Editing Semigroup (section)

== Structure of semigroups ==
For any subset ''A'' of ''S'' there is a smallest subsemigroup ''T'' of ''S'' that contains ''A'', and we say that ''A'' '''generates''' ''T''. A single element ''x'' of ''S'' generates the subsemigroup {{math|{{mset| ''x''<sup>''n''</sup> | ''n'' ∈ '''Z'''<sup>+</sup> }}}}. 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 semigroup|monogenic]] (or [[Cyclic semigroup|cyclic]]). If a monogenic semigroup is infinite then it is isomorphic to the semigroup of positive [[integer]]s 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 that 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 (ring theory)|ideal]] and at least one idempotent. The number of finite semigroups of a given size (greater than 1) is (obviously) larger than the number of groups of the same size. For example, of the sixteen possible "multiplication tables" for a set of two elements {{math|{{mset|''a'', ''b''}}}}, eight form semigroups{{efn|Namely: the trivial semigroup in which (for all ''x'' and ''y'') {{math|1=''xy'' = ''a''}} and its counterpart in which {{math|1=''xy'' = ''b''}}, the semigroups based on multiplication modulo 2 (choosing a or b as the identity element 1), the groups equivalent to addition modulo 2 (choosing a or b to be the identity element 0), and the semigroups in which the elements are either both left identities or both right identities.}} whereas only four of these are monoids and only two form groups. For more on the structure of finite semigroups, see ''[[Krohn–Rhodes theory]]''.