Cyclic 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 is denoted by $\langle a \rangle$. The set of elements of $\langle a \rangle$ is . There are two possibilities for the monogenic semigroup $\langle a \rangle$:
* ''a'' ^''m'' = ''a'' ^''n'' ⇒ ''m'' = ''n''.
* There exist ''m'' ≠ ''n'' such that ''a'' ^''m'' = ''a'' ^''n''.
In the former case $\langle a \rangle$ is isomorphic to the semigroup ( , + ) 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.

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 $\langle a \rangle$. 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'' )
* $\langle a \rangle$ =
* ''K''_''a'' = 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 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 $\langle a \rangle$ 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.{{cite book, author=Peter M. Higgins, title=Techniques of semigroup theory, year=1992, publisher=Oxford University Press, isbn=978-0-19-853577-5, page=4

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

See also

* 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

Algebraic structures
Semigroup theory