Let be a cyclic semigroup. Then as a set, .
If all powers of are distinct, then is (countably) infinite.
Otherwise, there is a least integer such that for some . It is clear then that the elements are distinct, but that for any , we must have for some , . So has 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 non-isomorphic cyclic semigroups with elements: these correspond to the different choices of in the above (with ).
The integer is called the index of , and is called the period of .
Then generates a subsemigroup of the full semigroup of transformations , and is cyclic with index and period .
|Date of creation||2013-03-22 13:07:30|
|Last modified on||2013-03-22 13:07:30|
|Last modified by||mclase (549)|