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.
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