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
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