The monogenic semigroup generated by the
singleton set {
a} is denoted by \langle a \rangle. The set of elements of \langle a \rangle is {
a,
a2,
a3, ...}. There are two possibilities for the monogenic semigroup •
am =
an ⇒
m =
n. • There exist
m ≠
n such that
am =
an. In the former case \langle a \rangle is
isomorphic to the semigroup ({1, 2, ...}, +) of
natural numbers under
addition. In such a case, \langle a \rangle is an
infinite monogenic semigroup and the element
a is said to have
infinite order. It is sometimes called the
free monogenic semigroup because it is also a
free semigroup with one generator. In the latter case let
m be the smallest positive
integer such that
am =
ax for some positive integer
x ≠
m, and let
r be smallest positive integer such that
am =
am+
r. The positive integer
m is referred to as the
index and the positive integer
r as the
period of the monogenic semigroup \langle a \rangle . The
order of
a is defined as
m+
r−1. The period and the index satisfy the following properties: •
am =
am+
r •
am+
x =
am+
y if and only if m +
x ≡
m +
y (mod
r) • \langle a \rangle = {
a,
a2, ... ,
am+
r−1} •
Ka = {
am,
am+1, ... ,
am+
r−1} is a
cyclic subgroup and also an
ideal of \langle a \rangle. It is called the
kernel of
a and it is the
minimal ideal of the monogenic semigroup \langle a \rangle . The pair (
m,
r) of positive integers determine the structure of monogenic semigroups. For every pair (
m,
r) of positive integers, there exists a monogenic semigroup having index
m and period
r. The monogenic semigroup having index
m and period
r is denoted by
M(
m,
r). The monogenic semigroup
M(1,
r) is the
cyclic group of order
r. The results in this section
actually hold for any element
a of an arbitrary semigroup and the monogenic subsemigroup \langle a \rangle it generates. ==Related notions==