In mathematics, a monogenic semigroup is a semigroup generated by a single element. [1] Monogenic semigroups are also called cyclic semigroups. [2]
In mathematics, a monogenic semigroup is a semigroup generated by a single element. [1] Monogenic semigroups are also called cyclic semigroups. [2]
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 :
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 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 . The order of a is defined as m+r−1. The period and the index satisfy the following properties:
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 it generates.
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.
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