A
semi-group
generated by one element. The monogenic semi-group generated by an element
is usually denoted by
(sometimes by
)
and consists of all powers
with natural exponents. If all these powers are distinct, then
is isomorphic to the additive semi-group of natural numbers. Otherwise
is finite, and then the number of elements in it is called the
order of the semi-group
,
and also the
order of the element
.
If
is infinite, then
is said to have
infinite order.
For a finite monogenic semi-group
there is a smallest number
with the property
,
for some
;
is called the
index of the element
(and also
the index of the semi-group
).
In this connection, if
is the smallest number with the property
,
then
is called the
period
of
(of
).
The pair
is called the
type
of
(of
).
For any natural numbers
and
there is a monogenic semi-group of type
;
two finite monogenic semi-groups are isomorphic if and only if their types coincide. If
is the type of a monogenic semi-group
,
then
are distinct elements and, consequently, the order of
is
;
the set
 |
is the largest subgroup and smallest ideal in
;
the identity
of the group
is the unique
idempotent
in
,
where
for any
such that
;
is a
cyclic group,
a generator being, for example,
.
An idempotent of a monogenic semi-group is a unit (zero) in it
if and only if its index (respectively, period) is equal to
1; this is equivalent to the given monogenic semi-group being a
group
(respectively, a
nilpotent semi-group).
Every sub-semi-group of the infinite monogenic semi-group is finitely generated.