monogenic semigroup

The monogenic semigroup generated by the singleton set {a} is denoted by $\backslash langle\; a\; \backslash rangle$. The set of elements of $\backslash langle\; a\; \backslash rangle$ is {a, a², a³, ...}. There are two possibilities for the monogenic semigroup {{nowrap|$\backslash langle\; a\; \backslash rangle$:}}

• a^m = aⁿ ⇒ m = n.
• There exist m ≠ n such that a^m = aⁿ.

In the former case $\backslash langle\; a\; \backslash rangle$ is isomorphic to the semigroup ({1, 2, ...}, +) of natural numbers under addition. In such a case, $\backslash langle\; a\; \backslash rangle$ 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 $\backslash langle\; a\; \backslash rangle$. 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)
• $\backslash langle\; a\; \backslash rangle$ = {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 $\backslash langle\; a\; \backslash rangle$. It is called the kernel of a and it is the minimal ideal of the monogenic semigroup $\backslash langle\; a\; \backslash rangle$.{{Cite web|url=http://www.encyclopediaofmath.org/index.php/Kernel_of_a_semi-group|title=Kernel of a semi-group - Encyclopedia of Mathematics}}{{Cite web|url=http://www.encyclopediaofmath.org/index.php/Minimal_ideal|title=Minimal ideal - Encyclopedia of Mathematics}}

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 $\backslash langle\; a\; \backslash rangle$ 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 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.{{Cite web|url=http://www.encyclopediaofmath.org/index.php/Periodic_semi-group|title=Periodic semi-group - Encyclopedia of Mathematics}}{{cite book|author=Peter M. Higgins|title=Techniques of semigroup theory|year=1992|publisher=Oxford University Press|isbn=978-0-19-853577-5|page=4}}

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

• Cycle detection, the problem of finding the parameters of a finite monogenic semigroup using a bounded amount of storage space
• Special classes of semigroups

Category:Algebraic structures

Category:Semigroup theory