Semirings and compactifications. I

Author:
Douglas Harris

Journal:
Trans. Amer. Math. Soc. **188** (1974), 241-258

MSC:
Primary 54D35; Secondary 54C40

MathSciNet review:
0365492

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Abstract: With each infinite cardinal is associated a topological semiring , whose underlying space is finite complement topology on the set of all ordinals less than , and whose operations are the natural sum and natural product defined by Hessenberg. The theory of the semirings of maps from a space *X* into is developed in close analogy with the theory of the ring 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 for sufficiently large will be the Wallman compactification of *X*. The classes of -entire and -total spaces, which are respectively analogues of realcompact and pseudocompact spaces, are examined, and an -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 -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.

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DOI:
https://doi.org/10.1090/S0002-9947-1974-0365492-4

Article copyright:
© Copyright 1974
American Mathematical Society