The ultrafilter algebra on a finite discrete space

If an ultrafilter on a nonempty finite set contained no singleton, [L1] would put the complement of every singleton into it; their finite intersection is empty, impossible for a filter. Thus it contains some singleton and is principal.

It cannot contain two distinct singletons because their intersection is empty, so the principal point is unique.

By [L2], ${\eta }_X$ is therefore a bijection and ${\xi }_X={\eta }_X^{-1}$. In the discrete topology [L3], an ultrafilter converges precisely to the point whose singleton it contains, so ${\xi }_X$ is the unique-limit map.

The equation ${\xi }_X{\eta }_X=1_X$ is immediate. Since ${\eta }_X$ is bijective by step 3.1, $\beta ({\eta }_X)$ is bijective with inverse $\beta ({\xi }_X)$. The monad unit law in [L4] says ${\mu }_X\beta ({\eta }_X)=1_{\beta X}$, so uniqueness of the inverse gives ${\mu }_X=\beta ({\xi }_X)$. Composing with ${\xi }_X$ yields ${\xi }_X{\mu }_X={\xi }_X\beta ({\xi }_X)$, and ${\xi }_X$ is an algebra.

If $X=\mathrm{\varnothing }$, no ultrafilter exists, so $\beta X=\mathrm{\varnothing }$ and the unique empty map is an algebra. For $X=\{0,1\}$, steps 1.1 and 2.1 give exactly ${\eta }_X(0),{\eta }_X(1)$ and step 3.1 gives their displayed values.