Semirings and $T_{1}$ compactifications. I

Abstract:

With each infinite cardinal ${\omega _\mu }$ is associated a topological semiring ${{\mathbf {F}}_\mu }$, whose underlying space is finite complement topology on the set of all ordinals less than ${\omega _\mu }$, and whose operations are the natural sum and natural product defined by Hessenberg. The theory of the semirings ${C_\mu }(X)$ of maps from a space X into ${{\mathbf {F}}_\mu }$ is developed in close analogy with the theory of the ring $C(X)$ of continuous real-valued functions; the analogy is not on the surface alone, but may be pursued in great detail. With each semiring a structure space is associated; the structure space of ${C_\mu }(X)$ for sufficiently large ${\omega _\mu }$ will be the Wallman compactification of X. The classes of ${\omega _\mu }$-entire and ${\omega _\mu }$-total spaces, which are respectively analogues of realcompact and pseudocompact spaces, are examined, and an ${\omega _\mu }$-entire extension analogous to the Hewitt realcompactification is constructed with the property (not possessed by the Wallman compactification) that every map between spaces has a unique extension to their ${\omega _\mu }$-entire extensions. The semiring of functions of compact-small support is considered, and shown to be related to the locally compact-small spaces in the same way that the ring of functions of compact support is related to locally compact spaces.