Monogenic semigroup

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

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 $\langle a \rangle$:

• a ^m = a ⁿ ⇒ 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:

• a ^m = 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} }
• K_a = { a^m, 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 $\langle a \rangle$.

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.

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