6

ELEMENTARY CONCEPTS

CH.

1

A consists of a single element a, then we call L(a) =

AS1^,

R(a) =

aS1,

and

J (a) = / S

1

^

1

the principal left, right, and two-sided ideal of 8, respectively,

generated by a.

A semigroup S is right simple if and only if a8 = 8 for every a in 8. For if

aS ^ 8, then aS is a proper right ideal of 8; and if R is a proper right ideal of

8, and aeR, then aS ^ RQ 8, so aS ^ S. But to say that aS = 8 for every

a in S is equivalent to saying that, for every a and b in #, there exists xinS

such that ax = b. Combining this with the dual proposition, and recalling

the Weber-Huntington axioms for a group, we see that a semigroup is a group

if and only if it is both left and right simple.

EXERCISES FOR §1.1

1. (a) If e is an idempotent element of a left cancellative semigroup S,

then e is a left identity element of 8.

(b) A cancellative semigroup can contain at most one idempotent

element, namely an identity element.

2. (a) If S is a cancellative semigroup, so is

S1.

(b) Let 8 be a left zero semigroup with \S\ 1. Then S is right

cancellative, but

81

is not.

3. Let a be an element of a semigroup 8, and let A = {x : axa = a, xeS}.

If A # • , then Aa [aA] is a left [right] zero subsemigroup of 8. (Bruck

[1958], pp. 25-26.)

4. A left zero semigroup S is left simple, and each element of 8 forms by

itself a right ideal of 8.

5. Let S be a semigroup such that if ab = cd (a, b, c, d in S) then either

a = c or b = d. Then # is either a left zero semigroup or a right zero semi-

group. (Thierrin [19526].)

6. If S is a semigroup having a right zero element, then the set K of right

zero elements of S is a right zero subsemigroup of 8, and is a two-sided ideal

of 8 contained in every two-sided ideal of S.

7. The right zero elements of ^x are just the "constant" transformations,

mapping every element of X onto a single fixed element of X. There are no

left zeros in &x if |X| 1.

8. Let K be the set of right zero elements of a semigroup 8, and assume

K # • . Then S £ &~K if and only if (i) xa = xb (a, b in S) for all x in K

implies a = 6, and (ii) if a is any transformation of K, there exists a in # such

that xa = #a for all x in i£. (Malcev [1952].)

9. An element a of ^x is idempotent if and only if it is the identical

mapping when restricted to Xa.

10. Let X be a finite set of cardinal n. Then ^~x contains the symmetric

group @x of degree n. If a e$~x, define the rank r of a to be |Xa|, and the

defect of a to be n — r.

(a) If jS is an element of ^x of rank r n, there exist elements y and 8