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
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
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'',
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
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