Editing Stone–Čech compactification (section)

===A monoid operation on the Stone–Čech compactification of the naturals===
The [[Natural number|natural numbers]] form a [[monoid]] under [[addition]]. It turns out that this operation can be extended (generally in more than one way, but uniquely under a further condition) to ''β'''''N''', turning this space also into a monoid, though rather surprisingly a non-commutative one.

For any subset, ''A'', of '''N''' and a positive integer ''n'' in '''N''', we define

:<math>A-n=\{k\in\mathbf{N}\mid k+n\in A\}.</math>

Given two ultrafilters ''F'' and ''G'' on '''N''', we define their sum by 

:<math>F+G = \Big\{A\subseteq\mathbf{N}\mid \{n\in\mathbf{N}\mid A-n\in F\}\in G\Big\};</math>

it can be checked that this is again an ultrafilter, and that the operation + is [[associative]] (but not commutative) on β'''N''' and extends the addition on '''N'''; 0 serves as a neutral element for the operation + on ''β'''''N'''. The operation is also right-continuous, in the sense that for every ultrafilter ''F'', the map

:<math>\begin{cases} \beta \mathbf{N} \to \beta \mathbf{N} \\ G \mapsto F+G \end{cases}</math>

is continuous.

More generally, if ''S'' is a [[semigroup]] with the discrete topology, the operation of ''S'' can be extended to ''βS'', getting a right-continuous associative operation.<ref>{{Cite book|last1=Hindman|first1=Neil|title=Algebra in the Stone-Cech Compactification|last2=Strauss|first2=Dona|date=2011-01-21|publisher=DE GRUYTER|isbn=978-3-11-025835-6|location=Berlin, Boston|doi=10.1515/9783110258356}}</ref>