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
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
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, means GuO, i.e.,
G with a zero element adjoined. By a group with zero we mean 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 =
With the analogous definitions, we see that the
right ideal of S generated by A is A u AS =
and that the (two-sided)
ideal of 8 generated by A is A U SA u AS U SAS =
If, in particular,
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