Monogenic semigroup

In mathematics, a monogenic semigroup is a semigroup generated by 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 ${\displaystyle \langle a\rangle }$. The set of elements of ${\displaystyle \langle a\rangle }$ is { a, a2, a3, ... }. There are two possibilities for the monogenic semigroup ${\displaystyle \langle a\rangle }$:

• a m = a nm = n.
• There exist mn such that a m = a n.

In the former case ${\displaystyle \langle a\rangle }$ is isomorphic to the semigroup ( {1, 2, ... }, + ) of natural numbers under addition. In such a case, ${\displaystyle \langle a\rangle }$ 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 xm, 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 ${\displaystyle \langle a\rangle }$. 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 + xm + y ( mod r )
• ${\displaystyle \langle a\rangle }$ = { a, a2, ... , a m + r − 1 }
• Ka = { am, a m + 1, ... , a m + r − 1 } is a cyclic subgroup and also an ideal of ${\displaystyle \langle a\rangle }$. It is called the kernel of a and it is the minimal ideal of the monogenic semigroup ${\displaystyle \langle a\rangle }$.[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 ${\displaystyle \langle a\rangle }$ 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.