Editing Semigroup (section)

== Examples of semigroups ==
* [[Empty semigroup]]: the [[empty set]] forms a semigroup with the [[empty function]] as the binary operation.
* [[Semigroup with one element]]: there is essentially only one (specifically, only one [[up to]] [[isomorphism]]), the singleton {''a''} with operation {{nowrap|1=''a'' · ''a'' = ''a''}}.
* [[Semigroup with two elements]]: there are five that are essentially different.
* A [[null semigroup]] on any nonempty set with a chosen zero, or a [[null semigroup|left/right zero semigroup]] on any set.
* The "flip-flop" monoid: a [[semigroup with three elements]] representing the three operations on a switch – set, reset, and do nothing.
* The set of positive [[integer]]s with addition. (With 0 included, this becomes a [[monoid]].)
* The set of [[integer]]s with minimum or maximum. (With positive/negative infinity included, this becomes a monoid.)
* Square [[Nonnegative matrix|nonnegative matrices]] of a given size with matrix multiplication.
* Any [[ring ideal|ideal]] of a [[ring (algebra)|ring]] with the multiplication of the ring.
* The set of all finite [[string (computer science)|strings]] over a fixed alphabet Σ with [[concatenation]] of strings as the semigroup operation – the so-called "[[free semigroup]] over Σ". With the empty string included, this semigroup becomes the [[free monoid]] over Σ.
* A [[probability distribution]] ''F'' together with all [[convolution power]]s of ''F'', with convolution as the operation. This is called a convolution semigroup.
* [[Transformation semigroup]]s and [[transformation monoid|monoids]].
* The set of [[continuous function]]s  from a [[topological space]] to itself with composition of functions forms a monoid with the [[identity function]] acting as the identity. More generally, the [[Endomorphism|endomorphisms]] of any object of a [[Category (mathematics)|category]] form a monoid under composition.
* The product of faces of an [[arrangement of hyperplanes]].