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subsemigroup

A subsemigroup is a basic concept in semigroup theory. Let (S, ·) be a semigroup. A subset T ⊆ S is called a subsemigroup if T is nonempty and closed under the semigroup operation: for all a, b ∈ T, a · b ∈ T. In other words, (T, ·) forms a semigroup in its own right.

Relation to other notions: If S is a monoid (a semigroup with an identity element e), a

Examples: In the additive semigroup of integers (Z, +), the nonnegative integers (N, +) form a subsemigroup. The

Generated subsemigroups: For a subset A ⊆ S, the subsemigroup generated by A, denoted ⟨A⟩, is the

Basic properties: The intersection of any collection of subsemigroups is a subsemigroup, but the union of subsemigroups

subsemigroup
need
not
contain
e,
and
thus
is
not
necessarily
a
submonoid.
A
subsemigroup
that
does
contain
the
identity
is
a
submonoid.
even
integers
2Z
form
a
subsemigroup
of
(Z,
+).
In
the
multiplicative
semigroup
of
natural
numbers
(N,
×),
the
set
of
all
powers
of
2,
{1,
2,
4,
8,
…},
is
a
subsemigroup.
smallest
subsemigroup
containing
A.
It
consists
of
all
finite
products
of
elements
of
A
(with
repetition).
If
S
is
a
monoid,
often
the
identity
is
included
in
the
generated
substructure.
need
not
be.
Subsemigroups
are
essential
for
understanding
the
internal
structure
of
semigroups
and
have
applications
in
areas
such
as
automata
theory
and
algebraic
combinatorics.