1.1

BASIC DEFINITIONS

3

such) is a subgroupoid -4 of S containing A and contained in every other

subgroupoid of S containing A. We call -4 the subgroupoid of S generated

by A. The subgroupoid (A} can also be described as the set of all elements

of S expressible as finite products of elements of A. If (A} = S, then A will

be called a set of generators of S. If S is a semigroup, then any subgroupoid

of S is also a semigroup, and we shall use the term subsemigroup rather than

subgroupoid.

If S is a groupoid, the cardinal number \S\ of the set S is called the order

ofS. If \S\ is finite, we can exhibit the binary operation in S by means of its

Cayley multiplication table as for finite groups, and this is often a useful

picture even for infinite S. The Cayley table is a square matrix of elements

of S, the rows and columns of which are labelled by the elements of S, such

that the element in the a-row and 6-column (a, 6 in S) is the product ab.

An element a of a groupoid 8 is said to be left [right] cancellable if, for any

x and y in S, ax = ay [xa = ya] implies x = y. A groupoid S is called left

[right] cancellative if every element of S is left [right] cancellable. We say

that S is cancellative (or is a cancellation groupoid) if it is both left and right

cancellative.

Two elements a and b of a semigroup S are said to commute with each other

if ab = ba. If this is the case, the third "law of exponents'',

(ab)n

=

anbn,

holds. A semigroup S is called commutative if all of its elements commute

with each other. An element of a semigroup S which commutes with every

element of S is called a central element of S. The set of all central elements

of £ is either empty or a subsemigroup of 8, and in the latter case is called

the center of S. If a\, a^, • • •, an are elements of a commutative semigroup S,

and cf is any permutation of the set {1, 2, • • •, n}, then

a\$aL4* • -an$ = a\a^' • -an.

This is easily proved by induction on n.

An element e of a groupoid S is called a left [right] identity element of S if

ea = a [ae = a] for all a in S. An element e of S called a two-sided identity

(or simply identity) element ofS if it is both a left and a right identity element

of S. We note that if S contains a left identity e and a right identity/, then

e = / ; for ef = f since e is a left identity, and ef — e since / is a right identity.

As a consequence of this, we see that exactly one of the following statements

must hold for a groupoid 8:

(1) S has no left and no right identity element]

(2) S has one or more left identity elements, but no right identity element;

(3) S has one or more right identity elements, but no left identity element;

(4) S has a unique two-sided identity element, and no other right or left

identity element.

An element z of a groupoid 8 is called a left [right] zero element if za = z

[az = z] for every a in S. An element z of S is called a zero element of S if it