1.1 BASIC DEFINITIONS

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The first published system of group axioms of this nature is that of J.

Pierpont (Galois theory of algebraic equations. II, Annals of Math. 2 (1900-

1901), 22-56; see p. 47); he postulated a two-sided identity element e and a

two-sided inverse a* for each element a of the set: aa' = a'a = e.

By a subgroup of a semigroup S we mean a subsemigroup T of S which is

also a group with respect to its binary operation. This is equivalent to

saying that

I7

is a subsemigroup of 8 such that if a, b e T then there exist

x, y in T such that ax = ya = 6. From this it is evident that a subset T of a

semigroup 8 is a subgroup of S if and only if aT = Ta = T for every a in T.

(Example: if X is a set, &x is a subgroup of ^p. )

The identity element e of a subgroup

J7

of a semigroup S is an idempotent

element of S; it need not be the identity element of 8.

If G is a group, then, by the convention given above, G° means GuO, i.e.,

G with a zero element adjoined. By a group with zero we mean G° where G

is a group. For example, let R( •, + ) be a ring. Then R( •) is a semigroup

which we call the multiplicative semigroup of R( •, +) . Evidently R( •, +)

is a division ring if and only if R( •) is a group with zero.

By the dual of a proposition or concept we mean the proposition or con-

cept obtained by replacing every product ah in the statement thereof by 6a.

Thus "left identity" and "right identity" are dual concepts. The dual of

Dickson's definition of a group is that a group is a semigroup G containing a

right identity e such that every element of G possesses a right inverse with

respect to e. The Weber-Huntington definition is self-dual.

Let d(A) denote the dual of a proposition A. If a proposition has the form

"A implies B" then its dual has the form "d(A) implies d(B)". Clearly, if

one is true then so is the other. Many nonself-dual theorems mill be estab-

lished in the book, and their duals will be taken for granted without comment.

If A and B are subsets of a groupoid S, then by the set product AB of A

and B we shall mean the set of all elements ab of 8 with a in A and b in B.

If A = {a} then we also write AB as aB, and similarly if B = {6}. Thus

AB = (J{^ : beB} =\J{aB : aeA}.

By a left [right] ideal of a groupoid S we mean a non-empty subset A of S

such that SA c J. [yl# c ^4], By two-sided ideal, or simply ideaZ, we mean

a subset of # which is both a left and a right ideal of S. A groupoid S

is called left [right] simple if 8 itself is the only left [right] ideal of S. Likewise

S is called simple if it contains no proper (two-sided) ideal.

If A is any non-empty subset of a groupoid 8, the intersection of all left

ideals of 8 containing A (8 itself being one such) is a left ideal of S containing

A and contained in every other such left ideal of 8. We call it the left ideal

of 8 generated by A. If 8 is a semigroup, the left ideal of 8 generated by A

is simply A USA =

SXA.

With the analogous definitions, we see that the

right ideal of S generated by A is A u AS =

AS1,

and that the (two-sided)

ideal of 8 generated by A is A U SA u AS U SAS =

S^AS1.

If, in particular,