Monogenic semigroup
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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 {a} is denoted by . The set of elements of is {a, a2, a3, ...}. 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, a2, ... , a m + r − 1}
- Ka = {am, 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
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
- ^ Howie, J M (1976). An Introduction to Semigroup Theory. L.M.S. Monographs. Vol. 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. Vol. 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.