Monogenic semigroup

[2] The monogenic semigroup generated by the singleton set {a} is denoted by

is isomorphic to the semigroup ({1, 2, ...}, +) of natural numbers under addition.

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 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

A related notion is that of periodic semigroup (also called torsion semigroup), in which every element has finite order (or, equivalently, in which every monogenic subsemigroup is finite).

[5][6] An aperiodic semigroup is one in which every monogenic subsemigroup has a period of 1.