Semigroup - Structure of Semigroups

Structure of Semigroups

For any subset A of S there is a smallest subsemigroup T of S which contains A, and we say that A generates T. A single element x of S generates the subsemigroup { xn | n is a positive integer }. If this is finite, then x is said to be of finite order, otherwise it is of infinite order. A semigroup is said to be periodic if all of its elements are of finite order. A semigroup generated by a single element is said to be monogenic (or cyclic). If a monogenic semigroup is infinite then it is isomorphic to the semigroup of positive integers with the operation of addition. If it is finite and nonempty, then it must contain at least one idempotent. It follows that every nonempty periodic semigroup has at least one idempotent.

A subsemigroup which is also a group is called a subgroup. There is a close relationship between the subgroups of a semigroup and its idempotents. Each subgroup contains exactly one idempotent, namely the identity element of the subgroup. For each idempotent e of the semigroup there is a unique maximal subgroup containing e. Each maximal subgroup arises in this way, so there is a one-to-one correspondence between idempotents and maximal subgroups. Here the term maximal subgroup differs from its standard use in group theory.

More can often be said when the order is finite. For example, every nonempty finite semigroup is periodic, and has a minimal ideal and at least one idempotent. For more on the structure of finite semigroups, see Krohn–Rhodes theory.