, u) Simi larly, G If o [0 , u) a H{u) • u E G' u H (u) is • If [0, u) Thus, G , a • 0 u � H (u) Clearly, such that an 1 is S:. A o G be !. compact groupoid � ! � � !. � 0 � u i s any subgroup of and which obeY! the � � G c ontaining E. H (u) , there exi s t s an u 1 • Proof. i C:. [0, u) i s any proper ideal of A which !! irreducibly eonnected between If by The orem 3 o lo • Corollary 3 . 3 . cellation law. u) C: [0, u € xi • u H (u) o • If id e al by Theorem 3.4. If x €. Thus, x G '\ u 1 H (u) • u xi E. o G \u si nce IV CHAP TER APPLICATIONS Monothetic Oroupoids Definition h . l . exi s ts an E G a a J and A De finition 4. 2 o there exi s ts in e G a is dense a A group?id G a o c ompac t m n o the ti c subgroupoid Le t f., G a A. Since • A c�llection is c alled a generator G of G • c ontains a o b e the c ollection of all groupoid , is a G [ 2] . fi ni te Bwords " of all topological groupoid A there G is monothetic iff groupoid a iff c an be expres sed as c alled a gen erator of element �! c ompact The orem 4 . 1 . Proof . is topo� o gi c al The o to be cyclic every element of G s uch that the G in s aid is _ s u�h that finite 1tword11 in a _ T is a finite "Words" in groupoid, and A is dense in T . De�ni tion 4 • .3. CLn Let There • a:re lL.n1 , •n2 , • · a manner bm+1 ' • • • • • • for ·. , all • • a groupoid and , a nN ( n) �� • ' • • b1 • , • 'Ihus we ob tai n [ 8 .] . &nm C ) , n a1N ( 1. ) and • . • } in this section &nN{n) , to po ss ib le sequences ar e the • this lnu(n) a ' l. 1 (n+ ) , • • • independent of the CL , n the elemen ts of {bns n e I} • element , of · G letter a } the se { �.N ( i) , se qu en c e bm an Denote ltelabel • if are the terms a a in wor� s _ 'bm-tN (n+ l.) The theorems l.n2 • •n ' 2 that i f , G be {c?ll�ction or all n letter (2n - 2) s N �n} nl {n 1)! such words &n , 1 • , Let ip • • a:r.ry way by a2N ( ) , :a sequence in such then the terms a (n+1) N (n+�) " . way we assign i .e . , �n1 , the theorems are 4.2. Theorem �bn�!l � Then the Proot . e. there exis ts · n m E. � and 0 0 clusters e is an idempotent of A about is dense i Theorem ·, groupoid G property (2 � n :n � e, and e • bn • e 1 I e , there exi sts k for all {bn sn e Th n i.eo g for some 4 o .3 o !, n G k bk I Thus , o nece s sary � e 0 { r} € I about e • Then a and as sume I} � G �end-group, · P p of U Q and G {bt+J - oc) ·· 1•1 e • sufficient c ondition for ! monothetie G 2 .3 . Henc e • CL has property {an & n f, 1 } I i s a semi-group by Lemma all the G • 2.1. Clearly a is a € a � o !?.!. ! monothetie groupoid � .! unit � sup where P � Q � disj oint , closed , - � e ach c ontain infini tely manz finite words Q" Since • � � !El. G b7 • Let 1 G � 4 . 4o P be a generator of • large st inte ger a such clusters at Since P and G o Denote the unit of Q" are open, each c ontains ele- To establish the conclusion for there is p But this c ontradic ts the 0 bn: n c of a are unique by Lemma Proof . 1 > I - If the generator · 1s �� -mat· G generator 1 • 'lhe orem - I f, p k such that 0 e. clusters at mono thetic topologic al semi-group has a generator wi th property empty. k e , given any o a _ . to be a monothetic semi-group [11 1 is that its generator have - - a ;} ar - - - Proo:r. powers or G Otherwi se , there exists an open set � � G each idempot'ent €. r/ bt � that for any positive inte ger fac t that !!_ bm such tha� > k and any open set } generator From this it i s clear, if we are given any o N.�xt, . . we S'!ppo se • � be a monothetic groupoid I Suppose positive in te ger e G Let net in k Q such that o But, 'btt €. {bt+i}:.1 P • P , suppose '!hen the net clusters at 1 by Thus Q i s closed and we have a con tradie · T o establish the concl us io n for Q , suppose there exists a largest !heorem 4. 2 . tion. - .... -· k inte ger an� - s i nce 1 E. Q .. . such that clusters at 1 E. � Henc e • but the latter net is in Q {��-1 i s a net in {'b)cbtc+i}Cl0i• 1 cluste s at bk . The n the net • by' c ontinuity P E btc Thus • r s i nce P P Theor�� , J.t- . 5'. ia a closed ideal - - Proof, Q of Let -a of V (b) a �ere erl � t b j • e Q n E __ _ . I -· e of • to � � �me V(a) a�l p�sitive integers '} � n _ De � ni �ion of _ 0 . _ _, · Lemma . . Q1.£b1 ti impli e s 4.1. � A C: Proof' , j _€ I 4. 4!. If . n} CD Let . � n Q1 {bi �1 . . 2.. ·n} We f'!rs t assume C e b +ny n n} V(ab) a · V(a) But • for all V(b) bn+�E far V(ab) Q . is a left ideal oi' B a subse t is letter words with lett er s in • V (ab) Clearly, • !. c ompac t groupoid with ! 0 be Suppose • • • _Q1on g � only i! 00 bj Hence i s a groupoid and G }. 'bn+Dy and a G n z. V(a) V (b) Q [bi ai C\ {bi i � ij 'lhus deno te s the totality or .. Q. fbiai· � n} �en there exist neighborhoods ab E n and Similarly, it i s right ide al or 1fl ab C\. {bi•l n y- i s a pos i ti ve inte �r - ��·� Q1 {�a i and b b respectively such that and E we • ll G !J! ! c ompac t monothetic groupoid is an arbitrar.y nei ghb orhood and , · i s closed and hava a contradiction. P . A C 0 , ----bi l i 2: Cl, { n} B � and A • A , of G , • a � �.G. Then AA • A A • A! 36 CD n rl G n•1 � � Sinc e • is a two sided ideal, A 'lhen there exists ! i e c ompact and x £ 2, a•(/t :J G 2m x A A C Then k Let Since x • bk � n} a sueh that , en) "V(a n) !(V(c1} , E N m_r, • be a . No ,., c1 n x e, G {bn} T(c1) , £ br a W_�br1 1 .bra' c1, c2, G n • • • • • • , c1, , A Thus, • r1} � • n • • , en e X --;y {b i t i � o {bnt D since �� , n} • • • en , , en in G We denote Suppose o � I} • G such that bk ci E. • by {b ii 7 n} .. . b e an arbitra.I7 neighborhood or , , and V (e2 ) e • Therefore there exis ts c1, c 2 , e2 , n} We will first show. that o , {bi i ·� E � bn,.· n., , o • • , ,en) ) • • • AA • {bit i nl n• . and suppose k <:. where en C V{W{e1, ca, v(o n) ) • • •, • Thus, as asserted above no ments generator or Ci en) ) c2, 1 Hence • V{c1) , V(c 2 ),p , • lhen there exists neighborhoode of W (el, ea, C V imPli es A C Q {bi t i � A". Let x • bk • n•1 such that it may easil)" be CIO is a word w1 th letters "'V(W{c1, c 2 , 'br1 AA ,. . • • • na Hence G G i s !· compac t �thetie groupoid, then • . . ., aod n} � x � a n , there exist elements -· .. m G Sinc e • � A V have a contradiction and we is not a cluster point or W(e1, ca, Let "?- If S I� • {b1 t i x " A , . Proof . £ I ' n• 1 The orem 4 . 6 . 0 an C Hence, • [btli n A . n- 1 rf1 c AA E. x ar gument1 such that m . . Suppose o is closed, '\here exists an open se t A shown there exi s ts an inte ger X such that A B VV [18.) • By USe of an elementary net X ·c A AA • respectively such that • en) ) , , .P {bi t i � • n} bt bi where where and ct e., {bi a i � is a But there exi sts o , . , bn+ny f, V{ c i ) brn) • , t o o o1 � n br E.. n o V(c n) • Renee have a contradi cti on. we n} 1 Thus , each o r the ele • i < n • Therefore bk • bp ·· p � n where k. '::;> btc e Thus, �--�} {bi 81 � n} and we 37 have a c on tr a diction . Hence an C bt i i fpr all n and C r'\ an • � n1 As above, on C ;; b1 t i � n 1 n•1 '� n 1 h-1 � �--------� n•1 � � n} " Hence A C G , A • A , and AA • A 'lhus, ()1G0 • (l1 n• \.: ni mpli es A C ... by L emma 4 1 n.{bi li o · fb1 ti Q {b.-1-�-�--n}... i s a monotheti c semi-group, If G · 1 sided ideal of • Koch An Example h. l . s emi-group is is the minill"..a.l two In this special case, Theorems 4. 2, h.4, and 4 . 6 G [19 ] . have been proved by eye lis .. Q (bi : 1 � �} G ( [10 ] example and [111 �ection 3 ] ) of a finite cyclic groupoid which is not 1, 2, 3 a a wi th the following multiplication .. table . Where X may be of G generator 2, 1, � ,j iff and generator. i 2 1 2 .3 3 X 2 3 X 1 3 X X 1 or 3 �� ClearIT,. �t; and (12) 3 , 1 (23) 1 a is • An Example 4. 2 . &t • •j 1 • j • example of an infini te cyclic groupoid which is not Define the following multiplication& &t•i • &t+1 • Clearly (a1 a2) a3 , a1 (a2a3) and a ia j •1 = a1 if is a If we use the discrete topolo gy in Example 4.1 and Example 4. ? we obtain monotheti-c groupoids which - Example h • .3 . are . n_ot monothetie semi-groups . Consider the interval [0, 1 ] of real numbers under the ueual topology mu�tipli c ati on on the tl?n'� . �e !J. !��� - is not potent then _ - • • .. A _ J � We e, . a the following multiplications and I} right is the usual note that 1� C 0 A ... �- S: . Q{bi ti '?- n} ple 4.�. C onsid er the l!Xam They form a {bna n • t, •· t e !eme�t �r G l , · and AA • A 1 • integers with the usual topology 1mder a gen erat or G be Let a of this A groupoid . Zero topologic al groupoid with a zero and such that an � • monothetic topological groupoid which is not a . One is De.fini tion 4 . S . an a Let vi th generator I} , A £ Topological Groupoi d s With be ab 2 where �e • real multiplication. is a compact mono theti c gr oupoid topological s emi - gr oup a b monothetic semi -group and which has zero . as its only idem- �- -� �ubtra�.�ion . ao is unique .for all n ! If . an � 0 , i . e . , if for e'!e:ry neighborhood U of 0 , there exists a positive inte ger � • �(U, a) such that an € U for all n � m , then a i s termed a nilpotent element or �fini tion 4.6. groupoid G with consists e.ntirely or G is a sh�t in a ni lpotent. !. !!!,! left (ript, � sided) !!!!!.! or a z ero is a {ri ght, left !!!,! semi-� A semi-group (sub-groupoid) of 0 of nilpotent potent e lem ents . Lelllfta h. 2o � {A� a X E. /\} sided) nil ideals or ! topologic al sided) ideal of two elements , _ .fam:i lz � whi ch {sub-grou.poid ) of nil- � riglit (lett, � ! �· Then A • hl A ). a right (left, two-s id e d) nil id eal of o . r--' - -Theorem h. 7o If G i s � elementwise bi c ompact sroupoid with ie- also -- G consisting entirely be !. gr oupoi d a t opolo gic � !!:_21 - �very - ....._._ - -- right (left, �-sided) � - A of ! G is either ! nil id e al tains � con id-.potente prol'ided �-.!!!:! 3: tL : · Prt:10t. Let A be a o.. in A �� �;�e� t .. a r/ 39 every element � non-nil right ide al of that an .,L+- 0 si nc e G • tar: A h!! pro:per- -:then there all a E. A is , an the powers . .t>! a are unique by Lem.ma 2 . ) . N ow, it we consider subsets \·� {&n�.Jl e:, l} , !y • {ai i � v} ( V • 1., 2, 3, : ) , and D /'4 B; ! .?'. is a sub group of a and BD c.;;. D b7 7heorem 2 .k. Let be the identity or n Ir e • ·o D (0) and an--+- o � . But., �is !�n�adicts the as81211lpti that . an 11 ,.... 0 '!hue e is a non-sero . • ideapotent. Since BD c.=. D , ae e. D .. and.. denote by (ae). 1 the in• ..... . nree ot ae in 'D Since A i a a right ide al, e ·• (ae) (ae) -1 E . J. H�me A It i a the s ame vi th the . e�tain�.. a �on-zero idempotent e or �he lett ideal. � 1heorem· 4.8. ! closed subgronpoid CJs. � !! elementwise bioomj>ac t voupoid � ! .!!!:2 !! ei tb:er ! � subgroup id � contains .�-.!.!::! idem pot4inti j;ro"Vided each element !! � has propertj � : ProOf. nte existence .�t· · the non-tter o i �empotent e i s established as in the �o� C?t Theoren 4. �. . On� t�en n o te s that e C D C · tf� 01 De�nition k . 7 � topolog1cal . groupoid wi th a zero i s said to be . an N •groupoid it 1 te nilpotent el e�n �s form an open set. L- k . J. � G � !. topological gro poi d � !. � � !.!.! a � .!!!. el nt or � � p�pertz a . !! an !! ! nilpotent elem�tlt. :rc,r· iome poai tiye integer n , theh !' itaelt !!_ !. nilpotent eleJiel'lt. Proot. All the powers ot a are unique b)" Lemma 2 .3. Let b aD and let tJ be an bi trary neighborhood ot 0 ; !hen there exist n•igh-'borhoods Vi (1 • l, !, , n - ) ot zero such that W1a1 C lJ 1 t ·• •_. . • • _ ' e , • • on 1 • we I .. .. I•-' ,. ,, • .. • '"' • "' . • . c ase o _ • . •.. . u eme . ' • • ...J ar• • � . • • • •• . • (i • 1, 2, !\ (i • • • - 1) , n 1, 2, = • 40 , , n • Now, sinee b .,.. � 0 o such tha t 1) - M* Let • If � � M . 1 we vri te k � _., � 0 • I we + i have e • a G C: 0 • 1 If · Ah �-lA • • • •_, 1heorem 4.9. o . o' Mn_-1) 0 :=. i � n aU m akn Wi ai M and let b� • C U e u • 6 etc o 1 G 2! !. topologic al groupoid with ! 0 in Let !,.! G !!.!! • !, nei ghborhood is an element �f N , there exists a positive inte ger p� E V0 a U(p) . of p q an ��- 4.10. ·The orem An element b Proof. G and p Let , u (p) E. .. x(yb) of a C N • that there a • If nilpotent. Hence, • G a topological • groupoid for all is N such that C: V0 q"fl of V0 x and G is said to be G y �n [14] . 'l'he radical 2,! .! bioom.paet N-groupoid !! � E!:2. vided each ,nilpotent element € E: -v0 qD a zero i s called the r �dieal or G right a ssoe i ati ve iff (:x;y)b x� � V(pn) 1he j oin o f all left nil ideals of 'Defi niti �n 4.10. any U (p) n · Definition 4 . 9 � w1 th = " 4� .3 and s _ a. nilpotent by Lemma N-gr�upoi�. groupoid � - _ _ /'-. such that any element of U(p) , then cf is n Recallin g the abo-ve convention, one can e asily see - neighborhood 1hua�. is , 0 , we which consi sts en �ire!z 2.f nilpotents, �en G is � N-groupoid. _ Proof . We denote the set of all nilpotent elements of G by • • is p q· 1) a is a nilpotent . If is + Then If i • Henc e • n (M* = 1 [18 ]. - Wi Mo implies Y � 1 which every element has property (l., . impli es b ¥" € 1 � �A 0 Mi. a topological groupoid and " � "'AA , we shall use the following notation t A • A A Definition 4.8 . A kn • akn+i aknai bkai • aU have i - If' • max(Mo, M1 , = there exist inte gers Mo such that and there exists a positive illteger bv � U � "V :J R ·x�a !.!. right assoeiative o be the radical of C R C N , where G N � and let a € R o Then far i s the set or all nilpo te nt G elements of �d e�l The above re lati on is valid since • ��re exi sts a ne i ghb orhood x� f_or each �d -� Vl\(a) €. 1 t • GJ 1, 2, v,� GV(a) C U V(a) • k . . .. G , . of , right associative . Thus, · Example h..S. on [0, l ] s a o b • ab G is of x:A :tJ.vCk""-) , ·1 .. 1 , 2, ·v(a) o o ., k . C · • G • QvA1 (a) 'lhus • Let 2 t * C R � md .. - u a, b E. is_ �ormed in wh i ch But, · Define the following multipli c ation - �ot � N-gr �� oid . is open. be the close d unit interval of real [0, 1 ] for all C [0, 1] where the multiplication on the right is the usual multiplication of . real numbers . .. groupoi d G N Sin c e G is • so that such th�t a V (a) e., a numbers under the usual topology. • C C i-(V.Cx�1a) C N • If we put L* • GV ( a) . i s . a left nil i de al since e ach nilpotent eleme�t is * t then. V(xA ) C V(x� a) a} & (x" ) IX,_ 'V(XA 1) Mv(�� i ) 'V(a� T(xAa) one can .f'lnd a f'inite sube ollee ti on k {v corresponds tok 1_ (k) such that V(x)\)V"(a) neighborhood of a . } _ i s open by assumpti on, find peighborhoods ean such that U V(x�) X�tG � �e den o� by V ( a) where a N x).. a of -v�xAa) Hence, one • of N bi c omp ac t and {Ycx>.i) 4.2 . Si nce G by �rt�e or Lemma or is a left nil R 0 it A topological is the only n:i �po tent element . we consider [0, 1 ] with the He nc e , discrete topology and th ! ab ove multiplic ation, we form a topo lo gi c al groupoid G' which i s a_� N-groupoid. We next c on s i der bri etl:r the concept of boundedness in topolo gic al .. .. groupoids w1 th a zero . Derlniti- on 4 . .11. A subse t B of a top ologi c al groupoid with a . zero i � ri ght bounded iff tor 8!JY neighborho�d U of � there exis ts a neighborhood .. V or :1.�17 �etined, and bounded. a 0 such that ·- · VB C U • tert boundedness i s simi- s e t is bounded iff it is ri ght bounded and left ! bicompact subset � Theorem 4.11. . h2 � ! topological groupoid ! ·!,!!! is bounded . Proof . B Let be a bieompae t subset of bitr&l")" neighborhood or Since B Let • W (O) Hence 4Cx) i s bican:pae t and tion {vC�_) : ki u 1, 2, Qwx1 (0) W(O)B C: U(O) b ounde d . We finally Si nc e • O:x: :x: £ B k} • • • , "! or Wx1(o) where B and .:J is 0 • of x and Wx (O) V(x) nei ghborhoods 0 G for or 0 x U(O) Let • € B be an ar- there exi s t , such that Wx (O)V (:x:) C U(O) . , there exi sts a finite subcolleo- B {v(x) :x } t. B such Simi larly, ri ght bounded . c on sid er the special c ase of a :k{v<xt> � that V(xi) corresponds to k , i B 1, 2, • B. • • •, k is left bic omp ac t groupoid with a zero and a unit which is irreducibly connected b e twee n 0 u and and which obeys the c ancellation law. Theorem 4 .12. which � !. bicompact groupoi d with !. � � !. � !.!!, irr e ducibly connected between 0 � u � � obeys _!:!!! cancel• lat:ion law� '!hen --- - - [0! x) (1)' ·- i s ! nil i de al of prope rty (�) � ... (3) property virtue of ""' �im:l1arly, � ideal (1) a � Let .. G J e. z 1heorem 3 . 1 and [01 x) � provide d !;!:! y r/ X � G provi de d !!!._ 4 U E. (O,x) have .. - C [O, x) and By Lemma 3 . 10 [0, x) Corollary . E.. (0, x] have o • 3 . 2o for all [01 x) y • � radic al � G !.!._ open sided ideal of (2) G a. [ o, x ] !!!.. !. Proof . .. G Let If a .j 0 , If a • a € G o is a nil 0 , C [O,x) If by this result is tri vi al . Thus , [0, x) i deal by [ 0, x] i s an ideal. a[O,x) is a C oroll ar y 3 . 2 • x .j u , [0, x] two- is a • 43 nil ideal by Corollary 3 . 2 . Le t (3) potents of such that L* • 1 a e R and C N Hence, t* and G one may find , GV(a) is U V(a) GV(a) Lemma 3 . 1.4 . __ ....�e If • T{a) 1 - . '!hen G be the radic al of R nei ghborhood • a of V(a) a as in the proof of 'lheorem h .lo. C N and be the set of nil N left nil i deal of a C R a let by virtue of G e, Y(a) C R note that i f every element of G Thus , • has property CZ R , is open. i s an G N-semigroup . Definition 4 . 12 . A topological groupoid ha� arbitrarily small ideal nei ghborhoods of- U 0 c.. ontain s -· - an � G wi th 0 iff a z ero i s said every neighborhood ide al which is a nei ghborhood of 0 '!his i s • Chapter 3 ] analogous to the notion of arbitrarily small subgroups [ 1.3, . . _ . which _ . .!! o. .. .. . If !!_!, Proof . __ then two e le� ts !h� ax • b a and groupoid A a!_ld exactly o�e such that ax • Definition 4 . 15. iff tor any two elements , � y a b and . � G b [7 , p. A groupoid a and G obep � �- is called a quasi group iff for any there exists exactly one A groupoid irr for any two elements G and Quasi Groups b in G . E. 0 .!.1!!. u � ! .!!!:2 � .! � result follows immedi ately from Lemma ) . 10 . , , D�finition 4.14. x i s !. compact voupoi d • arbitrarily smi.ll bie ompaet ideal neigb.borhooda 0 !!.!!, .Definition 4 . 1.3 � that G_ irre�c ibly c onnected be tween cellation or I 'lheorem 4 . 13 . _ to b s uch that 0 i s � alled in G , ya a in i s � alled. G , b E 0 such [ 7, p . 986 ] . � � quasi � there exists exactly one 986 ] . 0 • :X: a right � guasi group there exists exactly one e y G b ya • such that Definition h. 16 . ax lation law i ff . 44 [ 7] . G A groupoid • ay imp lies X • said is X y for all G satisries the right cancellation .!!! iff xa s �ti sfy to y , the � and a S 1 • ya implies x u cancel G J and y for all G s at i s fi e s the c ancellation law iff it satisfies x. , y ,.- and a � G the ri ght cancellati on law and the left cancellation law. o We now state a well lm own theorem from the theory of quasi groups • . !_ finite cancellatio�_ groupoid is a quasi_ group • [ 7] 1'heorem 4 . 1.4. __ Under certain additional hypotheses, we will extend the the or em to bi- compact groupoids . � G be Lemma 4.4. �ubset £! G uJ b �d W} an� B (�) be • !!!1 W • ! topologlcal groupoid [bc&)8bw • !!!. index G � system !!!.<! w � B* be !. bic omp act .!!!!!. � e, � � _e., r � Then, there exists b c {awtwt w� • a • Let At • • of • B Let family or subse ts of B* B t1 , • . • �. �t2 .' �tn are borh ood n e � gh Let • • • Ato • {vt(a) & t Vt (a) n the set of elements of elements of At • /\. Let Proof. of _w} €1 A • . � €- T} Clearly, whose elements B .. {Bt g t E, wi th the fi nite correspondin g Vt0 (a) Vt 0 (a) (l resp onds to Ate, • (€ /\ , �t ,; D have the T} T) <2..- � � By • Bt , we denote same indices with tho se We will show that o o B o is B a For let At 1 !h en 1 At 2 , Hence, there e4l_sts a such that Vt0(a) C Qvtt (a) deno te by Bt0 the subset of B which n _ (\ Then, 1 t is clear th at q 1 Bt0 C 1•1Bti A C B* B suppo se intersection property. subse ts of A t0 E... f., G be a complete nei ghborhood system , B tn be any finite number of sets in the We • � � ab lJ w � subsets � G . w} whose elements correspond � � � � system W {a.wi a A • and· • cor- • 4S B Thus has t� -/ we have 0 neighborhood of ab o Y(b) such or b b ·e Since Q� C2, property n hand quasi n Proof . CD . : -/ r•1 . 0 E, I If p . • Q� CD • for p • o { r•1 • qG As • r • 1, 21 • A " • element G !!!!. is .! · -- p for all • i, piG 'lhua, E, o . o } � r o Le e., G :J p 2Q ::J 'lhen, since o � qx(x e ' r • 1, 2, , let qx ;T • e: 00 G) • p9 o • Th en • But, (\ �G r•1 C qO G o Let there is a g Hence , G1 € let p' (b € a ce... show that and and C\,PiG • {gi i e. I} lfr p 1 • qg N ext , we will pG) Let o pr+1 ra . f'\ prG r• 1 • or one can find But , pr+irx E qx €. q {pO) (x E • any eleme nt be €. � Lennna 4 o 41 • f"'\p�Q • qG i•i o Qpi a where is bieompac t a n on-ne gative inte ger . _ show that Qpfa CZ) - G is • • 0 'nlerefore , o ., . I} ir where 0 0 . o o By � • :J pG G that V (q) :i: C V (qx) such 1, 2., 2, that arbi trary neighborhood of qx we will E.t b e fo re , V(b) Hen c e • � We intend to sh ow that • (:l q of Next i (pllo) • (ppn) G pn{pG) {� ii _, qG C I.} {fcta i pi z i C • pig1 , qg E q·{pO) r . T) . 0 " follows it () P; be an for n ia _ i• 1 '!hen p ' �d · P �. that G 1 for Q:) -� qG is !. bi c ompac t · groupoid !!! which every q e, ( Bt · n n Vk (a) A a e.. k 1 € G Put P r • V(qx) € V{q) {a Ak 'bt o and €. a neighborhood V q ) ::> we le t and • • Le t • , and let tr�- ako € th at a ( E /\ where Vk (a) N ow, if (I ab e, � of neighborhoods Vk(a) c group provlded p First, we show · · qG . plla :J (\ J5; 0 be an arbitrary V(ab) �- vhich sati sfie!_ the left c anc ell atio n l aw, !!!! f or_ !11 ::> -/ , le t md C V( ab ) Vk (a) V(b) . Lemma 4.So . !! _ _ ,!!!:!. Th e n there exi s t that •ko ' bto e. -v(ab ) � Q� n b -v(b) n BJc . an element •ko su�h there exi s ts . Let o i s bicompact B* the finite intersection property, and since C G such qG �d r.lpi (pa) a> Y(qx) � be . an • qx arbitrary nei ghborhood of such that V(q) x ot q r 1, 2, • pr+i:l!'x ntus E. • • • rirr (pG) pn (pG ) . • we show that P' • , i {!i Clpi (pa) • 1, 2, pn+lQ • �+ir+la C • 'lhus, • Q_JI' (pO) C\.pl'(pO) qx _e pi gi ' C V(qx) S i�c e • one can find a Then, o and for r p1 e: pG exists g ' e o • 2, 1, a.pi(pG) q(pG) C gi ' , where pr (pG) Let for V(qx) positive inte gers . • neighborhood V(q) E pr+irx for all pr+lo C q(pO) , • • • 46 • n , o . . • Finally, Q.p1(pa) Then !Jet I} • {p1 i • P f. :c ¥ c per pG such e I} '.nlen there that {i11.1. t i e_ I} " ;.- p 1 :thus, p 1 • q(pG) But, Qrcpa) Henc e , Qpi (pa) • q(pa) �d Qpi (pa � C q (pa) Qpi+lo Qpia Thus, q(pG) qG and pG by virtue or the �d Qt .• qg 1 :i I 0 E • 1 ... E Ill qg1 • • • • there exiets x • has property G � lation law, this Lemma • • lett cancellation law . --...&. 'lhus, if such that G ie b ax • G and a --- 1 a left hand quasi:.group. every element the ri gh t c ancellation law, the n ---- group 'f!ovide d p €. G � � !!!, n � I . �. "!he proof of thi s G b in B y virtue of the left cancel- • is any G !:. bicompact groupoid � � Q, and which s atisfie s !!_ !_ right � quasi • we are given x i s unique and 4.6. !_! € � A A (QpD) p • G (p llp) lemma is an al - . • ogous to A ( Op) pD � G � !!!_ the proof of the previ ous lenma. Theorem !l!!. pr operty 4 . 15. fZ ,!lli!. which gua�� group pr ovi de d • (pp'h) G for Proof. If G !!_ !. bic ompac t all s ati s fi e s � (Gpn) p A p € G m /\ ( Op) pD • and for all groupoid !!! � every element c anc e ll ati o n law, G (ppD) A n � I and � G !! A /\ n p (pDG) - p (pG) !. • '!'his result follows immediately .f'rom Lemma 4 o .$ and Lemma 4 o 6 o Example h o 6 . �� er thi � 47 Consider the following multiplic ation table multiplication 1 1 1 3 2 3 2 3 2 G ) 1 1 {1, • 2 3} 2, is a finite groupoid which satl s fi�e the � an� ellation law, in which every element has proper vhich £ p (Op�)p G Example Define � a_ (CJp)� • . M.d all n 4. 7. O�p;n) a e. I Le t E not a . ' semi -gr-oup Ex�mple ... . .. 4.8 . Let C 12 C · z E (xx • i� the d� �ter througn 4.9. Let P relative topologyo 1' and that the line x) � x • a x o y o • topology. midpoint or the segment r y be the parabola On P, yz • z_ if z_ ie with the �r C the · �ge becomes a emi-group although it is a quasi- � oup . .� define (7 x o y • z • x2) in E 2 a�thougn it is wi th the by the demand that is paraliel to the tangent at P the x Under this multiplic ation Under this multiplication whieh is not a semi-group For usual becomes a topologic al groupoid which is define topological � ou� id which is not usual semi-group. is not" a by defining ·and in £or all be the circumference of a circle in On __ Example (p�) G (l, , •Ithough it ie a quasi group . ueual relative topolo gy . y • be the Euclidean plane vith the Under this multiplic ati on • ot 2 E2 �ultiplio �tion on Q But, • p(p1Ja)• .;n(pG) and tY x ( set becomes a topologic al groupoid a quasi group . algebraic counterparts of :these example s, see [2�, p. 22h·] . BIBLIOGRAPHY BIBLIO CRAPHY . 1. •. \2. 3. Albert, A . A . "Quasi �oups 1 I ". Trans actions Ameri c an Mathema tic al 54 {1943) , PP • 507-519 • .. _ S o c iety, Vol . Etherin gton, I . M. H. 11Groupoids w:l th addit i ve endomorphisms 1 " Americ an Mathematical Monthly, Vol. 65 (1958) 1 pp . 596-601 . Fauc e tt, W . Proceedin idempotents . " Vol . 6. (1955) pp. h. . Frink, 5. Oarrieon1 6. o. matical pp . . 9. • eelf-di stributi ve � 62' (1955) , G. N. WQuaai- groups . " 4 74-487. . Haus:ni�n, - B n al of Jac obson, N . N ostrand, Kelley, J. 10 . K och, R : 11. Koc h , R . J; sity, . _ .A. . ·and·· o . Ore . , _ J. Americ an Mathe Ma themati c s , Vol . 41 (1940) , Elementary Topolo gy• 1951. York t John Americ an J our N�A . "'1heoey o f Quasi-group s , " pp . Lectures in Abstract Ali!bra, Inc . , 983-1004 . I. New York ! D . V an - Ge n9ral T opology. L. 1955 . Annals of Mathem ati c e yol . 59 (1937) , • eyste:m8 , •� 69 7-707 o pp . . Hall . ... D . W. and o . · L-. Spencer . and Sons , Inc . , 195$. _ a. 141-14 "Stimmetric and Monthly, Vol � 7. r. two of the Americ an Mathematical Society, "Compact s emi groups irreducibly c onne cted be tween M. "On topol o gi c al 195!. New York s D . V an N ostrand, Inc . , semi groups . " Dissertati on, Tulane Um ver •on monothetic semigrOu.ps . w P roc ee din gs ··of the Ame ri c an Mathematic al Society, "Vol . 8 , J�57) , · pp • ., 397-401 . 12 . Koeh, R. . 13 . Montgomery, D � and--L . Zippin. Iopologieal 'l'ransfonnati on Groups o . Inte rs eie nce Publi shers, Inc o , 1955 . York 3 J . and A . D . Wallace , "Maxi mal i deals in compac t semi groups . • Duke �1athematical Jour nal , Vol. 21 (1954) , pp . 681-685. lh. Pios tert,- -1' . s . and A . L . Shi elda . •On a c omp act mani fold wi th boundar.y. " (1957) , pp . �17-�3 . lS . Nnmakara, 16 . lhnnukar a; K. Yumagata the structure of s emi groups on Annals of Mathematics, Vol . 65 b!collpact semigi-oupa ." X. "'n Okayama Uni versi t;r, Vol . l (19 52) , pp . Mathemat:t e al JOU.rnal 99-ioB. "11Qn bic omp aet send. groups with a - zero . Uni-versity, N atural S ci ence ) , Vol o 4 ( New . " o:r -� Bulletin of (1951) , pp . 46S-4lf. 17. Rees, D . " On eemi gr oup s o " Proc ee dings of sophieal Society Vol. 'J6 (I94o), pp . � t�e Cambridge 398-460. Philoao- . _ 16 . Wallaee , - A . D. 'ItA note on aob s, I . " Anais da Ac ademia Bruileira da Ci enci aa, Vol . 24 (1952) , PP o 329-J34 o 19 . Wallace , A. - D o •A note on mobs 1 II o • Anaie da Ac ademia Brasileira da Cieneias , Vol . 25 (195�) , pp . 335=336. 20 � Wallaee , A . D. An Outline for Algebraic Topologz, I . a i ty, 1952 . n.. Wilder, R. L. 2�. 2�3§. : of Mani folds . • '!he Americ an Mathematic al Public ations, Vol . lillY o New Yorka (1949) 1 "Topology Soci� Collpquium pp . Tulane Univer . .. . .: Stain, S . K . "On the foundations of quasi- groups . " Tranaat:.ti.ons of the American Mathemati cal Sooie ty, Vol o 8 5, (19513) , 228-256.