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Concept
Monogenic semigroup
Summary
In mathematics, a monogenic semigroup is a semigroup generated by a single element. Monogenic semigroups are also called cyclic semigroups.
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 :
am = an ⇒ m = n.
There exist m ≠ n such that am = an.
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:
am = am+r
am+x = am+y if and only if m + x ≡ m + y (mod r)
= {a, a2, ... , am+r−1}
Ka = {am, am+1, ... , am+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 .
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).
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