Patch-generated Frames and Projectable Hulls

13 years 4 months ago

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This article considers coherent frame homomorphisms h : L −→ M between coherent frames, which induce an isomorphism between the boolen frames of polars, with M projectable, and such that M is generated by certain complemented elements of M. This abstracts the passage from a semiprime commutative ring with identity to its projectable hull. The frame theoretic setting is investigated thoroughly, ﬁrst without any assumptions beyond the Zermelo-Fraenkel axioms of set theory, and, subsequently, assuming that algebraic frames are spatial. The culmination of this eﬀort is the result that the spectrum of d-elements of M is obtained from that of L by reﬁning the given hull-kernel topology to the patch topology. The second part of the article relates the projectable hull to the (von Neumann) regular hull, in a variety of contexts, including that of f-rings. For a uniformly complete f-algebra A, it is shown that the maximal -ideals of A that are traces of real maximal ideals of the regu...

Anthony W. Hager, Jorge Martínez

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