§ 6.1 O-MINIMAL ZERO IDEALS 7

The further verification of the fact that SpM = T and SXM = U is straight-

forward.

EXERCISES FOR §6.1

1. Let C be a non-empty subset [right ideal] of the semigroup S = S°.

Then Ac is a right [two-sided] ideal of S.

2. Let i ? b e a zero right ideal of the semigroup S = S°. Then R c: AR

and the (two-sided) ideal AR is the (pR o pg

l)-class

which is the zero element

of 8lipR

op*1).

3. Let U be the union of all the 0-minimal zero right ideals of the semi-

group S = S°. Suppose that U ^ • • Then U is a zero two-sided ideal of

S. Let R be any 0-minimal zero right ideal of S. Then U £= RAR.

4. The extension S of R constructed in the proof of Lemma 6.3 does not

come under the types considered in §4.4, when R2 = 0, since then R is not

weakly reductive. Nevertheless, it might be conjectured that we could

modify the methods of §4.4 by replacing the (now no longer one-to-one)

natural mapping of R into its translational hull R by some isomorphism cf.

It is possible to find such isomorphisms in the present case when |JR| 2.

Let T = T° be a semigroup, and let T* = T\0. Denote the elements of

Rfhyrf(reR). Let S = Rf u T * and 8 = R u T * (where i? is the trans-

lational hull of R). Suppose that an operation (o) is defined in S so that

S(o) is an extension of Rf by T. Then, for each A in T*, there exist linked

left and right translations A^ and p^, respectively, of R such that

Aorcf) = (rAx)£ and (rtyoA = (rpA)f.

Then 6: A-+(\A, pA) is a partial homomorphism of T* into R and, by

Theorem 4.19, determines an extension S(*) of R by T given by

(AB, HAB^OinT,

\{A0)(B0), i f ^ £ = 0 i n T ;

^4*f=(.40)r; r*^4=r(^4#); r*s = rs

where ^4, BeT* and r, 5 G 5 .

$(o) will be a subsemigroup of £(*) if and only if

(1) AoB = (A6)(B6) whenever AB = 0 in T;

(2) (r\A)f = {A6)rf and (rpA)/ = (rf)(A6) for all rin R and ^ in T*.

In the extension given in the construction in the proof of Lemma 6.3,

XA = £ for every A in JP*, where r£ = 0 for every r in R, while {/^: i e T * }

is transitive on R\0. For no isomorphism f of R into i? will (2) be satisfied,

and consequently the extension used in the proof of Lemma 6.3 is not

obtainable by the proposed modification of the method of §4.4.

5. Let R = R° be a semigroup with

R2

# 0. If also the semigroup of

right translations of R is transitive on JR\0, then

R2

= R.