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Infinite-dimensional idempotent analysis: the role of continuous posets

Abstract : Idempotent analysis involves the study of infinite-dimensional linear spaces in which the usual addition is replaced by the maximum operation. We prove a series of results in this framework and stress the crucial contribution of domain and continuous lattice theory. Two themes are considered: integration and convexity. In idempotent integration, the properties of domain-valued maxitive measures such as regularity are surveyed and completed in a topological framework; we provide a converse statement to the idempotent Radon-Nikodym theorem; using the Z generalization of domain theory we gather and surpass existing results on the representation of continuous linear forms on an idempotent module. In tropical convexity, we obtain a Krein-Milman type theorem in several ordered algebraic structures, including locally-convex topological semilattices and idempotent modules; in the latter structure we prove a Choquet integral representation theorem: every point of a compact convex subset K can be represented by a possibility measure supported by the extreme points of K. The hope for a unification of classical and idempotent analysis is considered in a final step. The notion of inverse semigroup, which fairly generalizes both groups and semilattices, may be the right candidate for this; in this perspective we examine "mirror" properties between inverse semigroups and semilattices, among which continuity. The general conclusion broadens this point of view.
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https://pastel.archives-ouvertes.fr/pastel-00666633
Contributor : Paul Poncet <>
Submitted on : Sunday, February 5, 2012 - 9:36:08 PM
Last modification on : Wednesday, March 27, 2019 - 4:08:31 PM
Long-term archiving on: : Sunday, May 6, 2012 - 2:22:22 AM

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  • HAL Id : pastel-00666633, version 1

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Paul Poncet. Infinite-dimensional idempotent analysis: the role of continuous posets. Functional Analysis [math.FA]. Ecole Polytechnique X, 2011. English. ⟨pastel-00666633⟩

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