Monogenic semigroup explained

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

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

• a^m = aⁿ ⇒ m = n.
• There exist m ≠ n such that a^m = aⁿ.

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 .^{[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

\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.^{[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. Book: Howie, J M. An Introduction to Semigroup Theory. Academic Press. 1976. L.M.S. Monographs. 7. 7 - 11. 0-12-356950-8.
2. Book: A H Clifford. G B Preston. The Algebraic Theory of Semigroups Vol.I. American Mathematical Society. 1961. Mathematical Surveys. 7. 19 - 20. 978-0821802724.
3. Web site: Kernel of a semi-group - Encyclopedia of Mathematics.
4. Web site: Minimal ideal - Encyclopedia of Mathematics.
5. Web site: Periodic semi-group - Encyclopedia of Mathematics.
6. Book: Peter M. Higgins. Techniques of semigroup theory. 1992. Oxford University Press. 978-0-19-853577-5. 4.