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Apn. Univ. Ferrara - Sez. VII - Sc. MaL Vol. XXIII, 59-77 (1977) Boolean subsets of semigroups. JOSEF ESCttGF~LLEt~ (*) 1. - Introduction. The minimal prime ideals of 2 x, the power set of X, considered as a ring, correspond to the ultrafilters on X. Filters, which are of the form {A c X: x ~ A} for some fixed x ~ X, are called prineipa~ (see WlLLA~I) [24]) ; a non-principM filter is called ]ree. There is of course no hope to find explic- itly free ultrafilters on an infinite set; perhaps there are none: their existence is shown with the aid of the axiom of choice. We try in this paper to obtain an other description of the filter struc- tare on a set in the following way: Choose first a ring (or semigroup) A, where the prime ideals are in some sense well known. Then seek a corre- spondence between the prime ideals of A and the filters on X. In this paper we do this by embedding X in A in a suitable--<~ Boolean ~)--way. This means that we construct a surjective semigroup homomorphism ~: A -+ 2 z. Then the minimal (completely) prime semigroup ideals of A associated to the kernel of ~ correspond biunivocally to the ultrafilters on X (see Corol- lary 2.17). After doing this, we can hope that it could sometimes happen that we do not so well know X, but that we <~ know ~>the ultrafilters on X. 2. - Ultrafilters and Boolean sets. We shall always consider any ring as a semigroup with respect to its multiplicative structure. In particular we consider the power set of a set a semigroup under interaction of sets. (*) Indirizzo dell'autore: Istituto di Matematica dell'Universith di Ferrara, via Savonarola 9 - 44100 Ferrara.

Boolean subsets of semigroups

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Page 1: Boolean subsets of semigroups

Apn. Univ. Fer ra ra - Sez. VII - Sc. MaL

Vol. X X I I I , 59-77 (1977)

Boolean subsets of semigroups.

J O S E F E S C t t G F ~ L L E t ~ (*)

1 . - I n t r o d u c t i o n .

The m i n i m a l p r i m e idea ls of 2 x, t h e p o w e r se t of X, c ons ide r e d as a

r ing, c o r r e s p o n d to t h e u l t r a f i l t e r s on X . F i l t e r s , w h i c h a re of t h e f o r m

{A c X : x ~ A} for some f ixed x ~ X , a r e ca l led prineipa~ (see WlLLA~I) [24]) ;

a n o n - p r i n c i p M f i l ter is ca l led ]ree. T h e r e is of course no hope to f ind expl ic -

i t l y f ree u l t r a f i l t e r s on a n in f in i t e s e t ; p e r h a p s t h e r e a re n o n e : t h e i r ex i s t ence

is shown w i t h t h e a id of t h e a x i o m of choice.

W e t r y in t h i s p a p e r to o b t a i n a n o t h e r d e s c r i p t i o n of t h e f i l te r s t ruc -

t a r e on a se t in t h e fo l lowing w a y : Choose f irst a r i n g (or semigroup) A ,

where t h e p r i m e idea ls a re in some sense wel l k n o w n . T h e n seek a corre-

spondence b e t w e e n t h e p r i m e idea l s of A a n d t h e f i l ters on X . I n t h i s p a p e r

we do th i s b y e m b e d d i n g X in A in a suitable--<~ B o o l e a n ~)--way. This

means t h a t we c o n s t r u c t a s u r j e c t i v e s emig roup h o m o m o r p h i s m ~: A -+ 2 z.

Then t h e m i n i m a l ( comple te ly ) p r i m e s emig roup idea l s of A a s soc i a t e d to

t he k e r n e l of ~ c o r r e s p o n d b i u n i v o c a l l y to t h e u l t r a f i l t e r s on X (see Corol-

l a r y 2.17). A f t e r do ing th is , we c a n h o p e t h a t i t cou ld s o m e t i m e s h a p p e n

t h a t we do n o t so wel l k n o w X, b u t t h a t we <~ k n o w ~> t h e u l t r a f i l t e r s on X .

2 . - U l t r a f i l t e r s a n d B o o l e a n s e t s .

W e shal l a l w a y s cons ide r a n y r i n g as a s emig roup w i t h r e s p e c t to i t s

m u l t i p l i c a t i v e s t r u c t u r e . I n p a r t i c u l a r we cons ider t h e p o w e r set of a se t

a s emig roup u n d e r i n t e r a c t i o n of sets .

(*) Indirizzo del l 'autore: Is t i tu to di Matematica dell 'Universith di Ferrara, via Savonarola 9 - 44100 Ferrara .

Page 2: Boolean subsets of semigroups

60 JOSEF ESCHGFALLER

DEFI:~ITI0~ 2.1. A groupoid is a set S together with a mapping S• Let X be a set. A plan on X is a mapping ~ : S - - > 2 z, here S is some

other set (i.e., ~ is a relation between S and X). I f S is a groupoid and ~ is a homomorphism of groupoids, then r is

called a division (see ESCEGF;tLLE~ [9J). We shall often write X~ instead of ~(s). Hence r is a division if and only

if X ~ n X t = X ~ for all s , t ~ S . The kernel of T is also called the kernel of X and shall be denoted by

ker X; often shortly by N. Explici t ly N : = {s ~ S: X~ = 0}. N is an ideal of S, i.e., S N S c N. I t may happen tha t N ~- 0.

See section 3 for examples of plans and divisions.

DEFINITION 2.2. A plan T: S-->2 x is called Boolean, if ~ is onto, i.e.,

if every subset A c X has the form A ~ X8 for some s E S. We shall then also say tha t X is Boolean.

Each division ~ induces in an obvious way a topology on X (see ESCHG- ~:~LLE~ [9]); if T is Boolean, this topology is discrete. Usually the given

set X has some other structure we want to s tudy and the division we con-

sider are constructed in a way which is related to this other structure; if

the topology which comes from the division is not trivial, this means that we made the given structure << continuous )); if the division is Boolean, we

are ra ther pursuing an algebraic and set-theoretical s tudy of the struc-

ture X.

DEFINITION 2.3 (see e.g. WILLARD [24]). Let X be a set. A nonvoid

family 5 of subsets of X is called a filter basis, if

(i) 0 ~ 5 ,

(ii) if A, B ~ ~-, then there is D a 5 such that D c A ~ B.

A filter basis ~- is called a filter, if also

(iii) if A c 5 and A c B, then B e 5 .

An ultra/liter is a filter which is maximal with respect to set-theoretical inclusion among the filters on X.

DEFINITION 2.4. Let S be a groupoid. An ideal P of S is called a co/ace, if its complement S ~ P is a subgroupoid of S. We avoid the term prime

ideal because it creates some terminological difficulties for the case tha t S is also a ring. The complement of a eofaee is called a /ace of S (see PET~ICH [20]) ;

also the te rm filter is in use. A subset F c S is therefore a face of S i f f for

Page 3: Boolean subsets of semigroups

BOOLEAN S U B S E T S OF S E M I G R O U P S 61

s, $~ S:

st ~ F <::> s ~ F a n d t ~ F .

I f 1 ~ S, t h e n each face r 0 of S c o n t a i n s 1.

DEFINITION 2.5. L e t N be a subse t of t h e se t S. A subse t B c S is

ca l led disjoint f r o m N if N n S = 0.

DEFINITION 2.6. L e t F be a face of t h e g r o u p o i d S a n d l e t h r be a subse t

of S. W e s a y t h a t F is an N - m a x i m a l /ace of S, if F is m a x i m a l a m o n g faces

of S d i f f e ren t f r o m S a n d d i s j o i n t f r om N (in p a r t i c u l a r F (~ Z r = 0).

A coface P of S is ca l led an N - m i n i m a l co/ace of S, if P is m i n i m a l a m o n g

cofaces of S d i f f e r en t f rom 0 w h i c h c o n t a i n s

H e n c e F is a n N - m a x i m a l face iff S ~ F is an N - m i n i m a l coface.

I ~ L ~ K 2.7. B y a s t a n d a r d a r g u m e n t one can show t h a t each face of S,

d i s jo in t f r o m ~ , is c o n t a i n e d in a n N - m a x i m a l face of S. W e shal l n o t

need th i s f ac t a n d refer to KI sT [17] for a proof .

]~E~_~n]; 2.8. A n u l t r a f i l t e r ~- on a se t X is t h e r e f o r e t h e s a m e t h i n g

as a 0 - m a x i m a l face of 2 x, w h e r e 0 :--~ O.

LEMMA 2.9. L e t S - + 2 x be a d iv i s ion w i t h k e r n e l N .

1) L e t F b e a s u b g r o u p o i d of S ~ ' i t h F (~ N - - - - 0 . T h e n t h e collec-

t i on {Xt: t c F} is a f i l ter bas is on X . W e d e n o t e t h e a s s o c i a t e d f i l ter

b y F0, h e n c e F o : ~ {A c X : t h e r e ex i s t s t ~ F such t h a t X~ c A}.

2) L e t 9 7 be a f i l ter bas i s on X. Def ine 97~:~--{s~ S : t h e r e ex i s t s

A ~ 97 such t h a t A c X~}.

T h e n 976 is a face of S d i s j o i n t f r o m hr.

PR00F. 1) S ince X~ n X t = X~t a n d F is a s u b g r o u p o i d d i s j o i n t f r o m

hr, F~ is a f i l ter .

2) W e h a w e t h e second a s s e r t i o n : A s s u m e s, t E ~-~, s ay A c X~ a n d

B c X t w i t h A , B c 9 7 . T a k e D ~ 9 7 such t h a t D c A ( ~ B . T h e n D c A n

n B c X~ n X t ~ X ~ , hence st ~ 976. A s s u m e st ~ 97~, s a y A c X ~ for some

A e 9 7 . T h e n A c X ~ t - - - - X ~ A X ~ , h e n c e A c X ~ a n d A c X t , t h e r e f o r e

s, t c976 . H e n c e 5 6 is a face of S.

W e h a v e s t i l l to show t h a t 5 o n h r ~ 0 . T a k e d ~ N . T h e n X~-----0,

hence d r 97~.

Page 4: Boolean subsets of semigroups

62 J O S E F ESCHGFALLER

DEFINITI0~ ~ 2.10. A plan S--> 2 x is called complemented, if for every s ~ S there exists s ' ~ S (not necessarily unique) such tha t X, , -~ X \ X ~ . s' is called a complement to s.

REZ~AnK 2.11. Every Boolean plan is complemented.

DEFII~'ITI0~ 2.12. Let S be ~ groupoind. A face F of S is called local if for every s ~ S such tha t s ~ F and every complement s' to s holds s~CF.

REMARK 2.13. Let 2x--> 2 x be the trivial (identity) division on X. I t

is obviously Boolean. The complement (here unique) to A ~ 2 x is X ~ A .

I t is known (see WILLARD [24]) tha t a filter ~ on X is an ultrafilter if and

only if for A c X and A ~ ~- always X ~ A e ~-. In our terminology, the

ultrafilters on X are exactly the local faces of 2 x which do not contain

0 ~ 0 .

PnOP0SlTI0~ 2.14. Let S--> 2 x be a complemented division with ker-

nel N. Let F be a face of S. Then 17 is an N-maximal face of S if and

only if F is local and disjoint f rom 5 r.

PROOF. 1) Assume tha t F is N-maximal. By definition F (~ N = 0.

Assume s e S and s ~ F . Let s' be ~ complement to s.

Assume first

(A) There exists to ~ 3 r such tha t to S E _IV.

We show tha t then

(B) ts' e~ N for all t ~ F .

For, if ts' ~ N , then 0--~ Xts,--~ Xt (~ X~, and since Xs, ~ X ' ~ X s , it fol-

lows X t ~ , X s = O or X t c X~, hence Xt to= X t ~ X t c X s n Xt~ Xsto-~ O, therefore tto~ N, a contradiction.

We have shown tha t (A) implies (B).

Denote now by E~ the set {a c S: there exists t c F such tha t Xts, c Xa}. We show tha t F1 is a face of S and disjoint from N: Recall tha t always

Xtr ~-X~t ~nd X,.st----X,,.t (we need commuta t iv i ty and also associativity

only in 2x). Assume a, b~E~, say

Xts, c X~ and Xrs, c Xb for r, t c F .

Then

X~8'r~, c X a (~ Xb ~ Xab �9

Page 5: Boolean subsets of semigroups

BOOLEAN SUBSETS OF SEMIGROUPS 63

B u t

X~,,~, ~ X~, , , - - - - X ~ n X~, : X~. , , ;

hence X,.,,,cXab a n d aba f t ; t h e r e f o r e F~ is a s u b g r o u p o i d of S.

A s s u m e ab E F~, s a y X,~, c X ~ for some t ~ F .

Then X~, c X , a n d Xt~, c Xb, s ince Xao ---- X , ~ X~. H e n c e a, b ~ F~; t he re fo re F~ is a face of S.

I f d e N, t h e n X~----0, t h e r e f o r e n e v e r X ~ , c X ~ for t ~ F b y (B).

Hence N ~ F , ~ 0.

C lea r ly F c F~; s ince F is an N - m a x i m a l face of S, i t fol lows F ~ F~.

~ o w X~,~X~,, hence s '~F~cF . Al l th i s ho lds , if (A) is sa t i s f ied ; b u t if ts ~ N for a l l t ~ F , t h e n we

show in t h e s a m e m a n n e r t he c o n t r a d i c t i o n s ~ F , s ince s is also a comple -

m e n t to s ' .

The re fo re F is local .

2) A s s u m e n o w t h a t F is a loca l face of S a n d d i s j o i n t f r om N. W e

w a n t to show t h a t F is N - m ~ x i m a t .

A s s u m e t h a t t h e r e is a face F~ of S such t h a t F~ (~ N ---- 0 a n d F ~ F~.

L e t s be un e l e m e n t of F ~ F . T h e n s ' e F for some c o m p l e m e n t s ' to s, s ince F is local . T h e r e f o r e

0 ~ X~ ~ X~, ~ X~ , or ss'~ N; t h i s is a c o n t r a d i c t i o n , s ince ss'~ F~ F c F~.

P~0P0SITI0~ 2.15. L e t T: S - - > 2 x b e a d iv i s ion w i t h k e r n e l N .

1) A s s u m e t h a t T is c o m p l e m e n t e d a n d t h a t 5 is a n u l t r a f i l t e r on X.

T h e n 5 ~ is an N - m a x i m a l face of S.

2) A s s u m e t h a t T is Boo l ean und t h a t F is ~n N - m a x i m a l face of S.

Then F~ is a n u l t r a f i l t e r on X .

P~0OF. 1) A s s u m e (1). B y L e m m a 2.9 5 6 is a face of S d i s j o i n t f r om N.

A s s u m e t h a t F1 is a face of S such t h a t F l n N ~ 0 a n d 5 ~ F 1 .

A s s u m e s E F I ~ 5 ~. T h e n A ~ X ~ for a l l A ~ 5 , t h e r e f o r e X , ~ 5 . B u t

5 is ~n u l t r a f i l t e r t he r e fo re X ~ X ~ e 5. Since X is c o m p l e m e n t e d , t h e r e

exis t s ~ c o m p l e m e n t s ' to s, i .e. , X~,-~ X ~ X ~ . B u t X ~ X ~ 5, t h e r e f o r e

s ~ E 5 ~ C Ft. A g a i n 9----X~ n X~,----X~,, a n d ss'~ N, ~ c o n t r a d i c t i o n .

2) A s s u m e (2). W e k n o w b y L e m m a 2.9 t h a t F is a f i l ter on X .

A s s u m e A c X , ~ n d A ~ F ~ . W e h a v e to show X ~ A e F 6. A has t h e f o r m

Xs for some s c ~, s ince X is Boo lean . N o w A ~ F~, t h e r e f o r e s ~ F . L e t

Page 6: Boolean subsets of semigroups

64 JOSEF ESCHGFALLER

s' be a c o m p l e m e n t to s, i.e., Xs. ~-- X ~ A . B y Propos i t i on 2.14 21 is local,

therefore s' c F .

Therefore X ' - , A --~ Xs. ~ 2'6.

PI~OPOSlTIO~ 2.16. Ler ~: S--+2 X be a division wi th kerne l N.

(1) L e t F be a face of S d is jo in t f r o m N. T h e n 2" c _~so. Hence , if 2 ' is an N - m a x i m a l face, we ob ta in F = 2"oo.

(2) L e t 5 be a filter on X. T h e n 5 ss c 5 . Assume t h a t X is Boolean.

T h e n 5 ~ = 5 .

PROOF. (1) Assume to2". T h e n X~cF~, therefore t ~ F 6~. B y L e m m a 2.9,

2"~ is a face of S dis joint f rom N. (2) 1) Assume A c 5 ~ . T h e n there exists s ~ 5 ~ such t h a t Xs c A. s ~ 5 o means t h a t the re exists B ~ 5 such

t h a t B c X s . T h e n B c A and A ~ 5 , since 5 i s a filter. 2) Assume A E 5

and A ~ X a . T h e n a t 5 ~, hence A ~ 5 ~.

COROLLAI~Y 2.17. L e t S - + 2 x be a Boolean division wi th kernel N. Then the m a p 2" ~ F s is a b i jec t ion be tween the set of N - m a x i m a l faces

of S and the set of ul trafi l ters on X.

N o w the l a t t e r set is the Stone-(~eeh-compact i f icat ion fiX of the dis- crete set X ; if X has some addi t iona l s t ruc tu re which in some w a y is

ref lected in S, we can hope t h a t also the N - m a x i m a l faces of S con ta in in fo rmat ion a b o u t this s t ruc tu re and it m a y therefore useful to consider

the po in t s of fiX as N - m a x i m a l faces of S r a the r t h a n as 0 -max imal faces of 2 x, because in 2 x there is on ly place for the card ina l i ty of X and no o ther

proper t ies of it.

I~E~At~K 2.18. W e note a t this po in t a pecu l ia r i ty of some noncom- m u t a t i v e semigroups : L e t N be an ideal of the c o m m u t a t i v e semigroup S

and T a subsemigroup of S which is disjoiI!t f rom N. T h e n T is con ta ined

in a face F(T) which is dis joint f r o m N : Define 2"(T) : ~ T u (a ~ S: there is b E S such t h a t abc T}. I f S is no t c o m m u t a t i v e , this m a y fail to be

t rue ; consider for example the semigroup 2 : = (P(X), ~ ) , where X is some

set, P ( X ) : = X • {0} and

(x', z) :---- [ (x, z) if y ~-- x ' , (x, Y)

I 0 if y:/=x'.

Take N ----- 0; t h e n each (x, x) is an N - m a x i m a l subsemigroup of S, b u t

no t a face, since (x, y)(y, x ) = (x, x). I n fact , we can on ly say t h a t each

semimul t ip l ica t ive sy s t em (see R e m a r k 4.7) T c S, dis joint f r o m the ideal

Page 7: Boolean subsets of semigroups

BOOLEAN S U B S E T S OF S E M I G R O U P S 65

~V, is con ta ined in a (minimal) (~ semiface ~) F(T ) dis joint f r o m N, where

F ( T ) : = T U ( a e S : there are u, v c S such t h a t u a v c T } . The ad hoe

t e rm semiface means :

i) a, b ~ F (T ) implies axb c F (T ) for some x ~ S (i.e., F(T) is again

a semimul t ip l ica t ive sys tem),

ii) ab c- F (T ) implies a, b c F(T) .

I t m a y of course h a p p e n t h a t F(T) is no t a face, b u t t h a t there is a face dis joint f r o m h r which conta ins F(T) .

The above r e m a r k means t h a t in such semi~ 'oups S t he (~ points ~> of S are no t its 0 -max ima l faces, b u t its 0 -max ima l subsemigroups and in order

to ~( realize )> S we have to work r a t h e r wi th va lua t ions t h a n wi th characters .

This makes howe ve r the i r ~( poin ts )> less visible. The fol lowing corol lary to P ropos i t i on 2.14 holds :

P~OPOS[TIO~ 2.19. L e t S- ->2 X be a c o m p l e m e n t e d division wi th ker-

nel N. L e t T be an N - m a x i m a l subgroupo id of S and a s s u m e - - t h i s is a

h e a v y c o n d i t i o n I t h a t T conta ins an N - m a x i m a l face F .

Then T : 2~.

PROOF. A s s u m e s ~ T ~ F . T a k e a c o m p l e m e n t s' to s. Then s ' ~ F , hence ss' ~ N n T, a cont radic t ion .

3. - N a t u r a l l y d i v i d e d B o o l e a n s e t s .

])EFINITIO~ 3.1. L e t S be a semigroup. I f t e S, t h e n t he principal le/t

ideal gene ra t ed b y t is the set

s i t : = {t} u s } .

As always,

S I : : S ~ j ( 1 } (if l o S , t h e n S ~ = S ) .

S is called 1-duo~ if each pr incipal left ideal of S is also a r igh t ideal and

each pr inc ipa l r igh t ideal is also a left ideal. I t is easy to see t h a t this is t he case if and only if S i t :-- tS 1 for each

t E S .

PROPOSITIOn" 3.2. L e t S be a semigroup. Le t X be a subset of S. Con-

sider the fol lowing plans on X :

i) Xs :----- (x e X : there exists u ~ S 1 such t h a t sux = x}. This is called

the natural plan of the subse t X ; its kernel is called the natural kernel of X.

Page 8: Boolean subsets of semigroups

66 JOSEF ESCHGFALLER

ii) X ~ : = {x~ X: there exists u E S 1 and an integer k > 0 such that sux = xl'}. This is called the radical p lan of the subset X; its kernel is

called the radical kernel of X.

I f S is 1-duo, e .g , if S is commutat ive , then both are divisions.

P~ooF. Assume i). I f x ~ X s n X t , say s u x = x and t v x = x , then x = = s u x = s u . t v x . I f S is 1-duo, there is w ~ S 1 such tha t u t = t w . Hence

x = s t . w v x ~ X ~ . Assume x ~ X ~ t , say s tux = x. Obviously x ~ X ~ . Also

x = s tux = t vux ~ X t for some v e ~1, The proof for case ii) is similar.

DEFI~ITIO~ 3.3. Let X be a subset of the semigroup S. If the natural plan on X is a division S--> 2 x, we call X a natural ly divided subset of S.

I f S itself is a natural ly divided subset of S, S is called a natural ly

divided semigroup. Since 2s--~2 x is a homomorphism for each X c S, in

this case each subset X of S is natural ly divided. I f the radical plan on X is a division, we call X a radical subset of S.

S is culled a radical semigroup, if S is a radical subset of itself.

Proposit ion 3.2 shows tha t 1-duo semigroups and hence all commuta- tive semigroups are both natural ly divided and radical.

D]~FI~ITIO~ 3.4 (see LJAPI~ [19]). Let S be a semigroup. An element

a ~ S is called regular, if there is some element ~ e S such that a~a = a.

I~otice tha t ~ is in general not uniquely determined.

I f there exists ~ e S such tha t a~a = a and a~----~a, then a is culled

completely regular.

A semigroup (or ring) is called (completely) regular, if all of its elements are (completely) regular.

I~E)fARK 3.5. Let x be a regular element of the semigroup S. Assume t E S and let k > 0 be ~n integer. Then the following statements are equi-

valent:

i) there is u ~ S ~ such tha t t u x - - - - x ~,

ii) x ~ ~ tS ~.

I f x is completely regular, these s ta tements are independent from k > 0. Hence, if X is a subset of S, which consists of completely regular ele-

ments of S, then the natural and the radical plan on X coincide and are given by

X t = { x ~ X : there exists u c S 1 such that s u x = x } = X ~ t S 1.

Page 9: Boolean subsets of semigroups

BOOLEAN S U B S E T S OF S E M I G R O U P S 67

PROOF. i ) ~ i i ) is obv ious .

A s s u m e ii), s a y x k = t v for some v ~ S 1.

T h e n t v . ~ x ~ x * . ~ 2 x ~ - - x k, so we m a y t a k e u ~ v~.

I f x c S 1, t h e n also x * ~ t S 1.

A s s u m e t h a t x * c t S 1 for some k > 1 a n d t h a t x is c o m p l e t e l y r egu la r .

Then x k ~ t v a n d

X ~-I --~ X ~ - 2 X ~ X ~ - X ~ X ~ t V X ~ t ~ ~ .

I~E~ARK 3.6. L e t S be a s e m i g r o u p w i t h 0 a n d X a subse t of S. L e t N

be t h e n a t u r a l k e r n e l of X .

T h e n N ~ 0 if a n d o n l y if

i) 0 ~ X ;

ii) for each t =A 0, e S t h e r e a r e x ~ X a n d u ~ 81 such t h a t t u x : x .

I ~ ] ~ R K 3.7. L e t A be a c o m m u t a t i v e r i ng w i t h 1. A s s u m e t h a t t h e r e

is a s u b s e t X c A w i t h n a t u r a l k e r n e l 0.

T h e n l~ad A ---- 0.

The c o n v e r s e is n o t t r u e ; cons ide r A-- - -7 , .

H e r e l%ad A d e n o t e s t h e J a c o b s o n - r a d i c a l of A (see ATIYAH - MAcDo-

nAlD [1]).

PROOF. L e t X be such a set . T a k e s-A 0, E R a d A. T h e n Xs =~ 0, t he r e -

fore t h e r e a r e u c A a n d x ~ X such t h a t s u x : x . T h e n ( 1 - - s u ) x : O .

B u t 1 - - s u is i n v e r t i b l e , s ince s m l~ad A.

T h e r e f o r e x ~ - 0 . This is imposs ib l e .

I~EMARK 3.8. I f S - + 2 x is u d iv i s ion , t h e n each x E X def ines ~ coface

P~ : ~ {s ~ S : x ~ X,} ; see ESCttGFJ~LLER [9]. I f S --~ A is a r ing , one e~n

ask, w h e t h e r /)~ is ~ r i n g idea l ; in t h i s case also k e r X ---- ~ P , is a r i n g

ideal . T h e fo l lowing ho lds (see a lso 4.8): x~x

L e t X b e a subse t of a 1 -duo r i n g A. A s s u m e x c X a n d t h a t t h e fol-

lowing s e p a r a t i o n p r o p e r t y ho lds (as e.g. in 2x): I f u c A a n d u x ~= x ,

t h e n t h e r e is d ~ A such t h a t d x ~ - x a n d d u x - - - - O .

T h e n P~ is a ( c o m p l e t e l y p r i m e , see belo~v) idea l of A .

PRooF. A s s u m e s , t c P , a n d s - - t ~ P , . Then t h e r e is u ~ A 1 such t h a t

( s - - t ) u x = x or s u x = t u x + x .

Since t e P ~ , t u x =A x . B y h y p o t h e s i s t h e r e ex is t s d E A such t h a t d x = x

a n d d t u x = O.

T h e n d s u x ~ d t u x -~- d x - ~ x . Since A is l - d u o , d s ~ s d ' for some d' E A 1,

t he r e fo re s d ' u x = x a n d x ~ X~, a c o n t r a d i c t i o n .

Page 10: Boolean subsets of semigroups

68 ffOSEF ESCHGFALLER

I~EM,~RI< 3.9. L e t A be a r i n g w i t h 1 a n d z: A-->2 z be a d iv i s ion such

t h a t each P~ is a n i dea l of A. A s s u m e X~ -~ X . A s s u m e also a - - a 2 e ke r

for a l l a ~ A .

T h e n ~ is c o m p l e m e n t e d .

P x o o F . A s s u m e a e A . W e show t h a t 1 - - a is a c o m p l e m e n t to a.

I f a ~ P ~ a n d 1 - - a e P ~ , t h e n I ~ P ~ , s ince P~ is an idea l , a c on t r a -

d ic t ion .

I f a ~ P~ a n d I - - a ~ P~, t h e n a - - a ~ ~ P~.

This is n o t pos s ib l e , s ince a - - a ~ ~ k e r ~ ~ ~ P~. H e n c e X~_~ ~ X ' ~ X ~ . X E ~

DEFII~ITIOlX 3.10. L e t S b e a s e m i g r o u p w i t h 0. W e d e n o t e for a subse t

Y c S b y annL Y t h e l e f t a n n i h i l a t o r of Y a n d b y a n n , Y t h e r i g h t ann ih i -

l a t o r :

a n n L Y : ~ { s e S : s y : O for a l l y E Y } .

A r i n g is ca l led a Baer ring, i f i t c o n t a i n s 1 a n d for each s u b s e t Y c A t h e

l e f t a n n i h i l a t o r of Y is g e n e r a t e d b y a n i d e m p o t e n t (as a l e f t i dea l ) ; see

e.g. KAPLANSKY [15] or JA~OWITZ[13].

L e t S be a s e m i g r o u p w i t h 0. W e cal l a subse t X c S a Baer-subset, if

for e ach s u b s e t Y c X t h e r e is t E S such t h a t annL Y = t S 1.

I f S i t se l f is a B a e r subse t , we cal l S a weakly-Baer semigroup.

D]~YZ~ITZO~ 3.11. Two e l e m e n t s s, t of a s emig roup S w i t h 0 a re ca l led

orthogonal, if st ~ O.

PROPOSITION 3.12. L e t S be a s e m i g r o u p w i t h 0 a n d X a s u b s e t of S .

A s s u m e :

1) X is a B a e r subse t .

2) X cons i s t s of pa i rwi se o r t h o g o n a l e l ements .

3) i) E a c h e l e m e n t of X is r e g u l a r a n d x2ve 0 for a l l x ~ X , or ii) no

e l e m e n t of X is n i l p o t e n t .

T h e n in case i) t h e n a t u r a l p l a n on X is B o o l e a n a n d so is t h e r a d i c a l

p l a n in case ii).

PI~OOl~. A s s u m e Y c X . Since X is a B a e r subse t of S, t h e r e is t ~ S

such t h a t annL ( X ~ Y ) ----- tS 1.

W e show Y = Ir t , whe re X t in case i) re fers to t h e n a t u r a l p l a n a n d

i n case ii) to t h e r a d i c a l p lan .

a) A s s u m e y ~ Y. T h e n for e ach x ~ X ~ Y b y o r t h o g o n a l i t y y x = O,

t h e r e f o r e y c annL ( X ~ Y ) ~- tS ~.

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B O O L E A N S U B S E T S O F S E M I G R O U P S 69

I n case i) we obtain y E X t b y R e m a r k 3.5. Assume ii). We have y = tu for some u E S 1, hence y ~ tuy, i.e., again

y ~ X t .

b) Assume i) and x e X " ~ Y . Since x~=/= 0, x ~ a n n L ( X ~ : Y ) - - - - t S 1, i.e.,

x~X,.

Assume ii) and x ~ X ~ Y and x E X ~ , say tux ~ x ~.

Then xk~ annL ( X ~ Y ) , hence x~+l~ 0, a contradiction.

DEFI~ITIO~ 3.13 (see K_APL(~SKu [15]). Le t S be a semigroup with 0. An idempoten t e of S is called minimal , if e ~ 0 and for every idempoten t /=/=e wi th / ~ e S n S e holds ] - -~0 .

I f therefore also ] is a min imal i dempo ten t =/= e and e / = ]e (e.g., if bo th e and ] are in the center of S), t hen e] = O.

COROLI.A~V 3.14. Le t S be a semigroup with 0. Assume t h a t S is weakly Baer. Denote b y X the set of all central minimal idempoten t s of S.

Then the na tu ra l p lan on X is Boolean.

P]~OP0SlTIO~ 3.15. Le t A be a (real or complex) Banach algebr~ and X a countable set of elements of A, which are:

a) comple te ly regular in some semigroup H which contains A as a subsemigroup (as H - - - - R ~ contains C(X) for a space X ) ,

b) mu tua l ly orthogonal, bu t r 0.

Consider the p lan Xa :-~ {x ~ X : there is u ~ H 1 such t h a t aux = x}.

Then this p lan is Boolean; it is easy to see t ha t i t is also a division, if A is contained in the center of H.

PI~OOF. Assume Y c X.

a) I f Y----0, then Y----X0.

b) Assume Y ve 0, say Y = {y~, y~, ...}, where all y~ are distinct for

dist inct indices.

Define co

: = , ! �9 ' n= l

This is a well defined element of A.

We show Y----- X~.

i ) T ke T h e n yo= (lln )(WllYolI) and a).y.= for

: = " I!YolI"

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70 J0SEF ESCHGF~LL~R

Since y~ is completely regular in H, there exists u e H such that

y,~uy~ = y,, and uy~ = y n u .

Then a2uy~ a , ~ y , u ~ i.e. Xa. = = y , ,u = y,~, y , , e

ii) Take x ~ X ~ Y . Then x a - ~ O.

I f x ~ X ~ , s~y a u x = x , then x ~ : x a u x -~ O, ~ contradiction.

DEFI~'I~IO~ 3.16. Let us call a ring A a le]t Y - r i n g , if every left ideal of A is the left annihilator of some element of A.

I~]~)[ARK 3.17. Such rings were considered in YOHE [25]; see also JAE- GEI~MA~ - KttI~I?A [12].

YOHE characterizes commutat ive Y-rings with 1 as the direct sums of

completely pr imary principal ideal rings. Completely pr imary means local ~ i th nilpotent maximal ideal.

PROPOSlTm~ 3.18. Let A be a left Y-ring with 1 and X a subset of A. Assume further:

1) A is 1-duo;

2) i) x2r 0 for each x ~ X or ii) no element of X is ni lpotent;

3) X consists of pairwise orthogonal elements;

4) each nonzero left ideal of A contains an element of X.

Then in ease i) the natural division on X is Boolean and so is in case ii) the radical division.

PnooF. Assume Y c X . I f Y = X, then Y = X1. Assume Y r X. Then there exists a ~ A such tha t A ( X ~ Y ) = annL a. We show Y = Ya.

a) Assume x c X ~ Y . Then x a = O . Assume x c X a , say x ~ = a u x .

Then x k + ~ = x a u x = O. Therefore x ~ X , in bo th cases i) and ii).

b) Assume y E Y and y a = O . Then y ~ A ( X ~ Y ) , i.e., y = d ~ x ~ +

+ . . . - ~ d,x,~ for some d ~ A and x ~ X ~ Y .

Hence y~ = 0, since the elements of X are pairwise ortogonal and y r x~ for all i. This contradicts both i) and ii), hence y a ~= O.

By hypothesis 4) there is an element z ~ X such tha t z ~ A y a , say z =- u y a . Then z 2 = u y a u y a ~- u y . u y a . v = u y z v for some v ~ A ~, since A is

1-duo. But z ~ r 0, hence y z r 0 and z = y by orthogonality. Now y = u y a implies y2 = u y a y :/-O, therefore a y ~= 0 and there is z ' ~ X

such that z ' e A a y , say z ' = u ' a y ; again z ' 2 = - u ' a y . z ' ~ O , hence y = z' and

y = u ' a y = a u " y ~ X ~ .

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BOOLEAN SUBSETS OF SEMIGROUPS 71

COROLLARY 3.19. Le t A be a 1-duo left Y-ring, in weich eve ry non-

zero left ideal conta ins a cent ra l min ima l idempoten t . Then the set of all cent ra l min ima l i dempo ten t s of A is a Boolean subset

of A wi th respec t to the na tu ra l division.

4. - Other e x a m p l e s o f n o n - c o m m u t a t i v e div is ions .

I ~ E ~ R K 4.1. I f the semigroup S is no t c o m m u t a t i v e , t he na tu ra l p lan

is in general no t a division. I t is m u c h more difficult to ob ta in divisions

over a n o n - c o m m u t a t i v e semigroup; in some sense howeve r t h e y are of even

g rea te r in teres t t h a n c o m m u t a t i v e divis ions: the semigroups which <( occur in n a t u r e ,) are n o n - c o m m u t a t i v e ; a divis ion S -> 2 ~ is a sor t of abeli~nizing

process and there fore no t so easy to achieve if S is no t c o m m u t a t i v e : b u t t h e n it should give pe rhaps of ten surpr is ing in format ion .

RE~IARK 4.2. L e t H be a semigroup and F a face of H and let S be a

semigroup ~nd X a set. Assume ~: S • X -> H is a m a p p i n g which satisfies

(*) T(s, x) ~(t, x) = ~(st, x) for all s, t e S and x c X .

T h e n Xs : = {x e X : z(s, x) ~/v} defines ~ division S --> 2 x.

EXA~PLES 4.3. (1) L e t S be a semigroup and let X be ~ set of cent ra l

i d e m p o t e n t s of S. Then T: S • defined b y ( s , x ) ,~ ->sx satisfies ( . ) .

(2) L e t A be a r ing and deno te b y . the Jacobson-c irc le-product in A

(see e . g . v . d . W . ~ ] ~ R D ~ [23] or J~COBSO~- [11]), i.e., a . b :---- a + b - - ab.

L e t X be a set of cent ra l i d e m p o t e n t s of A. T h e n ~: (A , . ) • .) which sends ( a , x ) to a * x satisfies ( , ) .

PRooF. a) A s s u m e (1). Then

z(s , x)7:(t, x) = s x ' t x -~ st "x ~ = s t ' x = T(st, x) .

b) I n case (2) we have to show

(ab) , x ---- (a * x)(b * x) .

This is s t r a igh t fo rward .

REMARK 4.4. E x a m p l e 4.3(1) gives m a n y n o n - c o m m u t a t i v e divisions,

which are no t v e r y in te res t ing h o w e v e r in t h a t t he face p r o p e r t y implies

Page 14: Boolean subsets of semigroups

72 JOSEF ESCHGF&LLER

t h a t the kernel is ei ther all of S (if X c S ~ F ) or equal to S ~ F , if X (~ n F4= o.

DEFINITIO~N 4.5. A scale is a quasiordered groupoid (S, < ) such tha t ab <-< a and ab < b for all a, b ~ S (see ESCHGF:~LLER [9]).

REMARK 4.6 (see [9]). Le t S be a semigroup. Define a < b if there is b ' e S such tha t a ~ bb'.

I f S is 1-duo, t hen (S, 4 ) is a scale. I f A is 1-duo ring, then (A, . ) is a 1-duo semigroup; hence also ( A , . , < )

is then a scale, if we define a < b if b . b ' = a for some b 'EA.

R E ~ K 4.7. We collect here some facts f rom non-commuta t ive algebra (see e.g. BEHI~E~S [3], RENAULT [21], REN.~ULT [22]).

Let A be a ring.

1) An ideal p of A is called prime, if for ideals a, ~ of A with a b c p one of a c p or b o p holds.

I t is known tha t this is equivalent to the following requirement :

A ~ is a semimultiplicative set, i.e., if a, b e A and a r p, b Cp, then there exists x e A such tha t axb r p.

2) An ideal p of A is called (left) primitive, if there exists a maximal left ideal m of A, such tha t ~ is the largest ideal contained in m.

A left pr imit ive ideal is not necessarily r ight primitive, bu t bo th of t he m are usually called primitive. A primit ive ideal is prime.

3) An ideal p of A is called completely prime, if it is also a cofaee, A ~ p is a subsemigroup of A.

4) Without requir ing commuta t iv i ty or uni ty let us call a ring A integer, if 0 is a completely prime ideal of A.

5) A semigroup with 0 is called reduced, if it has no ni lpotent ele- ments ~ 0.

6) Each pr ime ideal of a ring A contains a minimal pr ime ideal.

7) Le t A be reduced. Then each minimal prime ideal of A is comple- t e ly prime.

8) I f A is a commuta t ive ring with 1, then the cofaees of A are exact ly the unions of prime ideals of A (see KAPLA~-SKY [14]).

RE~rA~K 4.8. a) Le t S be a semigroup with 1 and N an ideal of S.

Page 15: Boolean subsets of semigroups

BOOLEAN SUBSETS OF SEMIGt~OUPS 73

Le t P be a semicoface of S (i.e., S ~ P is a semimult ipl ieat ive set and _P is a semigroup ideal of S) which contains N. Then:

a) /~ is min imal among the semicofaces of S which contain N if and

only if for each x e _P there exist Sl, ..., s,+l ~ P such t ha t s~s~x . . , s, xsk+~ e N.

b) Le t A be a ring with 1 and N a (ring-) ideal of A.

Le t p be a p r ime ideal of A. Then:

i) p is min imal among the pr ime ideals of A which contain N if and only if for each x e p there exist sl, . . . , s , + ~ r such t ha t s~xs~x...

....s~xs~+~ e N.

ii) Therefore by a), the min imal p r ime ideals to N are the minimal

semicofaces to N.

iii) Assume there is a eoface P of A such tha t N c / ~ c p .

By ii) /~ = p; therefore in this case p is also complete ly prime.

PROOF (compare I=~ENAULT [ 2 1 ] o r BOURBAKI [4]). We have to show only a) and b) i); the proof is a lmost the same, only t ha t in case b) we deal

wi th r ing ideals. Wri te S z A and P - ~ O . Assume x E P. The case x E N is t r ivial . Assume x ~ N. Define T to be the set of all expressions SlXS2X...s~+l, where s j ~ P .

Assume T n N ~ 0. Le t a Z S l X . . . s k + l and f l~ - t l x . . . tm+l be two elements of T. Since P

is a semifaee, there exists y e A such t h a t s~+lytl~P, hence also ~yf leT . Therefore T is a semimult ipl icat ive set which does not contain 0. I n the r ing case, b y RE~.~.VLT [21], there is a p r ime ideal p ' ~ P ' with

P ' n T = 0 and N c P ' . I n the semigroup r define P ' : : S ~ F ( T ) (see 2.18), t ha t is, p ' z

: {a e S : uav r T for all u, v e S}. Again P ' (~ T ~ 0 and P ' is a semicoface which contains N.

But x e T, hence x ~ P ' . On the other side, if t ~ P ' and t ~ P , then t x ~ P ' n T = O . There-

fore P ' c P, hence P ' - - - -P , since P was minimal for N.

This contradicts x e P ~ P ' .

RE~IAR~ 4.9 (see CHACR0~X [19]). Le t A be a reduced ring with 1. Then

all idempoten ts of A are central.

Page 16: Boolean subsets of semigroups

~ 4 J O S E F ESCHGFXLLER

RE~A~K 4.10. Le t S be a semigroup of central idempotents of the semi- group H and X ~ subset of H. Then

X~ :~- {x ~ X : ex = x} is a division S -~ 2 x .

P~OOF. i) Assume e x ~ - x and d x ~ - x . Then e d x : e x ~ x .

ii) Assume e d x ~ x . Then e x : e 2 d x : e d x ~ . x and d x = d e d x = ~--- e d z x ~ x .

RE~AICK 4.11 (see LJAPIN [19], CARSO~ [6], BEHaENS [3], HOWlE [10J).

1) A semigroup S is called strongly regular, if for each a ~ S there is a+r S such tha t a~a + ~ - a +. E~ch completely regular semigroup (see 3.4) is s t rongly regular.

A semigroup S is called inverse, if for each a ~ S there is a unique a + r S such tha t aa + a -~ a and a + aa + ~ a +.

A completely regular inverse semigroup is ~lso called ~ Clil]ord semigroup .

A semigToup is inverse if and only if i t is regular and all idempotents commute .

A completely regular semigroup is inverse if and only if all idempotents are central (see LjAPI~ [19], p. 322).

2) A regular ring is strongly regular if and only if i t is reduced (see cA~so~ [6 ] ) .

A strongly regular ring A is reduced, regular, biregular ( that means, each principal ideal is generated by a central idempotent) and duo (see B ~ E s s - R.~PHAEL [ 5 ] ) .

3) Le t A be ~ biregular ring. Then:

i) A is semis imp le , i.e., R a d A = 0.

ii) Each pr imit ive ideal is maximal.

4) Le t A be a strongly regular ring and p an ideal of A. Then (see K E I ~ N I S O N [16]) :

p is prime<=>p is primitivec~>p is maximal<:>p is maximal one-sided.

PlCOPOSITIO~ 4.12. Let S be a completely regular inverse semigroup and X a set of idempotents of S. Denote for a ~ S by a + the unique inverse to a (i.e., aa+a ~ - a and a + a a + = a+). Then:

X ~ : = { a c X : a a + e ~ - e } defines a division S - + 2 ~ .

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BOOLEAN SUBSETS OF SEMIGROUPS 75

PROOF. Recall (ab) + ~ b+a + and t h a t all idempoten ts of S are central

(411.1). Assume e c X ~ ( ~ X b , i.e., a a + e - - bb+e ~ e. Then ab(ab)+e := abb+a+e -~

~-- abb +ea + ~ aa +e ~-- e, hence e~Xab. Assume e ~ X~b. Then e ~- ab(ab)+e ~ ab 'b+a+e, hence aa+e ~ aa+ab �9

�9 b+a+e -~ abb+a+e ~-- e, i.e., e e X ~ .

Also aa+ ~ a+a, hence e ~ a+ ea ~ a+.abb+a+ e . a ~-- a+aebb+a+ae ~- bb+ e,

hence e ~ Xb.

DE~N~TIO~ 4.13. Le t A be a r ing with 1 and denote b y B ( A ) the set

of all central idempoten ts of A. I t is known tha t B ( A ) is a Boolean ring with old mult ipl icat ion and

new addit ion e -~ d :--= e 4- d - - ed (see D)_u~s - HOF~V~A~ [8]). We say t ha t A or also the Boolean ring B ( A ) has proper ty (I) (see BADE -

CURTIS [2]), if for every pair of sequences (e~), (d~) f rom B ( A ) such tha t e i e j - ~ 0 if i :~ j and d, dj-----0 for i :~ ~ and e ~ d ~ : 0 foral l i, ~, there exists

] ~ B ( A ) such tha t ]e~-~ e~ and ] d , = 0 for all i. I f A is a commuta t i ve Banach algebra, there are topological characteri-

zation of p rope r t y (I) (see [2]).

PlC0P0SlTIO~ 4.14. Le t B be a Boolean ring and X an ~t mos t countable

or thogonal subset of B and assume 0 6 X. Assume tha t B has p rope r ty (I). Then the division X , : ----{x~X: e x ~ - - x } (see 4.10) for e ~ B is Boolean.

Pl~ooF. B y 4.10 this is a division, since B is commuta t ive . Assume

Y c X. Since X is a t most countable,

Y = {Y~, Y2, ...} and X ~ . Y - ~ (x t , x2, . . . } .

Since X is orthogonal , all p roducts x~x~, x~yj , y~yj vanish wi th excep-

t ion of x~ and y~. Since B has p r o p e r t y (I), there exists e ~ B such tha t e y ~ - y ~ , i.e.,

yiEXe, and ex , - - -O , i.e., x ~ X , .

REMAI~I~ 4.15. A large va r i e ty of divisions similar to those occurring in this paper can be obta ined b y considering topological semigroups.

For example , assume tha t S is a commuta t i ve topological semigroup

and X a set of idempotents , say, of S. Then the following is a division S - ~ 2 x :

X ~ : z { x e X : there is a net (u~) in S such t ha t s u ~ x - §

Pervenuto in Redazione il 23 dicembre 1976.

Page 18: Boolean subsets of semigroups

76 JOSEF ESCHGFALLER

RIASSUNT0

Se S -~ 2 x ~ un omormorfismo surgettivo di semigruppi, allora gli ultrafiltri su X corrispondono biunivoeamente agli ideali primi minimali del nueleo. Vengono dati degli esempi.

SUMMARY

If S ~ 2 x is a surjective semigroup homomorphism, then the ultrafilters on X correspond bimfivocally to the minimal prime ideals to the kernel. Some examples are given.

BIBLIOGRAFIA

[1] M. ATIYAH - I. G. M A C D O N A L D , Introduction to commutative algebral Addison- Wesley, Reading, Mass., 1969.

[2] W. G. BADE - R. C. CURTIS, Homomorphisms o] commutative .Banach algebras, American Journal of Math., 82 (1960), pp. 589-608.

[3] E. A. BEI~RnNS, Algebren. Hoehschultasehenbueh 97, Mannheim, 1965. [4] N. BOURBAKI, Commutative Algebra. [5] W. D. BURGESS - R. RAPfIAEL, Abian's order relation and ovthogonal completions

]or reduced rings, Pacific Journal of Math., 54 (1954), pp. 55-63. [6] A. B. CARSON, Representation o] regular rings o] ]inite index, Journal of Algebra,

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