On The Function Rings of Pointfree Topology
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The purpose of this note is to compare the rings of continuous functions, integer-valued or real-valued, in pointfree topology with those in classical topology. To this end, it first characterizes the Boolean frames (= complete Boolean algebras) whose function rings are isomorphic to a classical one and then employs this to exhibit a large class of frames for which the functions rings are not of this kind. An interesting feature of the considerations involved here is the use made of nonmeasurable cardinals. In addition, the integer-valued function rings for Boolean frames are described in terms of internal lattice-ordered ring properties
Abstract
The purpose of this note is to compare the rings of continuous functions, integer-valued or real-valued, in pointfree topology with those in classical topology. To this end, it first characterizes the Boolean frames (= complete Boolean algebras) whose function rings are isomorphic to a classical one and then employs this to exhibit a large class of frames for which the functions rings are not of this kind. An interesting feature of the considerations involved here is the use made of nonmeasurable cardinals. In addition, the integer-valued function rings for Boolean frames are described in terms of internal lattice-ordered ring properties
참고문헌 (8)
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- A. L. Foster, "Generalized "Boolean" theory of universal algebras II: Identities and subdirect sums of functionally complete algebras", Math. Z, 59, pp.191~199 (1953)
- L. Gillman, "Rings of Continuous Functions" (1960), Van Nostrand Reinhold Company, New York
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