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Monogenic semigroup
In mathematics, a monogenic semigroup is a semigroup generated by a set containing only 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 \( \langle a \rangle \) . The set of elements of \( \langle a \rangle \) is { a, a2, a3, ... }. There are two possibilities for the monogenic semigroup\( \langle a \rangle \) :
- am = an ⇒ m = n.
- There exist m ≠ n such that a m = a n.
In the former case \( \langle a \rangle \) is isomorphic to the semigroup ( {1, 2, ... }, + ) 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:
am = a m+r
a m+x = a m+y if and only if m + x ≡ m + y ( mod r )
\(\langle a \rangle \) = { a, a2, ... , a m + r − 1 }
Ka = { am, a m + 1, ... , a m + r − 1 } 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 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.[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 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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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