# Monogenic semigroup

In mathematics, a **monogenic semigroup** is a semigroup generated by a single element.^{[1]} Monogenic semigroups are also called **cyclic semigroups**.^{[2]}

## Structure[edit]

The monogenic semigroup generated by the singleton set { *a* } is denoted by . The set of elements of is { *a*, *a*^{2}, *a*^{3}, ... }. There are two possibilities for the monogenic semigroup :

*a*^{ m}=*a*^{ n}⇒*m*=*n*.- There exist
*m*≠*n*such that*a*^{ m}=*a*^{ n}.

In the former case is isomorphic to the semigroup ( {1, 2, ... }, + ) of natural numbers under addition. In such a case, 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 . The **order** of *a* is defined as *m*+*r*-1. The period and the index satisfy the following properties:

*a*^{ m}=*a*^{ m + r}*a*^{ m + x}=*a*^{ m + y}if and only if*m*+*x*≡*m*+*y*( mod*r*)- = {
*a*,*a*^{2}, ... ,*a*^{ m + r − 1}} *K*_{a}= {*a*^{m},*a*^{ m + 1}, ... ,*a*^{ m + r − 1}} is a cyclic subgroup and also an ideal of . It is called the*kernel*of*a*and it is the minimal ideal of the monogenic semigroup .^{[3]}^{[4]}

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 it generates.

## Related notions[edit]

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.^{[5]}^{[6]}

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

## See also[edit]

- Cycle detection, the problem of finding the parameters of a finite monogenic semigroup using a bounded amount of storage space
- Special classes of semigroups

## References[edit]

**^**Howie, J M (1976).*An Introduction to Semigroup Theory*. L.M.S. Monographs.**7**. Academic Press. pp. 7–11. ISBN 0-12-356950-8.**^**A H Clifford; G B Preston (1961).*The Algebraic Theory of Semigroups Vol.I*. Mathematical Surveys.**7**. American Mathematical Society. pp. 19–20. ISBN 978-0821802724.**^**http://www.encyclopediaofmath.org/index.php/Kernel_of_a_semi-group**^**http://www.encyclopediaofmath.org/index.php/Minimal_ideal**^**http://www.encyclopediaofmath.org/index.php/Periodic_semi-group**^**Peter M. Higgins (1992).*Techniques of semigroup theory*. Oxford University Press. p. 4. ISBN 978-0-19-853577-5.