cyclic semigroup

A semigroup which is generated by a single element is called a cyclic semigroup.

Let S=x be a cyclic semigroup. Then as a set, S={xnn>0}.

If all powers of x are distinct, then S={x,x2,x3,} is (countably) infinite.

Otherwise, there is a least integer n>0 such that xn=xm for some m<n. It is clear then that the elements x,x2,,xn-1 are distinct, but that for any jn, we must have xj=xi for some i, min-1. So S has n-1 elements.

Unlike in the group case, however, there are in general multiple non-isomorphic cyclic semigroups with the same number of elements. In fact, there are t non-isomorphic cyclic semigroups with t elements: these correspond to the different choices of m in the above (with n=t+1).

The integer m is called the index of S, and n-m is called the period of S.

The elements K={xm,xm+1,,xn-1} are a subsemigroup of S. In fact, K is a cyclic groupMathworldPlanetmath.

A concrete representation of the semigroup with index m and period r as a semigroup of transformations can be obtained as follows. Let X={1,2,3,,m+r}. Let


Then ϕ generates a subsemigroup S of the full semigroup of transformations 𝒯X, and S is cyclic with index m and period r.

Title cyclic semigroup
Canonical name CyclicSemigroup
Date of creation 2013-03-22 13:07:30
Last modified on 2013-03-22 13:07:30
Owner mclase (549)
Last modified by mclase (549)
Numerical id 6
Author mclase (549)
Entry type Definition
Classification msc 20M99
Defines index
Defines period