Monogenic semigroup

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

The monogenic semigroup generated by the singleton set { a } is denoted by ${\displaystyle \langle a\rangle }$. The set of elements of ${\displaystyle \langle a\rangle }$ is { a, a², a³, ... }. There are two possibilities for the monogenic semigroup ${\displaystyle \langle a\rangle }$:

In the former case ${\displaystyle \langle a\rangle }$ is isomorphic to the semigroup ( {1, 2, ... }, + ) of natural numbers under addition. In such a case, ${\displaystyle \langle a\rangle }$ 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.

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