The classical intermediate value theorem for polynomials with real coefficients is generalized to the case of polynomials with coefficients in a lattice-ordered ring that is a subdirect product of totally ordered rings. Several candidates for a generalization are investigated, and particular attention is paid to the case when the lattice-ordered ring is the algebra C(X) of continuous real-valued functions on a completely regular topological space X. For all but one of these generalizations, the intermediate value theorem holds only if X is an F-space in the sense of Gillman and Jerison. Surprisingly, for the most interesting of these generalizations, if X is compact, the intermediate value theorem holds only if X is an F-space and each component of X is an hereditarily indecomposable continuum. It is not known if there is an infinite compact connected space in which this version of the intermediate value theorem holds

Topics:
intermediate value theorem, IVT ring, IVT space, strong IVT ring, strong IVT space, f-ring, semiprime f-ring, 1-convex f-ring, maximal ideal, prime ideal, minimal prime ideal, F-space, zero-dimensional, strongly zero-dimensional, compact space, Lindelöf space, Stone-Čech compactification, connected space, component, continuum, indecomposable continuum, hereditarily indecomposable continuum, valuation domain, SV-space, Mathematics, Physical Sciences and Mathematics

Publisher: 'Wiley'

Year: 1996

DOI identifier: 10.1111/j.1749-6632.1996.tb36802.x/abstract

OAI identifier:
oai:scholarship.claremont.edu:hmc_fac_pub-2009

Provided by:
Scholarship@Claremont

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