CHAPTER 6

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 [1951], 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 [1942] 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 [1951]. These characterizations are

analogous to those of Croisot [1953], 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

either

A2

= A or

A2

= 0 (§2.5). Clearly, if A is nilpotent, then we cannot

l

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