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 formula_1. The set of elements of formula_1 is {"a", "a"2, "a"3, ...}. There are two possibilities for the monogenic semigroup formula_1:
In the former case formula_1 is isomorphic to the semigroup ( {1, 2, ...}, + ) of natural numbers under addition. In such a case, formula_1 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 "a" "m" = "a" "x" for some positive integer "x" ≠ "m", and let "r" be smallest positive integer such that "a" "m" = "a" "m" + "r". The positive integer "m" is referred to as the index and the positive integer "r" as the period of the monogenic semigroup formula_1. The order of "a" is defined as "m"+"r"-1. The period and the index satisfy the following properties:
The pair ( "m", "r" ) of positive integers determine the structure of monogenic semigroups. For every pair ( "m", "r" ) of positive integers, there does exist 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 formula_1 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.