4

ELEMENTARY CONCEPTS

CH.

1

is both a left and a right zero element of S. Any left zero of 8 must coincide

with any right zero of S, and hence the foregoing tetrachotomy holds if we replace

the word "identity" by "zero".

Let X be any set, and define a binary operation (o) in X by xoy = y for

every x, y in X. Associativity is quickly verified. We call X(o) the right

zero semigroup on X. Every element of X(o) is both a right zero and a left

identity. The left zero semigroup X(*) on X is defined by x*y = x. In

spite of their triviality, these semigroups arise naturally in a number of in-

vestigations, for example in Theorem 1.27 below.

A semigroup S with a zero element 0 will be called a zero or null semigroup

if ab = 0 for all a, 6 in 8.

Let S be any semigroup, and let 1 be a symbol not representing any ele-

ment of 8, Extend the given binary operation in 8 to one in 8 U 1 by defin-

ing 11 = 1 and la = a\ = a for every a in S. It is quickly verified that

S U 1 is a semigroup with identity element 1. We speak of the passage from

S to S U 1 as "the adjunction of an identity element to £" . Similarly one

may adjoin a zero element 0 to 8 by defining 00 = 0a = aO = 0 for all a in S.

Throughout the book we shall adhere to the following notation:

_ JS if S has an identity element,

\S U 1 otherwise;

QO — / ^ tf^

^as a zero e^emen^ and

\&\ 1

| S u O otherwise.

An element e of a groupoid $ is called idempotent \f ee — e. One-sided

identity and zero elements are idempotent. The converse is in general false,

but note Exercise 1 below, and Lemma 1.26. If every element of a semi-

group 8 is idempotent, we shall say that 8 itself is idempotent, or that 8 is a

band. Bands were introduced by Klein-Barmen [1940], who used the term

"Schief".

H. Weber (Lehrbuch der Algebra, vol. 2 (1896), pp. 3-4) effectively defined

a group as a semigroup G such that, for any given elements a and b of G, there

exist unique elements x and y of G such that ax = b and ya = 6. E. V.

Huntington (Simplified definition of a group, Bull. Amer. Math. Soc, 8

(1901-1902), 296-300) showed that it is not necessary to postulate the unique-

ness of x and y, that this followed as a consequence.

An equivalent definition of group was given by L. E. Dickson (Definitions

of a group and a field by independent postulates, Trans. Amer. Math. Soc, 6

(1905), 198-204), namely that a group is a semigroup G containing a left

identity element e such that, for any a in G there exists y in G such that

ya = e. Such an element y is called a left inverse of a with respect to e.

Dickson showed that e is also a right identity of G (and so is the unique

identity of G), and that every left inverse of a is also a right inverse, and is

unique. The inverse of a will, as usual, be denoted by a

- 1

. The unique

solutions ofax = b and ya — b are then x = a

- 1

6 and y —

ba~l.