Totally Ordered j9-Class Decompositions

8/14/2019 Totally Ordered j9-Class Decompositions

1/7

TOTALLY OR DE RE D J9-CLASS DE CO M PO SITIO NS

B Y K . H . H O F M A N N A N D P. S . M O S T E R T1

Communicated by E. Hewitt, June 8, 1964

Introduction. A n irreducible semigroup i s a compact connected semi

g roup wi th iden t i ty wh ich con ta ins no p roper compac t connec tedsubsemig roup con ta in ing the iden t i ty and mee t ing the min imal idea l .S ince every compact connected semigroup with ident i ty has an i rreducib le subsemigroup jo in ing i ts ident i ty to i ts min imal ideal , thes t ud y of the categ ory of i r reducib le semigrou ps is bas ic in an y que s tfo r in fo rmat ion abou t the s t ruc tu re o f compac t semig roups wi thiden t i ty. In a p rev ious ann ou nce m en t [4 ] , the au th o r s desc ribed thenotion of a hormos and conjectured that an i r reducib le semigroup isan i r reducib le hormos. (The s t ructure of the la t ter was complete ly

descr ibedwithou t p roo f sin tha t announcemen t . ) We a l so announced a number of categories of semigroups for which we are ableto ob ta in th is resu l t .

One d is t inguish ing feature of an i r reducib le hormos is that theD-class (and in fact even the -c l ass ) de com posi t ion is a to ta l ly orderedsemig roup in fac t an / - semig ro up [6 ] . I t ma y be con jec tu red th a t as t ud y of sem igroup s whose P-class deco mp osi t ion is to ta l ly orderedmight lead to a description of all ir reducible semigroups, and i t is thismotive which br ings us to the resu l ts o f th is announcement .

We shall give a br ief outl ine of the techniques we use to prove ourresu l ts . T he deta i ls wi ll ap pe ar in our for thcom ing book. F or term inolog y, the rea de r is referred t o[ l ] and [4] ,

1. Admiss ib le spaces . The following concept is of mere technicalinterest but i t is needed for the pr incipal results of this paper. Aspace X will be called admissible i f every compact subspace Y ofX for which there is a to ta l ly ordered compact connected spaceTand funct ion

FiYXTX F - > Ywith th e fo l lowing prop er t ies is necessar i ly a po in t :

(a) F(F(x, t, y), s , y) = F(x, tAs, y)for all #, yz Y, t, s:Twhere/ A s = m in [t, s} ;

1 This work was partially supported by NSF Grant GP 1877.

765


2/7

76 6 K. H. HOFMANN AND P. S. MOSTERT [November

(b) F(F(x , t, y), t, z) = F(x, t, z)for all x, y, zEY, t&T;(c ) F(x, 1, y)=x, F(x, 0, y)=y for all x, yGY, wh ere 0 and 1 de

no te the min imum and max imum e lemen ts o fT, r espec t ive ly ;(d) for each j G Y, F( -, -, y) is con t inuou s .

P R O P O S I T I O N 1. If X is a finite-dimensional compact space, then X isadmissible.2

The proof of th is resu l t uses the Vietor is mapping theorem on a

carefu l ly constructed subspace.T he re is an exam ple , due to Jo hn S ta l l ings , of a n inf in i te-d imen

sional nonadmiss ib le compact connected space.

2 . Clifford semigroups. A Clifford semigroup is a semigroup whichis a un ion of groups . The a lgebraic s t ructure has been to a large exte n t de ter m ine d by Clifford [2]. (See also [l , p. 122 ].) L etS be a compa ct Clifford sem igroup . ( I t i s no t assum ed th at 5 h as an id ent i ty . )I t i s known that the D-class decomposi t ion , which we denote byT, is

a semila t t ice in th is case , and that the natura l p ro ject ionTT: S>Tisa homomorph ism. I f we assumeT i s to ta l ly ordered by the natura lordering of the D-classes (i.e. ,D(x)t=*D(y) if SxS(ZSyS),* t h en T isa to ta l ly o rdered sem i la t ti ce . W e den o te by 0 the min imal e lemen tand 1 the m ax ima l e lemen t ofT. We sha l l say tha t S h as connectinghomomorphisms if for each "gap" [a, b] in T ( if the open interval]a, b[ i s em p ty, [a, b] is called a gap), the re is a hom om orph ism

: ir~l(b)>w~l(a ) such that a l l idempotents of7r~ l(a ) which are belowan idempo ten t o f7r~~l(b ) are in the image under / . (No t ice in par t icu

lar that if T i s connected , 5 t r iv ia l ly has connect ing homomorphisms. )

P R O P O S I T I O N 2. Let S be a compact Clifford semigroup whose D-classdecomposition is totally ordered and which has connecting homom orphisms. If the set E of idempotents is admissible, then there is a homomorphism

^ T T - K I ) XS/D-+S

with the following properties:

(i) \p(a, \)=a for each a(E.ir~l(\)\(ii) the following diagram comm utes

2 The dimension we use is the Cech cohomology dimension [3].3 In the semigroups we consider, it is the case that{%} V JS x ^Jx S d S x S . Hence,

the D- ( = /- ) class decomposition is determ ined byx=yD if and only if SxSSyS.


3/7

i 9 6 4 ] TOTALLY OR DE RED D-CLASS DECOM POSITIONS 76 7

T T- ^ I ) XS/D >S

\ /S/D

where TT ' is the natura l projection. In genera l, >p is not surjective.

F r o m th e f ac t t h a t E i s admiss ib le , one is enabled to prove thatth r o u g h each id em p o ten t eiS, there is one and only one to ta l ly

o rdered compac t semig roupTe such that 7r |Te i s an isomorphism on to[0 , ir(e)]. This then leads to the proposit ion af ter a ser ies of more orless t echn ica l though no t pa r t icu la r ly deep lemm as . Never the less ,we would rank the proof of this result and i ts application to the proofof Th eo rem 1 as a bre ak thr ou gh of non tr iv ia l p ropo r t ions . In p articular, i t sheds a great deal of l ight on the location of Koch's arcs4

in semigroups .

T H E O R E M 1. Let S be a compa ct C lifford semigroup with identitysuch that S/D is connected and suppose that the set of idempotents isadmissible. Then every I-semigroup containing the identity is in thecentralizer of the maxima l subgroup of the identity. At least one suchexists joining the identity to the minimal ideal, and no two such determine the same collection of D-classes.

R E M A R K . In pa rtic ula r, if 5 is a co m pa ct Clifford sem igrou p sucht h a t S/D i s con nected an d to t a l ly ordered , and such th a t the se t ofidem po te n ts is admiss ib le , then , th rou gh each idem po ten t of themaximal D-class , there is p recise ly one Koch 's arc . This is no longer

t rue in the absence of admiss ib i l i ty.3 . Semigroups over "gaps." Again in our s tudy of un i ta l semi

groups whose .D-class decomposit ion is totally ordered, we are forcedto consider semigroups without ident i ty. This t ime they ar ise f romlooking a t a "gap" between regular D-classes in the order ing , andhere we are ab le to ob ta in much bet ter in format ion . However, because of our inabi l i ty to handle the s i tuat ion where an arc of regularD-classes occurs, we are unable to apply the full power of this theorem. (Recal l that a regular D-class is one which contains an idem-p o ten t [ l ] . )

Le t 5 be a compac t semig roup wi th iden t i ty whose D-c lass decomposi t ion is to ta l ly ordered . LetX an d Y be compac t spaces and[ , ] : YXX-~>S be a con t inuo us funct ion . Sup pose, fur ther, th a t

4 A Koch's arc is an i d em p o te n t / - sem ig ro u p . I t was R . J . Ko ch [5 ] wh o in t ro d u ce dth e t ech n iq u e o f o b t a in in g su ch semig ro u p s i n co mp ac t semig ro u p s .


4/7

7 6 8 K. H. HOFMANN AND P. S. MOSTERT [November

the re a re d i s t ingu ished po in tsazX, a n d J G F su ch t h a t [b, a] I.T h en th e Rees p r o d u c t [S ] R== [X, S, F] with respect to the sandwich func tion [ , ] an d m ultip licatio n (x, s, y)(xr, s', y') (x, s[y, x']s', y) ad m i t s a h o m o m o r p h i s m : S-+R, and an order-p rese rv ing one-one map': S/D>R/D such that the following diag r a m c o m m u t e s :

7f I , 1 7r'S/D-+R/D.

T h e m o r p h i s m S, such that(i) Sl^S-\}{D{x)^Dl}C{[X,^, F ] ) ,(ii) if TTI'. S>S/i, 7r: S>S/D are the natural projections, then there

is an order-preserving map m:

^L/H^S/D such that the diagram

[X , 2 , F ] - > s

i (*, s, 3>)->7riO) I T

2/H S/Dm

comm utes, providing D(e) and D(l) are sub semigroups*Further, if Dx is a subsemigroup, then these are equivalent:

(a) ([X, 2 , F ] ) is a semigroup and w\im


5/7

i 9 6 4 TOTALLY ORD ERED D-CLASS DECOMPOSITIONS 76 9

The proof of this theorem follows from a sequence of rather technical , bu t no t par t icu lar ly deep , lemmas once we have the fo l lowingpowerfu l resu l t :

T H E O R E M 2. Let S be a compa ct semigroup with identity and maxim alsubgroup H. Suppose that there is a neighborhood of the identity withoutidempotents other than 1. If the orbit space of HXH acting on S under((h, k), y)>hyk~1 is an arc in some neighborhood of H-1 -H H , thenthere is a one-parameter semigroup : [0 , )>S such that f(t)(E.H if

and only if t Q, and such that the image of f is in the centralizer of H(i.e., f(t)h = hf(t) for allt< E [0, ) and hEH).

Th is theo rem is ac tua l ly con ta ined in Theo rem E wh ich weannounced in [4] . I ts p roof uses the s t ructure theory of compactgroups and their homological p roper t ies , the construct ion theory forone-paramete r semig roups , and an invo lved theo ry o f pe r iphera l i tywhich the authors developed for this purpose. Even a br ief outl ine ofits proof could not be given in the short space of a few pages.

4 . The main theorem. Since the s ta t ing of the main theorem for alarger c lass of semigroups than those with ident i ty requires l i t t leadditional effort, we give it in the following form. Because it is ar a ther comple te s t ruc tu re theo rem, the conc lus ions become a b i t involved. The reader may glance at the l is t of corollar ies, al l of whichfollow with almost no effor t f rom the main theorem, to get some ideaof its power.

M A I N T H E O R E M . Let S be a compact semigroup satisfying the following conditions:

(a) 52 = 5 ;(b) the union of all subgroups of S forms a semigroup;(c ) S/D is connected and totally ordered relative to the natural order

D(x)^D(y)ifSxSCSyS;(d) in each D-class, the components of the set of idempotents are ad

missible spaces;then the following properties hold for S:

A . The relation D is a congruence, the D-class projectionx : S-+S/D

is a homomorphism, and S/D is an I-semigroup.B . There are sub semigroup s Sr and Si such that(i) T(Si)=w(Sr) = S/D;(ii) if a2~azS/D, then 7r~1(a)r\Si (resp. 7T^1(a)r\Sr) is a direct

product of a group and a right- (resp. left-) zero semigroup;(iii) SSiSr](iv) S rr\Si~T, and T is a hormos such that ir(T)~S/D.


6/7

7 7 0 K. H. HOFM ANN AND P. S. MO STERT [November

C. If a2 = azS/D, then the semigroup ir~l(a) is a completely simplesemigroup.

D . If [a, b] is an interval in S/D which is an I-semigroup with onlythe two idempo tents a and b, then there is a compact connected abeliangroup A and a homomorphism

f: ]a, b]~>A

onto a dense one-parameter semigroup and a homomorphism mapping

Sab onto (7T_1

(]a, &]))*, whereSai = T- i( i ) X ({(r,f(r)):r G ]a, b]} W ({a} X A)),

such that

So* - * - * T- > ( k b])

ir f \ i /7T

[a,b]

is a comm utative diagram, where irf(x, y, z) =y.E . If [a, b] is an idempotent semigroup, then TT~1([a, b\) is a Clifford

semigroup and there is a homomorphism of the semigroup Sab intoT~ l([a, b]), where

Sab = v'Kb) X k b],

such that

^ - - ^ r ! ( [ f l , J ] )

[a,b]

is a comm utative diagram with T'(X, y) y.F . If a2 = a, b2 = b(ES/D, a


7/7

i964] TOTALLY ORDERED D-CLASS DECOMPOSITIONS 771

T h e s em ig r o u p T i s ob ta ined uniqu ely and Sr an d Si defined asSr = \j{R(f):tGT}9 Si = [}{L(f):teT}.

COROLLARY 1. If S is a compact connected semigroup with identityand the D-class decom position is a congruence, then S contains an irreducible hormos in the centralizer o f the maximal subgroup joining theidentity to the minimal ideal providing the set of idempotents is an admissible space. (In particular, if the set of idempotents is finite d imensional, then the statement is true.)

COROLLARY 2 . If S is a compact connected semigroup with identitysuch that xSQSx for every x, and the set of idempotents is admissible,then S contains an irreducible hormos in the centralizer of the maximalsubgroup and joining the identity to the minimal ideal.

REMARK. R ot hm an [7] has s tud ied t he case when th e L-class decom posi t ion of a co m pac t connected sem igroup is to ta l ly orderedunder the natura l o rder. S ince , as he has shown, th is implies thatS2 = S and the L- (-D-) class decomposit ion is a congruence, the

main theorem gives a ra ther complete p ic ture of th is case when theset o f idempotents sa t is f ies the admiss ib i l i ty condi t ion s ta ted there .( W h en xS = Sx for every x, the resu l t ho lds without these condi t ions[4]-)

COROLLARY 3 . If S is an irreducible semigroup whose D-class decomposition is totally ordered and such that the union of its maximal subgroups is a semigroup, then S is an abelian hormos, provided also thatthe set of idempotents is adm issible (or finite dimensional).

BIBLIOGRAPHY

1. A. H. Clifford and G. B.Preston, The algebraic theoryof semigroups,Math.Surveys No.7, Amer. Math. Soc, Providence,R.I. , 1961.

2. A. H. Clifford, Semigroups admitting relative inverses,Ann. of Math. (2) 42(1941), 1037-1049.

3. H . Cohen, A cohomological definition of dimension f or locally compact Hausdorffspaces, Duke Math. J . 21 (1954), 209-224.

4. K. H. Hofmann and P. S. Mostert, Irreducible semigroups,Bull. Amer. Math.Soc. 70 (1964), 621-627.

5. R. J . Koch, Arcs in partially ordered spaces,Pacific J. Math. 9 (1959), 723-728.

6. P. S. Mostert and A. L.Shields, On the structureof semigroups on a compactmanifold withboundary, Ann. of Math. (2) 65 (1957), 117-143.7. N . J . Rothman, Linearly quasi-ordered semigroups,Proc. Amer. Math. Soc.13

(1962), 352-357.8. A. D. Wallace, The Rees-Suschkewitsch structure theoremfor compact simple

semigroups,Proc. Nat. Acad. Sci. U.S.A.42 (1956), 430-432.

T H E TULANE UNIVERSITYO F LOUISIANA

Documents

Totally Ordered j9-Class Decompositions