Monogenic semigroup

In mathematics, a monogenic semigroup is a semigroup generated by a single element. Monogenic semigroups are also called cyclic semigroups.

Structure
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 $\langle a \rangle$: 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.
 * am = an ⇒ m = n.
 * There exist m ≠ n such that am = an.

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
A related notion is that of periodic semigroup (also called torsion semigroup), in which every element has finite order (or, equivalently, in which every mongenic subsemigroup is finite). A more general class is that of quasi-periodic semigroups (aka group-bound semigroups or epigroups) in which every element of the semigroup has a power that lies in a subgroup.

An aperiodic semigroup is one in which every monogenic subsemigroup has a period of 1.