Subdirectly Irreducible Semigroups Page: 2
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2
= al(a2---ar-ar+1---a)
= a1a2 --an.
Thus by induction, S satisfies the general associative law,
and so all parentheses may be omitted from products of ele-
ments of a semigroup.
Definition 1.3. A nonempty subset T of a semigroup S
is a subsemigroup of S iff T is closed under the operation
on S (if a,beT, then abeT).
Thus a subsemigroup T of a semigroup S, along with the
multiplication of S, is itself a semigroup since associa-
tivity is inherited from S.
Definition 1.4. A semigroup S is generated by a subset
G of S iff every element of S can be expressed as the product
of elements of G.
Definition 1.5. A semigroup S is cyclic iff there
exists asS such that S is generated by {a}.
Definition 1.6. If A is a nonempty subset of a semi-
group S, then the subsemigroup of S generated by
A is {a1a2- -e-an j aiEA, l<i<n; nEZ+}, where Z+ is the
set of all positive integers.
Lemma 1.7. If A is a nonempty subset of a semigroup S,
then the subsemigroup of S generated by A is the intersection
of all subsemigroups of S containing A.
n +
Proof. Let T ='{ II ai j nEZ; aA, in} and let
i=l
{G } {G subsemigroup of S I ASG}.
a~cYE
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Winton, Richard Alan. Subdirectly Irreducible Semigroups, thesis, December 1978; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc504365/m1/4/?rotate=270: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .