Monogenic semigroups

The monogenic semigroups are semigroups that are generated by a single element. All semigroups that are generated by at least two elements must contain distinct maximal proper subsemigroups corresponding to those elements. This means that all semigroups that contain only a single maximal proper subsemigroup are necessarily monogenic. Nonetheless, not all monogenic semigroups contain only one maximal proper subsemigroup.

The monogenic semigroups that contain only one maximal proper subsemigroup are called primary monogenic semigroups. Non-primary monogenic semigroups include cyclic groups that are of non-prime power orders. The prime power cyclic groups are called primary cyclic groups and hence they are primary monogenic semigroups. The primary monogenic semigroups can be used to generate the elements of any semigroup.