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The space S of all non-trivial real places on a real function field K|k of trascendence degree one, endowed with a natural topology analogous to that of Dedekind and Weber's Riemann surface, is shown to be a one-dimensional k-analytic manifold, which is homeomorphic with every bounded non-singular real affine model of K|k. The ground field k is an arbitrary ordered, real-closed Cantor field (definition below). The function field K|k is thereby represented as a field of real mappings of S which might...

Relatively complete ordered fields without integer parts

Mojtaba Moniri, Jafar S. Eivazloo (2003)

Fundamenta Mathematicae

We prove a convenient equivalent criterion for monotone completeness of ordered fields of generalized power series $[[F^{G}]]$ with exponents in a totally ordered Abelian group G and coefficients in an ordered field F. This enables us to provide examples of such fields (monotone complete or otherwise) with or without integer parts, i.e. discrete subrings approximating each element within 1. We include a new and more straightforward proof that $[[F^{G}]]$ is always Scott complete. In contrast, the Puiseux series field...

Remarks on the Pythagoras and Hasse number of real fields.

Alexander Prestel (1978)

Journal für die reine und angewandte Mathematik

Separating families for semi-algebraic sets.

Murray A. Marshall (1993)

Manuscripta mathematica

Separation by hyperplanes in finite-dimensional vector spaces over Archimedean ordered fields.

Gritzmann, Peter, Klee, Victor (1998)

Journal of Convex Analysis

Solution d'un problème d'Erdös, Gillman et Henriksen et application à l'étude des homomorphismes de C-(K).

Jean ESTERLE (1975/1976)

Seminaire de Théorie des Nombres de Bordeaux

Some euclidean properties for real quadratic fields

Daniel Shapiro, Raj Markanda, Ezra Brown (1986)

Acta Arithmetica

Some intersection and generation properties of convex sets

Robert E. Jamison (1977)

Compositio Mathematica

Some properties of algebras of real-valued measurable functions

Ali Akbar Estaji, Ahmad Mahmoudi Darghadam (2023)

Archivum Mathematicum

Let $M (X, 𝒜)$ ( $M^{*} (X, 𝒜)$ ) be the $f$ -ring of all (bounded) real-measurable functions on a $T$ -measurable space $(X, 𝒜)$ , let $M_{K} (X, 𝒜)$ be the family of all $f \in M (X, 𝒜)$ such that $coz (f)$ is compact, and let $M_{\infty} (X, 𝒜)$ be all $f \in M (X, 𝒜)$ that ${x \in X : | f (x) | \geq \frac{1}{n}}$ is compact for any $n \in ℕ$ . We introduce realcompact subrings of $M (X, 𝒜)$ , we show that $M^{*} (X, 𝒜)$ is a realcompact subring of $M (X, 𝒜)$ , and also $M (X, 𝒜)$ is a realcompact if and only if $(X, 𝒜)$ is a compact measurable space. For every nonzero real Riesz map $ϕ : M (X, 𝒜) \to ℝ$ , we prove that there is an element $x_{0} \in X$ such that $ϕ (f) = f (x_{0})$ for every $f \in M (X, 𝒜)$ if $(X, 𝒜)$ is a compact measurable space. We confirm...

Some remarks on ordered fields

Masayoshi Nagata (1974/1975)

Séminaire Dubreil. Algèbre et théorie des nombres

Spaces of orderings of fields under finite extensions.

Claus Scheiderer (1991)

Manuscripta mathematica

...-Strukturen.

Niels Schwartz (1978)

Mathematische Zeitschrift

Subfields of lattice-ordered fields that mimic maximal totally ordered subfields

Robert H. Redfield (2001)

Czechoslovak Mathematical Journal

Subgroups and hulls of Specker lattice-ordered groups

Paul F. Conrad, Michael R. Darnel (2001)

Czechoslovak Mathematical Journal

In this article, it will be shown that every $ℓ$ -subgroup of a Specker $ℓ$ -group has singular elements and that the class of $ℓ$ -groups that are $ℓ$ -subgroups of Specker $ℓ$ -group form a torsion class. Methods of adjoining units and bases to Specker $ℓ$ -groups are then studied with respect to the generalized Boolean algebra of singular elements, as is the strongly projectable hull of a Specker $ℓ$ -group.

Sui gruppi e corpi ordinati

Roberto Moresco (1977)

Rendiconti del Seminario Matematico della Università di Padova

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