MINIMAL IDEALS AND MINIMAL CONDITIONS
The first five sections of this chapter deal with semigroups containing
0-minimal ideals. The semigroups we treat will have a (two-sided) zero
element, usually denoted by 0, and at least one further element. Recalling
our conventions of Volume 1, a semigroup 8 has this property if S = S°.
We will use this as a convenient shorthand and use phrases such as: "let
S = S° be a semigroup" to convey that 8 is a semigroup, that \S\ 1, and
that 8 has a zero element. Again, as in Volume 1, the corresponding results
for semigroups with minimal ideals that follow from the results for semi-
groups with 0-minimal ideals (cf. §2.5) will only rarely be stated.
§6.1 looks at the various ways in which a zero semigroup can be embedded
as a 0-minimal ideal in a semigroup. §§6.2-6.4 culminate in structure
theorems for the left and right socles of a semigroup. The original ideas
and many of the results here are due to S. Schwarz , but we believe
that our main structure theorem (Theorem 6.29) for the union of the left
and right socles of a semigroup is essentially new. There is a strong analogy
with Dieudonne's theory  of the socle of a ring. In §6.5 are obtained
characterizations of the various 0-direct unions of 0-simple semigroups
which arise in the treatment of the socles. Here again the first discussion of
such decompositions is due to Schwarz . These characterizations are
analogous to those of Croisot , for semigroups without a zero element,
that we presented in §4.1.
The final section, §6.6, discusses various minimal conditions, such as ML,
MR, and Mj, which we met in Chapter 5. The exercises for §6.6 also contain
a discussion of some of the radicals that have been defined for semigroups.
The topic of radicals is taken up again in §11.6.
6.1 0-MINIMAL ZERO IDEALS
A preliminary objective of this section, Theorem 6.4, gives necessary and
sufficient conditions on a semigroup R with zero that R can be embedded as
a non-degenerate 0-minimal right ideal of some semigroup S. This is then
applied (Theorem 6.7) to the case R a zero semigroup, and an analogous two-
sided result (Theorem 6.9) is given. These results are new, and the theory is
far from being worked out; note the discussion at the end of the exercises.
Let A be a 0-minimal [left, right] ideal of the semigroup 8 = 8°. Then
= A or
= 0 (§2.5). Clearly, if A is nilpotent, then we cannot