Monogenic semigroup

Monogenic semigroup of order 9 and period 6. Numbers are exponents of the generator a; arrows indicate multiplication by a.

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

## Contents

* 1 Structure
* 2 Related notions
* 3 See also
* 4 References

## Structure

The monogenic semigroup generated by the singleton set {a} is denoted by ⟨a⟩. The set of elements of ⟨a⟩ is {a, a2, a3, ...}. There are two possibilities for the monogenic semigroup ⟨a⟩:

* am = an ⇒ m = n.
* There exist m ≠ n such that am = an.

In the former case ⟨a⟩ is isomorphic to the semigroup ({1, 2, ...}, +) of natural numbers under addition. In such a case, ⟨a⟩ 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 am = ax for some positive integer x ≠ m, and let r be smallest positive integer such that am = am+r. The positive integer m is referred to as the index and the positive integer r as the period of the monogenic semigroup ⟨a⟩. The order of a is defined as m+r−1. The period and the index satisfy the following properties:

* am = am+r
* am+x = am+y if and only if m \+ x ≡ m \+ y (mod r)
* ⟨a⟩ = {a, a2, ... , am+r−1}
* Ka = {am, am+1, ... , am+r−1} is a cyclic subgroup and also an ideal of ⟨a⟩. It is called the kernel of a and it is the minimal ideal of the monogenic semigroup ⟨a⟩.[3][4]

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⟩ 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.[5][6]

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

1. ↑ Howie, J M (1976). An Introduction to Semigroup Theory. L.M.S. Monographs. 7. Academic Press. pp. 7-11. ISBN 0-12-356950-8.
2. ↑ 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.
3. ↑ "Kernel of a semi-group - Encyclopedia of Mathematics". http://www.encyclopediaofmath.org/index.php/Kernel_of_a_semi-group.
4. ↑ "Minimal ideal - Encyclopedia of Mathematics". http://www.encyclopediaofmath.org/index.php/Minimal_ideal.
5. ↑ "Periodic semi-group - Encyclopedia of Mathematics". http://www.encyclopediaofmath.org/index.php/Periodic_semi-group.
6. ↑ Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press. p. 4. ISBN 978-0-19-853577-5.

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