Editing Semigroup (section)

== Special classes of semigroups ==
{{Main|Special classes of semigroups}}
* A [[monoid]] is a semigroup with an [[identity element]].
* A [[group (mathematics)|group]] is a monoid in which every element has an [[inverse element]].
* A subsemigroup is a [[subset]] of a semigroup that is closed under the semigroup operation.
* A [[cancellative semigroup]] is one having the [[cancellation property]]:{{sfn|ps=|Clifford|Preston|2010|p=3}} {{nowrap|1=''a'' · ''b'' = ''a'' · ''c''}} implies {{nowrap|1=''b'' = ''c''}} and similarly for {{nowrap|1=''b'' · ''a'' = ''c'' · ''a''}}. Every group is a cancellative semigroup, and every finite cancellative semigroup is a group.
* A [[band (algebra)|band]] is a semigroup whose operation is [[idempotent]].
* A [[semilattice]] is a semigroup whose operation is idempotent and [[commutativity|commutative]].
* [[0-simple]] semigroups.
* [[Transformation semigroup]]s: any finite semigroup ''S'' can be represented by transformations of a (state-) set ''Q'' of at most {{nowrap|{{abs|''S''}} + 1}} states. Each element ''x'' of ''S'' then maps ''Q'' into itself {{nowrap|''x'' : ''Q'' → ''Q''}} and sequence ''xy'' is defined by {{nowrap|1=''q''(''xy'') = (''qx'')''y''}} for each ''q'' in ''Q''. Sequencing clearly is an associative operation, here equivalent to [[function composition]]. This representation is basic for any [[automaton]] or [[finite-state machine]] (FSM).
* The [[bicyclic semigroup]] is in fact a monoid, which can be described as the [[free semigroup]] on two generators ''p'' and ''q'', under the relation {{nowrap|1=''pq'' = 1}}.
* [[C0-semigroup|C<sub>0</sub>-semigroups]].
* [[Regular semigroup]]s. Every element ''x'' has at least one inverse ''y'' that satisfies {{nowrap|1=''xyx'' = ''x''}} and {{nowrap|1=''yxy'' = ''y''}}; the elements ''x'' and ''y'' are sometimes called "mutually inverse".
* [[Inverse semigroup]]s are regular semigroups where every element has exactly one inverse. Alternatively, a regular semigroup is inverse if and only if any two idempotents commute.
* Affine semigroup: a semigroup that is isomorphic to a finitely-generated subsemigroup of Z<sup>d</sup>.  These semigroups have applications to [[commutative algebra]].