Monogenic semi-group

cyclic semi-group

A semi-group generated by one element. The monogenic semi-group generated by an element $a$ is usually denoted by $\langle a\rangle$ (sometimes by $[a]$) and consists of all powers $a^k$ with natural exponents. If all these powers are distinct, then $\langle a\rangle$ is isomorphic to the additive semi-group of natural numbers. Otherwise $\langle a\rangle$ is finite, and then the number of elements in it is called the order of the semi-group $\langle a\rangle$, and also the order of the element $a$. If $\langle a\rangle$ is infinite, then $a$ is said to have infinite order. For a finite monogenic semi-group $A=\langle a\rangle$ there is a smallest number $h$ with the property $a^h=a^k$, for some $k>h$; $h$ is called the index of the element $a$ (and also the index of the semi-group $A$). In this connection, if $d$ is the smallest number with the property $a^h=a^{h+d}$, then $d$ is called the period of $a$ (of $A$). The pair $(h,d)$ is called the type of $a$ (of $A$). For any natural numbers $h$ and $d$ there is a monogenic semi-group of type $(h,d)$; two finite monogenic semi-groups are isomorphic if and only if their types coincide. If $(h,d)$ is the type of a monogenic semi-group $A=\langle a\rangle$, then $a,\dots,a^{h+d-1}$ are distinct elements and, consequently, the order of $A$ is $h+d-1$; the set

$$G=\{a^h,\dots,a^{h+d-1}\}$$

is the largest subgroup and smallest ideal in $A$; the identity $e$ of the group $G$ is the unique idempotent in $A$, where $e=a^{ld}$ for any $l$ such that $ld\geq h$; $G$ is a cyclic group, a generator being, for example, $ae$. An idempotent of a monogenic semi-group is a unit (zero) in it if and only if its index (respectively, period) is equal to 1; this is equivalent to the given monogenic semi-group being a group (respectively, a nilpotent semi-group). Every sub-semi-group of the infinite monogenic semi-group is finitely generated.

References

[1]	A.H. Clifford, G.B. Preston, "The algebraic theory of semigroups" , 1 , Amer. Math. Soc. (1961)
[2]	E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)