Semigroup Formn, Vol. 1 (1970), 276-278.

A NOTE ON LEFT SIMPLE TOPOLOGICAL SEMIGROUPS Constantine Kassimates I and N.A. Tserpes i. Throughout

S

will be a topological semigroup

simple (i.e., Sx = S for all

x E S) and

f

which

will denote

is

left

a real val-

ued function on S. For fixed tion

a E S, ta denotes the right translata: x + xa, which is clearly "onto".For B ~ S, ~ ( B ) = { x ; xa E B

S is called a left group if S is right cancellative or (equivalently) if S has an idempotent element. For example the complex numbers with multiplication

#

0

x*y = xly I . Examples of an S without any idempo-

tents (and hence not a left group) are given in [i, p. 82]. It is known [I] that if S is a left group in the above sense, then there a (continuous) isomorphism group and

is

n: E x G ~ S, where E is a left zero semi-

G is a group. ~ may not be an homeomorphism unless G

is

a

topological group(every maximal subgroup of S is topological). The symbol

spt f

denotes the support of

S; f(x) ~ 0} so that by definition CZ f.

f

and CZ f denotes the set {x

spt f = CZ~f = closure in

S of

We define the following classes of real valued functions on S:

K(S) = { f;

spt f

is compact}

T(S) = { f; there is a ~ S such that Co(S)- { f; f is continuous and To(S)- { f; there is

a E S

spt f

such that

f a ~ f't a c K(S) } is compact} fa E Co(S) }

In this note we derive a characterization of

left groups in the class

of left simple semlgroups using the classes To(S) and Co(S ) associated with the semlgroup. It turns out that To(S ) constitutes the Invarlant part of Co(S) under composition on the right with the ta. The exist ence of such invarlant Subsets of Co(S) is fundamental in integration theory and the extension of harmonic analysis to semlgroups.(In defining an invarlant functional on Co(S ) a natural requirement is that for every

f ~ Co(S), a r S, f't a c Co(S). Also it turns out that either

I' Deceased.

'

....... S76

2

KASSIMATES et al.

T(S) = {0} or S is a left group. The question (posed by the first author) whether there is an example of a Hausdorff locally compact (nondiscrete) semigroup with the ta'S, a e S, closed mappings and with T(S) = {0}, is still open. 2. L ~

i. The

ta'S, a e S, in a left group are homemorphis m .

The proof follows since each right translation has a suitable right translation as inverse, and all the ta'S are continuous. Note that if e e S is idempotent, there is

then e is a right identity and for every a

b ~ S such that

LEMMA 2.

CE f =

ta(Spt fa ) = spt f fa

has

and

ab is also idempotent. for every

for all d e S such that

a ~ S. Moreover ,

ta i_ssclosed mapping

CZ fa= t-l(CZa f) and

spt fa =

ta(CZ fa) = CZ f and hence

taking

closures we obtain

CZ f

ta l(spt f)" Since ta

is

= ta(Spt fa) c spt f ;

by

ta(Spt fa) = spt f .

LEMMA 3. The following inclusions obtain: To(S ) ~- Co(S) c

To(S)

or

compact support.

One has onto,

ba = e

ta(Spt fa ) = spt f

e S

T(S)

=

K(S) and

.

oK(S)

LEMMA 4. To(S) and T(S) are invariant under composition on the right

with the

If

ta'S, a e S.

f e To(S ) , then

find

y

such that

fa ~ C~

for some

a e S; let

x a S; we can

fa = fyx = (fx)y" we have spt (fx )y compact; also (fx)y is continuous so that fx e To(S)" Similarly T(S). THEOREM 2. The

a = yx; since

followin~ statements are

eguivalent:

(i) S is a left group. (ii) To(S ) = Co(S ) c (iii) T(S) Proof. f e K(S) spt fb

(1) => (il) such that

by LEMMA 1 . (iii) => (i). There is f(a) # 0 and also there is

is compact. Let

(1) t al ( a) = tc(spt

K(S) = T(S) and one of these classes is #{0}.

~ {0}.

t~l(cz f)

c =

be such that

cz fa

c

b e S

a e S and

such that

ca = b. Then

tc[Spt (fa)c] = tc(Spt fca )=

fb ) 9

Hence

t-l(a) as a compact subsemlgroup has an idempotent [4] and so a S is a left group. CQROLLARY _i. Let S b_e_en0t a left group. Then no set of the form tal(b), a,b E S, i__sscompact. If

f (b) - i, f " 0

otherwise, spt fa .

is applicable.

277

t-l(b) a

and THEOREM 2 then

KASSIMATES et al.

3

COROLLARY 2. If S is locally compact Hausdorff, then the followin K statements

(ill)

are equivalent: (i) S is a left group. (li) Co(S) = To(S).

tal(b) i_~scpmpact for eyery

a,b e S.

THEOREM 3. Let the ta, a e S, be closed mappings. Suppose f u r t h e r that there is fo__r every

f # 0

in Co(S) such that

x e spt f. Then

Observe that

t-l(x) n a

spt fa ~is compact

S is a left Kroup. = restriction of ta on spt fa' is perfect

ta/sp t f a

map [2, p. 236]. Hence its domain, spt fa' is compact, and so To(S) #0. COROLLARY 3. Let the ta, a e S, be closed. Then Co(S ) = To(S ) and only if for each f e Co(S) there exists spt fa i~s compact for every

i._ff

a e S such that t-l(x) n a

x e spt f.

THEOREM 4. Let the ta, a e S, be closed maps; further assume

that

S islSt-countable, normal, admitting ~ complete uniform structure; (e.g., discrete, metric, Ist -countable paracompact, ISt-countable normal realcompact, et.c.). Then if S is not a left group, every of the form Proof.

set

t-l(b), a,b e S, .has non-empty interior. a ... Assume

A = t-l(b) has empty interior for some a,b e S.Then a

by [3, p. I0], A is countahly compact complete and hence compact. We can find

c e S such that

cb = a; hence, t~l(b) = t~l-t;l(b)

and

tbl(B) =

tc[tal(b)]; thus tbl(b) has an idempotent, a contradiction. REFERENCES

I. Clifford, A.H. and G.B. Preston, The alKebralc theory o_~fsemiKroups Vol. ii, American Math. Society, 1967. 2. DugundJl, J., Topology, Allyn and Bacon, Boston, 1966. 3. Morita, K. and S. Hanai, Closed mappings and metric spaces,Proc. Jap. Acad. 32 (1956). 10-14. 4. Numakura, K., On bicompact semigroups, Math. J. Okayama U., 1 (1955), 99-108. UNIVERSITY OF SOUTH FLORIDA TAMPA, FLORIDA 33620 Received by the editors

February 19, 1970.

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