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Monogenic semigroup

In mathematics, a monogenic semigroup is a semigroup generated by a single element. Monogenic semigroups are also called cyclic semigroups.

Structure

The monogenic semigroup generated by the singleton set {a} is denoted by . The set of elements of is {a, a<sup>2</sup>, a<sup>3</sup>, ...}. There are two possibilities for the monogenic semigroup

  • a<sup>m</sup> = a<sup>n</sup> ⇒ m = n.
  • There exist m ≠ n such that a<sup>m</sup> = a<sup>n</sup>.

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<sup>m</sup> = a<sup>x</sup> for some positive integer x ≠ m, and let r be smallest positive integer such that a<sup>m</sup> = a<sup>m+r</sup>. 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<sup>m</sup> = a<sup>m+r</sup>
  • a<sup>m+x</sup> = a<sup>m+y</sup> if and only if m + x ≡ m + y (mod r)
  • = {a, a<sup>2</sup>, ... , a<sup>m+r−1</sup>}
  • K<sub>a</sub> = {a<sup>m</sup>, a<sup>m+1</sup>, ... , a<sup>m+r−1</sup>} 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.

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 monogenic 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.

An aperiodic semigroup is one in which every monogenic subsemigroup has a period of 1.

See also

References