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Semigroup
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== 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'' Β· ''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 β 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 Ξ£ with [[concatenation]] of strings as the semigroup operation β the so-called "[[free semigroup]] over Ξ£". With the empty string included, this semigroup becomes the [[free monoid]] over Ξ£. * 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]].
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