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Keywords:
archimedean lattice; joinfit coreflection; infinitesimals; fitness conditions
archimedean lattice; joinfit coreflection; infinitesimals; fitness conditions
Summary:
This paper extends the notion of an archimedean frame to frames which are not necessarily algebraic. The new notion is called {\it joinfitness\/} and is {\it Choice-free\/}. Assuming the Axiom of Choice and for compact normal algebraic frames, the new and the old coincide. There is a subfunctor from the category of compact normal frames with skeletal maps with joinfit values, which is almost a coreflection. Conditions making it so are briefly discussed. The concept of an {\it infinitesimal\/} element arises naturally, and the join of suitably chosen infinitesimals defines the joinfit nucleus. The paper concludes with mostly Choice-free applications of these ideas to commutative rings and their radical ideals.
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